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194 | ELECTRICITY | In the upper half-plane above the $x$-axis, there is a magnetic field with intensity $B$ directed inward, perpendicular to the page. A charged particle with rest mass $m_{0}$ and charge $-q (q>0)$ is shot at high speed into the positive $y$ direction from the origin. It experiences a drag force equal to $q c B$ in magnitude, opposite to the direction of velocity. It is known that the charged particle comes to a stop after turning through an angle $\theta_{0} (\theta_{0} > \frac{\pi}{2})$ in the direction of velocity, and at point $A$ on its trajectory, the velocity is parallel to the $x$-axis ($\theta_{A} = \frac{\pi}{2}$). Now, a smooth track is laid along the path of the charged particle (taking $\theta_{0} = \pi$). Another particle with the same rest mass $m_{0}$ and charge $-q$ is launched from the origin along the inner side of the track with the same initial momentum, experiencing a drag force proportional to velocity, ${\vec{f}}_{0} = -q B {\vec{v}}$. What is the normal force $N_{A}$ exerted on the track by the charged ion when it reaches point $A$? Consider relativistic effects, with the speed of light in vacuum as $c$. | ||
547 | ELECTRICITY | For the electromagnetic cannon model, its structure consists of two parallel rails spaced $\mathit{l}$ apart, with one end connected to a power supply for energy, and the other end connected to a metal rod that can slide freely on the rails to form a circuit. In the situation where the circuit length $x$ is much larger than the spacing $\mathit{l}$ (but ignoring the delay in circuit signal propagation caused by the length), it can be assumed that the self-inductance coefficient $L$ of the circuit is linearly related to $x$, i.e., $L = A x + B$. $A$ and $B$ are two constants. The current flowing through the metal rod is $I$, and the permeability of vacuum is $\mu_{0}$. In fact, for different electromagnetic cannon configurations, the value of the Ampere force on the metal rod is actually different. Assume the rail is a thin-walled cylinder with a radius $r \ll \mathit{l}$. Under direct current conditions, it can be assumed that the current is uniformly distributed over the surface of the cylinder. Make an appropriate approximation and calculate the specific expression of the Ampere force on the metal rod. | ||
647 | MECHANICS | This problem discusses the motion and forces on a particle in a rapidly oscillating field. First, we discuss the one-dimensional theory in a rapidly oscillating field. Let $q$ denote the coordinate of the particle. For a particle in a potential field $V_{0}(q)$ with mass $m$, if there is a rapidly oscillating force $f( \boldsymbol{q},t)$ acting on it (the angular frequency $ \omega$ of $f$ is very high), its equation of motion is: $$ m{ \ddot{q}}=F_{0}=-{ \frac{ \mathrm{d}V_{0}}{ \mathrm{d}q}}(q)+f(q,t) $$ The core of the theory is to divide the actual motion into fast motion (high-frequency oscillation) and slow motion (average motion): fast motion is denoted as $x(t)$, and slow motion is denoted as $X$. Due to the high frequency, the response of $x$ is very small, that is $x \ll X$. Expanding the above equation: $$ m \ddot{X}+m \ddot{x}=- \frac{ \mathrm{d}V_{0}}{ \mathrm{d}q} \Big \vert_{X}- \frac{ \mathrm{d}^{2}V_{0}}{ \mathrm{d}q^{2}} \Big \vert_{X} \cdot x+f(X,t)+ \frac{ \partial f}{ \partial x} \Big \vert_{X,t} \cdot x $$ First, solve for fast motion: $ \begin{array}{r}{m \ddot{x}=- \frac{ \mathrm{d}^{2}V_{0}}{ \mathrm{d}q^{2}} \Big \vert_{X} \cdot x+f(X,t)} \end{array}$, which is evidently a forced oscillation. Considering the high-frequency approximation, that is $ \begin{array}{r} { m \omega^{2} \gg \frac{ \mathrm{d}^{2}V_{0}}{ \mathrm{d}q^{2}} \bigg|_{X}} \end{array}$, we obtain $$ x=-{ \frac{1}{m \omega^{2}}}f(X,t) $$ Thus, in fact, when studying rapidly oscillating forces, the second term on the right side of the equation does not have any practical effect. Next, solve for the slow motion. Substitute the result of the fast motion into the actual dynamical equation to obtain the slow motion. Extending this situation to three dimensions, if $ abla \times f=0$, the effective potential $F_{ \mathrm{eff}}=- abla V_{ \mathrm{eff}}$ can be written down. Please provide the expression of $V_{ \mathrm{eff}}$ in terms of some average value of $f$. | ||
549 | MECHANICS | Please solve the following physics problem, and enclose the final answer in \boxed{}: Consider a regular dodecahedron with side length $a$ in an environment without a gravitational field. At the center of the dodecahedron, there is a mass $m$. Each vertex of the dodecahedron is connected to the mass $m$ by a spring. Each spring has a natural length $l$ ($l$ is equal to the radius of the circumscribed sphere of the dodecahedron), and all springs have the same spring constant $k$. The mass of the springs is negligible. Find the eigenfrequency of small oscillations of the mass $m$. | ||
267 | THERMODYNAMICS | We divide medium molecules into two categories: one is non-polar molecules, such as carbon tetrachloride; the other is polar molecules. For polar molecules, the centers of positive and negative charges are separated, forming an intrinsic dipole moment $\vec{p}_{i}$, such as in a water molecule. Under the influence of an external field, the positive and negative charge centers of non-polar molecules shift, forming a dipole. This mechanism is called induction polarization. For polar molecules, the intrinsic dipole moment rotates to a stable equilibrium position under the influence of the external field. This process is called orientation polarization. In this problem, we discuss the orientation polarization of the medium.
Assuming the dipole moment of a water molecule is $p_{w}$ and considering a system with a number density of water vapor molecules $n$, the interaction energy between the water dipole moment $p_{w}$ and the applied electric field is assumed to follow the Boltzmann distribution. That is, the isotropic distribution probability is multiplied by the factor $e^{-\frac{E_{p}}{k T}}$, where $E_{p}$ is the interaction energy between the dipole and the external field, $k$ is the Boltzmann constant, and the system temperature is assumed to be $T$. Derive the polarization vector $P$ of the system under the condition of a weak external field $E$. | ||
350 | ADVANCED | A homogeneous semicircular ring with a radius $R$, bending stiffness $EI$, and mass $m$ is placed with its opening facing upwards on a rigid horizontal surface in a state of rest. Assume all deformations are small, and the bending stiffness $EI$ is defined as: $M=EI(k-k_0)$, where $k$ is the curvature at a point, $k_0$ is the original curvature, and $M$ is the applied external moment. What is the height of its center of mass above the ground? The answer should be retained to the first-order term of $\frac{R^2 m g}{EI}$. | ||
756 | ELECTRICITY | Unlike the motion of charged particles in an electromagnetic field, the motion of neutral atoms in an electromagnetic field has unique characteristics. By modeling the neutral atom as a homogeneous sphere with a relative dielectric constant of $ \varepsilon_{r}$ and a radius of $R$, we consider the motion laws of the atom in a specific situation. In a spatial rectangular coordinate system, there is a uniform magnetic field $B$ in the $+z$ direction, where if $B < 0$, it means the magnetic field direction is actually along $-z$. A uniformly charged insulating thin wire with a linear charge density of $+ \lambda$ is fixed on the $z$ axis. Now a neutral atom is placed at $x = r_{0} \gg R, y = z = 0$ and released with an initial velocity $v_{0}$ in the $+y$ direction. Ignoring the retardation effect and relativistic effect, and assuming the atom has a uniform mass distribution with a density of $ \rho$. If the atom moves in a circular motion, try to find the value of $v_{0}$. Given that the vacuum permittivity is $ \varepsilon_{0}$. | ||
196 | MECHANICS | In the graduate entrance exam of the University of Wisconsin's Department of Physics in 1975, there appeared a seemingly simple rigid ball collision problem: Two rigid spheres of the same material—where the radius of the lower sphere is $2a$, and the radius of the upper sphere is $a$—fall from a height of $h$ (measured from the center of the larger sphere) above the ground. Assume that the centers of both spheres always remain on the same vertical line and that all collisions are elastic. Under these conditions, what is the maximum height the center of the upper sphere can reach? (Hint: Assume the larger sphere first collides with the ground and rebounds before colliding with the smaller sphere.)
It is now known that the rigid sphere collision model described in this problem satisfies chaotic conditions. Moreover, for this chaotic problem, as long as basic physical laws (such as energy conservation and rigidity assumptions) are satisfied, all states can occur—it is merely a matter of how much time it takes. Determine the highest position $h_{max}$ the smaller upper sphere could possibly reach after a very long period of time following its release. | ||
340 | ELECTRICITY | Consider two infinitely long parallel cylindrical metal conductors located in an isotropic linear dielectric medium characterized by a dielectric constant \( \epsilon \), magnetic permeability \( \mu \), and electrical conductivity \( \sigma \). The radius of each conductor is \( r \), and their separation is \( d \), where \( d \gg r \). The capacitance per unit length between the two conductors is denoted as \( C_{l} \), the inductance per unit length as \( L_{l} \), and the electrical conductance per unit length as \( G_{l} \) (none of which are known quantities).
Now, consider constructing an infinitely long one-dimensional circuit network: the lower conductor serves as the ground line, while the upper conductor contains an infinite number of inductors \( L \). The right end of the \( n \)-th inductor and the left end of the \( (n+1) \)-th inductor define the \( n \)-th node.Each node is connected to the ground line through a capacitor \( C \) and conductance \( G \). Resistance and conductance are connected in parallel.Let the electric potential at the \( n \)-th node be \( V_{n} \).
Using Kirchhoff's equations, a recurrence relation for \( V_{n} \) and its time derivative can be derived. Let \( C = C_{l}dx \), \( L = L_{l}dx \), and \( G = G_{l}dx \); by continuous approximation of the recurrence relation, the governing partial differential equation for \( V(x,t) \) can be obtained.
We hypothesize a solution for the decaying electric potential wave:
$$
V(x,t) = V_{0}\cos(kx - \omega t + \varphi)e^{-px}
$$
Substituting this solution into the partial differential equation, \( k(\omega) \) and \( p(\omega) \) can be determined.
The group velocity is defined as:
$$
v_{g} \equiv \frac{d\omega}{dk}
$$
Solve for the group velocity \( v_{g}(\omega) \) of the decaying electric potential wave. | ||
717 | OPTICS | Let the refractive index of the core material of an optical fiber Bragg grating be $n_1 = 1.51$. In the fiber, a one-dimensional optical structure is constructed by periodically changing the refractive index of the core material, with the changed material's refractive index being $n_2 = 1.55$. This structure is made up of alternating layers of two refractive indices: the layers have refractive indices $n_2$ and $n_1$, and thicknesses $d_2$ and $d_1$ respectively, forming a periodic refractive index distribution with a total of N layers. Light is perpendicularly incident on this structure and, after passing through the alternating layers, undergoes reflection and transmission at each layer interface. In each period, the odd layers are medium layers with refractive index $n_1$ and thickness $d_1$, the even layers are medium layers with refractive index $n_2$ and thickness $d_2$, with the arrangement: $$n_1,\, n_2,\, n_1,\, n_2,\, \dots,\, n_1,\, n_2$$ To simplify the analysis, assume that absorption loss in the medium is neglected during the design process, and only consider single reflections of light at each layer interface. When light is perpendicularly incident from a medium with refractive index $n_1$ to a medium with refractive index $n_2$, it gets partially reflected and partially transmitted at the interface. Specifically, if the incident light's electric field strength is $E_0$, then: The electric field strength of the reflected light is: $E_r = \frac{n_1 - n_2}{n_1 + n_2} E_0$ The electric field strength of the transmitted light is: $E_t = \frac{2n_1}{n_1 + n_2} E_0$ Assuming that the wavelength of the incident light in vacuum is $\lambda = 1.06 \mu m$, if the device's amplitude reflectivity is required to reach $8\%$, what is the minimum total number of layers N? Hint: Use the coherent superposition of light without needing to use the Bragg reflector reflectivity approximation. | ||
726 | MECHANICS | A cannon located on the ground can fire shells in any direction at a fixed rate $u$. The horizontal firing direction is the x-axis, and the vertical direction to the ground is the y-axis. The boundary of the safe zone is defined as: if the target is within this boundary, the shell may hit the target; if it is outside this boundary, it cannot be hit regardless of how it is fired. Find the equation of the boundary of the safe zone in the air, written in the form $y = y(x)$. The gravitational acceleration $g$ is known.
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149 | THERMODYNAMICS | A vertical adiabatic container with a height of $2H$ and a volume of $2V$ is sealed at the bottom by an adiabatic piston. Initially, the container is divided into two equal-volume sections by an adiabatic partition of mass $m$. The partition rests on a support, and the contact point is sealed to prevent any leakage. Each section initially contains helium gas at pressure $p$ and temperature $T$. Using a force, the piston is slowly pushed upward. Consider the following question:
Find the temperature $T_{1}$ of the gas in the upper gas chamber when the piston reaches the support.
Hint: During the process, the partition will leak for a certain period. | ||
492 | ELECTRICITY | Solve the following problem using the principle of virtual work:
Two identical superconducting rings have a radius of \( a \), each carrying a current \( I \) and with self-inductance \( L \). These rings are coaxially and parallelly positioned with their centers separated by a distance \( D \). The magnetic field generated by the current in each ring within the other ring can be considered as a magnetic dipole field. Assume a virtual displacement \( \delta \) occurs in the distance between the rings, i.e., \( D \to D + \delta \). Based on the conservation of magnetic flux in superconducting coils, please derive the interaction force between the two rings from the perspective of energy. The vacuum permeability is given as \( \mu_0 \). | ||
235 | THERMODYNAMICS | The molar mass of a certain ideal gas is $\mu$, and it flows from left to right through a long straight adiabatic horizontal duct with smooth inner walls. The cross-sectional area of the duct is $S$. The internal energy of 1 mole of this gas at an absolute temperature of $T$ is ${\frac{5}{2}}R T$, where $R$ is the universal gas constant.
A heating device is fixed and placed in the middle of a duct to heat the gas with a constant power. Assume that the resistance of the heating device to the gas flow can be neglected. After the gas flow stabilizes, although the gas in the vicinity of the heating device has nonuniform states, it gradually becomes uniform as the distance from the heating device increases. In the uniform steady-flow region to the left of the heating device, the pressure of the gas is $p_{0}$, the temperature is $T_{0}$, and the flow velocity to the right is $V_{\mathrm{0}}$. Given that the pressure of the gas in the uniform steady-flow region to the right of the heating device is $p_{1}$, determine the temperature $T_{1}$ of the gas in that region. | ||
359 | MECHANICS | The rotational inertia of a planar object relative to its center of mass is given by $I=\rho^2 M$. On a smooth horizontal surface, there is a thin plate of arbitrary shape with mass $M$, and point $O$ is the center of mass of the plate, with a radius of gyration $\rho_0$. There is a beetle $A$ on the plate with mass $m$. A polar coordinate system is established with the center of mass of the plate as the origin, and this coordinate system is fixed relative to the plate. In this reference frame, the motion laws of the beetle $A$ relative to the plate are known, given by $\rho(t)$ and $\varphi(t)$. Using the conservation of linear momentum and angular momentum, determine the absolute angular velocity $\Omega$ of the plate at any given time.
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532 | ELECTRICITY | Equilibrium Problem of Charged Plates in a Capacitor
The bottom area of a cylindrical insulating adiabatic container is \( A \), and its height is \( 2L \). The container stores gas with an initial pressure of \( p_{0} \), temperature \( T_{0} \), and constant-volume molar heat capacity \( C_{V} \). There are 3 identical metal plates, each with the same area as the bottom area of the cylindrical container. Two of them are grounded and fixed at the two bottom surfaces inside the container. The third metal plate is parallel to the bottom surface and positioned in the middle of the container, dividing the container and the gas inside it equally left and right. Now, an electric charge of \( Q \) is rapidly introduced to the middle metal plate. It is assumed that the middle plate is neither leaking electricity nor gas, and can freely move along the axis of the container. The relative permittivity of the gas is approximately 1. \( L \) is sufficiently small, and the conductor plates conduct heat well enough that the gas on both sides always reaches thermal equilibrium. The ideal gas constant is denoted as \( R \).The dielectric constant in a vacuum is \( \varepsilon_{0} \). Let the mass of the middle plate be \( m \). If \( Q > 2\sqrt{\varepsilon_{0}p_{0}}A = Q_{c} \), find the period of small oscillations when the plate is given a small disturbance at the stable equilibrium position. | ||
414 | MECHANICS | When the cup is tilted at a certain angle, if a lateral impulse is applied, it will start to spin in place. Now, let's establish a model to study the situation where the cup spins stably in place. Let the gravitational acceleration be $g$, and model the cup as a cylinder with mass $m$, radius $r$, and height $h$, with its generatrix deviating by an angle $\theta$ from the vertical line. Place the cylinder on a horizontal surface, with the contact point labeled as A. Assume point A moves in a circular motion on the horizontal surface with an angular velocity $\Omega$, and the cylinder undergoes pure rolling. Try to study the radius $R$ of the circular motion of point A. For ease of calculation, take $h=2r$, and stipulate that $h$ does not appear in the final result.
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207 | ELECTRICITY | Two eccentric spherical conductors with radii $R_{1}$ and $R_{2}$ form a capacitor. The centers of the spheres are separated by a distance $d \ll R_{1}, R_{2}$. The dielectric constant in vacuum is $\varepsilon_{0}$, and $R_{2} > R_{1}$. Find the reciprocal of the capacitance of the system, retaining only the lowest-order non-zero term that includes $d$, and list relevant boundary conditions for an approximate solution. | ||
257 | THERMODYNAMICS | There is a vertically placed, sealed, adiabatic cylindrical container with a cross-sectional area of \( A \). Inside, there is a thermally insulated piston of mass \( m \). The thickness of the piston is negligible compared to the length of the cylinder. Initially, the piston is fixed at the center of the cylinder, dividing it into two chambers of equal length \( l \). Each chamber contains \( n \) moles of monatomic ideal gas at an initial temperature of \( T_{0} \).
In the following problem, you may assume that the friction between the piston and the cylinder wall is negligible, and \( l \) is much larger than the displacement of the piston (\( l \gg z \)).
Although the friction between the piston and the cylinder wall is small, after a long time it will eventually bring the piston to rest. Furthermore, due to the motion of the piston, the temperatures of the two chambers eventually become equal, and all the heat dissipated due to friction is converted into the internal energy of the gas. Determine the final temperature \( T_{f} \) of the gas inside the cylinder when the piston comes to rest. It is assumed that the heat capacities of the cylinder walls and the piston are negligible. | ||
27 | MECHANICS | On an incline, a snow rod made of snow rolls down the slope purely, absorbing the thin layer of snow on the slope into itself, causing the snow rod to become increasingly larger. We assume the snow rod always maintains a cylindrical shape, and its initial volume is very small and can be neglected. The snow is evenly distributed on the slope, with a surface density of $\sigma$, and will be absorbed as long as the snow rod passes over it. The density of the snow is $\rho$. It is known that
$$
\sin \theta = \frac{3}{7}
$$
where $\theta$ is the angle of the incline. Find the relationship between the velocity of the center of the snow rod and time. | ||
330 | ELECTRICITY | In electromagnetism, we often draw analogies between magnetic dipoles and electric dipoles. However, the force patterns in a non-uniform field differ between the two. Consider the force experienced by a charged rigid body moving in a non-uniform magnetic field. In space, there exists a homogeneous solid sphere with a radius of \( R \), mass \( m \), and uniformly distributed charge \( q \). Gravitational effects are neglected. A Cartesian coordinate system is established, and a magnetic field is present in space given by \( \vec{B} = (B_0 + \alpha x) \hat{z} \). Using the equation of motion for the center of mass of the sphere, consider the initial state where the sphere's center of mass is located at the origin, its velocity is \( (v_{x0}, 0, 0) \), and it has no angular velocity. It is known that the position of the first turning point in the \( x \)-direction is \( x = l \). Provide the expression for \( v_{x0} \). | ||
643 | OPTICS | In this problem, we study a simple \"gas-fueled rocket.\" The main body of the rocket is a plastic bottle, which can take off after adding a certain amount of fuel gas and igniting it. It is known that the external atmospheric pressure is $P_{0}$, and the initial pressure of the gas in the bottle is $P_{0}$, with a temperature of $T_{0}$. The proportion of the molar amount of fuel gas is $\alpha$, and the rest is air. Alcohol is chosen as the fuel, with the reaction equation in air as: $C_{2}H_{5}OH(g) + 3O_{2}(g) \rightarrow 3H_{2}O(g) + 2CO_{2}(g)$. Assume that the air contains only $N_{2}$ and $O_{2}$, all gases are considered as ideal gases, and none of the vibrational degrees of freedom are excited. It is known that under conditions of pressure $P_{0}$ and temperature $T_{0}$, the reaction heat is $-\lambda$ (enthalpy change for $1\mathrm{mol}$ of ethanol reacting under isothermal and isobaric conditions). Ignore heat loss, assume the reaction is complete, and that there is air remaining, with the gas volume remaining constant throughout. Find the pressure $P_{1}$ of the gas inside the bottle after the gas has completely reacted upon ignition. | ||
421 | OPTICS | Consider a Young's double-slit interference device and establish a rectangular coordinate system, with the direction of the slit connection as the $x$-axis, the direction of light propagation as the $z$-axis, and the $y$-axis determined by the right-hand rule. The object screen is located at $z=0$, the double slits are located at $(\pm a,0,0)$, and the light screen is located at $z=D$. The region $0\le z\le D$ is filled with a medium with a refractive index distribution of $n\left(z\right)=1+\sigma z$. The wavelength of the incident light is $\lambda$. The angles at which a light ray is emitted are represented by $\alpha$, $\beta$, and $\gamma$, which are the angles with the $x$, $y$, and $z$ coordinate axes, respectively.
Taking the approximation $\gamma\ll1$ (hence $\alpha,\beta\lesssim\pi/2)$, but with $\sigma D\sim1$, $\sigma a\ll\pi/2-\alpha$, and ignoring square terms of small quantities, write the function expression $f(x,y,p)=0$ on the light screen for the $p$th-order interference bright fringe (defined as the optical path difference being $\pm p\lambda$). Provide $f(x, y, p)$. The term with $x$ must be positive with a coefficient of 1; if there is another term where the sign can be either positive or negative, please mark the sign of that term as plus-minus $(\pm)$. | ||
427 | ELECTRICITY | Establish a 3D right-handed rectangular coordinate system $(x,y,z)$. There is a $4a\times6a$ resistor network on the xy-plane, with the endpoint $\mathsf{A}$ having coordinates (0,0,0) and endpoint B having coordinates (6a, 4a, 0). Each small square has a side length of $a$, and the resistivity is uniform. A current $I$ flows in from A, and a current $I$ flows out from B. Now, introduce a magnetic field $\mathsf{B} = (B_{0},0,B_{0})$ to the entire space. Try to solve for the magnitude of the torque $M$ acting on point A in the resistor network. | ||
185 | ELECTRICITY | The upper and lower plates of the capacitor are semicircular metallic plates $a$,$b$ with a radius of $ \pmb{R}$. The centers are separated by a distance $d \ll R$, and the centers of metallic plates $a$ and $b$ are located on the same vertical axis. The lower surface is horizontal, but due to manufacturing processes, the upper metallic plate is not strictly parallel to the lower surface. The straight edge of the upper surface is parallel to the lower surface, and the angle between the two planes is $ \alpha R \ll d$. Because the angle is very small, we can approximately consider that in the top view, both metallic plates are still strictly semicircular. There is an axis $c$ at the center of the upper metallic plate, perpendicular to the upper surface, around which the upper surface can freely rotate. In the top view, the angular displacement of the diameters of the circular discs on the upper and lower surfaces of the capacitor is $ \pmb{ \theta}$. Determine the capacitance $c( \theta)$ of this capacitor at this angular displacement (As an approximation, we assume that charges only exist on the directly opposing parts of the upper and lower surfaces, and no charges are present on the remaining parts). | ||
233 | MODERN | In high-energy physics, the detection of photons is a very important and meaningful task. To this end, consider a simplified model: A photon with energy $E_{0}$, which is much smaller than the rest energy $m_{e}c^{2}$ of an electron, will undergo $n$ random Compton scatterings within a material consisting of initially static free electrons, resulting in its energy decreasing to $E$. Try to determine the specific form of energy $E$ as it decreases with the number of scatterings $n$, given that the speed of light in vacuum is $c$. | ||
334 | MECHANICS | Consider the process of a stepladder collapsing and bouncing off the wall. The stepladder can be viewed as two uniform rods, each with length $L$, hinged together at the top $A$ with a smooth, lightweight hinge. The left rod has a mass of $m$, while the mass of the right rod can be neglected. The static and kinetic friction coefficients between the bottom end $B$ of the left rod and the ground are both $\mu$. The bottom end $C$ of the right rod makes smooth contact with the ground. Initially, the two rods are almost overlapping, with an angle of zero. The ground is horizontal, and the distance from endpoint $C$ to a vertical wall corner $O$ is $\sqrt{2}L$. The rods are released from rest, and the acceleration due to gravity is $g$.
To ensure that during the collapse process of the stepladder, the endpoint $C$ continuously slides to the right towards $O$ while endpoint $B$ remains stationary, find the minimum value of the friction coefficient $\mu$. | ||
224 | MECHANICS | Consider the wheel-slider transmission mechanism. There is a ring with radius $R$ on the horizontal plane, and there is an infinitely long smooth rod passing through the center of the ring. Looking from above, the intersection point of the rod and the right side of the ring is $P$. On the ring and the rod, there are two masses, $A$ and $B$, respectively, with a transmission rod between them. At time $t=0$, mass $A$ starts to rotate counterclockwise with a constant angular speed $\omega$ from the hinge $P$, while mass $B$ is on the right side of $A$. This movement drives the slider at the $B$ end of the transmission rod $AB$ to reciprocate along the track. The length of the transmission rod $AB$ is $l > R$. Calculate the acceleration of the slider at the $B$ end moving to the left at any time $t$. | ||
587 | MECHANICS | A homogeneous thin rod is placed on the track of a real-axis rotated hyperboloid. Given the rod’s length $L$ and mass $M$. The equation of the hyperbola in the plane is $$ {\frac{y^{2}}{a^{2}}}-{\frac{x^{2}}{b^{2}}}=1 $$ where $g$ is the known gravitational acceleration. If the hyperbolic surface is smooth and the rod is confined within the vertical xy-plane, find the slope $k$ of the rod at equilibrium ($k>0$) expressed in terms of $a$, $b$, and $L$ (under the condition that $aL>2b^2$).
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512 | MECHANICS | Collision is one of the simplest mechanical interactions between objects, representing a process of energy and momentum exchange. The chain reaction caused by one-dimensional collision cascades provides a simplified model for a series of natural phenomena dominated by continuous collisions. When studying such systems, the physical quantities we're interested in are the proportion of energy or momentum transmitted through the collision cascade and its dependence on the restitution coefficient and the mass ratio of the colliding objects. Between the initial ball of mass $M$ (fixed) and the terminal ball of mass $m$ (fixed), $n$ balls ($n \gg 1$) are placed (with masses $m_{i}$, $i = 1, 2, ..., n$, tunable). Only the first collision between neighboring balls is considered. All collisions are inelastic, characterized by a restitution coefficient $e$ (very close to 1).
Recently, Ricardo and Lee demonstrated that when the mass distribution of the cascading ball chain ensures that the mass of each ball $m_{i}$ is the geometric mean of the masses of its two neighboring balls, $\sqrt{m_{i-1}m_{i+1}}$, the transmission efficiency of kinetic energy and momentum between the initial and terminal balls is maximized. For an $n$-ball system, please write down the maximum transmission efficiency of kinetic energy between the initial and terminal balls $r_{K n}$; | ||
776 | ELECTRICITY | For conductors, reaching electrostatic equilibrium from a non-equilibrium state takes a certain amount of time, but generally, this time is very short, so this process is often neglected in conventional electrostatic analyses. However, many effects generated by this process are still not negligible. Below, we will make a simple analysis, considering that the charge movement speed is much less than the speed of light. There is a dielectric sphere with a radius of $R$, dielectric constant $ \epsilon$, and conductivity $ \alpha$ in a vacuum. At time $t=0$, a uniform electric field $E_0$ is suddenly applied. We assume that the dielectric sphere first reaches electrostatic equilibrium in a non-conductive sense and then begins to leak charge. Determine how much time passes during the leakage process before the current density at all locations is reduced to half of its initial value. | ||
479 | ELECTRICITY | In an infinitely large isotropic linear dielectric medium, there exists a uniform external electric field $\vec{E}_{0}$. The vacuum permittivity is known to be $\varepsilon_{0}$.
The dielectric is a solid with a relative permittivity $\varepsilon_{r}$. Within the dielectric, a spherical cavity with radius $R$ is carved out, and the cavity can be considered a vacuum. Then, an ideal conductor sphere with a radius slightly smaller than $R$ is placed within the cavity, such that the conductor sphere just does not come into contact with the cavity walls. The conductor sphere carries a net charge of $Q$.
Find the magnitude of the electrostatic force $F$ exerted on the conductor sphere. | ||
703 | ELECTRICITY | A point charge with mass $m$ and negative charge $-q(q>0)$ moves without resistance in a spatial magnetic field $B=B_0a/r (B_0>0, a>0)$ along the +z direction. Here, $r$ is the distance of the particle from the z-axis, $r= \sqrt{x^2+y^2}$. Assume the coordinate system is right-handed. At $t=0$, the charge is located on the x-axis with coordinates $x=a, y=z=0$, and its velocity is exactly along the +y direction with a magnitude of $v_0 (v_0>0)$. If the particle appears at $t>0$ with its velocity direction perpendicular to $ \vec r$ (where $ \vec r$ represents the radial vector from the origin to the particle), find the distance $r$（different from $r_0$） from the particle to the z-axis, assuming $qB_0a-mv_0>0$. Hint: Consider the differential equation of the particle's angular momentum $L$ and radial distance $r$. | ||
288 | THERMODYNAMICS | For electromagnetic radiation within a cavity, the method of studying gases can be used for investigation, and we refer to this as a photon gas. It is known that the internal energy of a photon gas with a volume $V$ and temperature $T$ can be expressed as $U = a V T^{4}$, where $a$ is a known constant. A thermal radiation field consists of radiation with various wavelengths. When a blackbody radiation cavity undergoes adiabatic expansion, not only does the radiation temperature change, but the radiation wavelength also changes correspondingly due to the Doppler effect. Therefore, there must be a certain relationship between the change in radiation wavelength and the change in temperature. Derive the relationship between the radiation wavelength and the temperature within a spherical blackbody radiation cavity undergoing adiabatic expansion. It can be assumed that the expansion rate of the cavity is much less than the speed of light. | ||
378 | ELECTRICITY | A very large metal conductor is shaped into an L-shaped conductor by removing a 90-degree dihedral angle. The edge of the dihedral angle is the z-axis, and its two half planes are the xz (x > 0) half plane and the yz (y > 0) half plane, respectively. The region where x > 0, y > 0 (i.e., the removed 90-degree dihedral angle) is vacuum. The L-shaped conductor is grounded, with its center point at the origin. An infinitely long, thin wire with a radius of r is parallel to the z-axis, located at x = a, y = b, -∞ < z < ∞. The vacuum permittivity is \(\varepsilon_0\). Calculate the capacitance per unit length in the z direction between the thin wire and the L-shaped conductor using \(\varepsilon_0, a, b, r\). Assume that the wire radius r is much smaller than a and b. | ||
462 | ELECTRICITY | ### Given
An isolated conducting ellipsoid with charge \(Q\) and the surface (Cartesian coordinate) equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1\). The surface charge distribution at electrostatic equilibrium is \(\sigma=\frac{Q}{4\pi abc}(\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}+\frac{z^{2}}{c^{4}})^{-1/2}\). The vacuum permittivity is \(\epsilon_0\).
### Problem
For a conducting thin circular disk with radius \(R\) and charge \(Q\), find the system electrostatic energy at electrostatic equilibrium. | ||
431 | ELECTRICITY | Given a superconducting coil with self-inductance $L$ and radius $b$, initially a current $I_0$ is applied at infinity. There is also a fixed superconducting sphere with radius $a$. The coil is moved from infinity towards the sphere such that the normal direction of the coil is consistent with the direction of the line connecting the center of the coil to the center of the sphere and points outward from the sphere's center. Till the distance between the center of the coil and the center of the sphere is $r$. From this point on, the movement process to be considered is as follows: The center of the coil is slowly translated along a circle with the center of the sphere as the center and radius $r$ (without changing the orientation of the coil) until the normal direction of the coil is perpendicular to the line connecting the coil and the sphere, at which point the line connecting the two has rotated by 90 degrees. Find the work done by the external force $A$ during this process. The vacuum permeability is known as $\mu_0$. | ||
287 | THERMODYNAMICS | When uncharged cells are in close proximity, fluctuations on the surface of the cell membrane can create repulsive forces between the cells. As a simplification, one can ignore the bending of the cell membrane's surface and treat it as a two-dimensional plane. The cell membrane can be approximately considered as an elastic membrane with a surface tension coefficient of $\alpha$ and a mass per unit area of $\sigma$. It is known that the cell membrane is in a thermodynamic equilibrium environment at temperature $T$, and the Boltzmann constant is $k_{B}$.
Assume there are two interacting cell membrane samples, each a square membrane of size $L \times L$, with the membranes parallel and at a distance $D (D \ll L)$. To describe the surface fluctuations of the cell membrane, use a coordinate system $(x, y)$ with the origin $O$ at the vertex of one cell membrane. Let $h(x, y)$ represent the surface fluctuation of the sample.
If $D \to +\infty$, the cell membrane under thermal equilibrium should undergo free oscillations. The surface vibration mode of the cell membrane should be a two-dimensional standing wave concerning the edges, with the following form:
$$
h(x,y)=h_{0}(t)\sin\left(k_{x}x\right)\sin\left(k_{y}y\right)
$$
Investigate when $k_{x}, k_{y}>0$, the minimum value mode of $|k_{x}|, |k_{y}|$.
The fluctuations between the two cell membranes must not intrude into each other, and the fluctuation of the lower cell membrane can be considered as $h \leq D/2$. The following identity is known:
$$
\frac{D}{2}=\sum_{m,n=1;m,n\text{是奇数}}^{\infty}\frac{4D}{mn\pi^2}\sin\left(\frac{m\pi x}{L}\right)\sin\left(\frac{n\pi y}{L}\right),~(x,y\in[0,L])
$$
As an estimate, geometric constraints can be independently considered for each $(k_{x}, k_{y})$ component.
Find the repulsive force $F$ between the cell membranes when $k_{B}T\ll\alpha D^{2}$.
[Hint] If $X, Y, Z, T, P$ are all state functions and satisfy the differential relationship:
$$
d X=Y d Z+T d P
$$
then:
$$
\left({\frac{\partial Y}{\partial P}}\right)_{Z}=\left({\frac{\partial T}{\partial Z}}\right)_{P}
$$ | ||
774 | MECHANICS | In a vacuum far from other celestial bodies, there exists a very thin gas cloud. Under the influence of gravity, the gas accumulates and eventually forms a spherically symmetric distribution upon reaching equilibrium. The density distribution is given by:
$$
\rho = \rho_0 e^{-r/r_0}
$$
After the gas cloud forms, a small meteorite core (with mass much smaller than the total mass of the gas cloud and density much greater than $\rho_0$) orbits the center of the gas cloud in a uniform circular motion with radius $R$. Now, the small meteorite core experiences a slight radial perturbation. Determine the precession angular velocity of the orbit after stabilization. Express the final result using the angular velocity $\omega$ during stable uniform circular motion and the dimensionless parameter $u=\frac{R}{r_0}$. | ||
172 | ELECTRICITY | Above an infinitely large grounded conductor plate at a distance $L$, there is a light rod with a length of $2R$, with charges $Q$ and $-Q$ at each end, respectively; both have a mass $m$. The center of the light rod is fixed and it can rotate freely. Consider only effects related to electrostatics.
Please analyze the frequency of small oscillations when the rod is vertical. | ||
525 | MODERN | Using the Lorentz transformation, find the velocity of a particle moving with velocity $\vec{v}$ in the S frame, as observed in the S' frame moving with velocity $\vec{v_{0}}$ relative to the S frame. Do not represent the result using a cross product. | ||
304 | MECHANICS | A smooth hemispherical bowl with a radius of $R$ is fixed in place.The rim of the bowl is horizontal.A uniform rod with a length of $L$ leans against the edge of the bowl. The lower end of the rod is supported by the bowl, while the upper end extends outside of the bowl. Considering the equilibrium condition, find the angle $\theta$ that the rod makes with the horizontal when it is in equilibrium. | ||
66 | MECHANICS | Connect a horizontal plane with an inclined plane using an arc with radius $R$. The angle between the inclined plane and the horizontal plane is $\theta$. A rope with linear density $\lambda$ and length $L$ is placed on the horizontal plane and the arc, with its right end just at the junction of the arc and the inclined plane. The gravitational acceleration is $g$, and $R \to 0$ with $R \ll L$. All surfaces are smooth, and the rope should detach at some position. Let the distance the rope slides down the inclined plane be $x$. First, consider that $R$ exists, calculate the result, and then let $R=0$. Try to determine which part of the rope detaches first (expressed the result with the angle $\varphi$ between the line connecting the point on the arc and the center of the circle and the vertical line). | ||
777 | ELECTRICITY | Consider two concentric thin metallic spherical shells, with the inner shell having a radius of $r$ and the outer shell having a radius of $R$. The space between the two shells is completely filled with a uniform medium characterized by a dielectric constant $ \epsilon$, conductivity $ \alpha$, specific heat capacity $c$, and density $ \rho$. Initially, the system is uncharged. At time $t=0$, a charge $Q$ is suddenly placed on the inner shell, while the outer shell remains initially uncharged. Assumptions: 1. The metallic shells are ideal conductors. 2. The heat capacity of the metallic shells can be neglected. 3. The system is adiabatic, meaning all Joule heat generated by the current flowing through the medium is completely absorbed by the medium, with no heat lost to the surroundings. 4. The dielectric constant $ \epsilon$ and conductivity $ \alpha$ of the medium remain constant throughout the process. Find: After the system reaches the final electrostatic equilibrium state (i.e., after a sufficiently long time, when the current stops flowing), what is the total temperature increase $ \Delta T$ of the medium? | ||
375 | OPTICS | A double convex thin lens, with both front and back interfaces being spherical surfaces, has curvature radii $R_1$ and $R_2$, respectively. The distance between the spherical surfaces is negligible, and the system is axially symmetric relative to the optical axis. The medium has a refractive index of $n$, and the system is placed in air (taking the refractive index of air as $n=1$). Consider a non-paraxial object point, with the distance between it and the optical center line ("object distance") being $u$, and the angle between its line to the optical center and the optical axis being $\theta$. Consider only the light rays refracted in the vicinity of the optical center that form an image, where the angle between the line to the image point and the optical axis remains $\theta$. Determine the distance between the image point and the optical center ("image distance") $v$, expressed in terms of $R_1, R_2, n,$ and $\theta$. | ||
542 | MECHANICS | We consider an inhomogeneous cylinder composed of two materials with different densities. The cylinder has radius \( r \) and length \( L \). The lower half is made of a material with density \( \rho \), while the upper half is made of a material with density \( c\rho \), where \( c \) is a parameter satisfying \( 0 < c < 1 \).
In solving the problem, we can utilize the fact that for a semicylinder of radius \( r \), its center of mass (CM) is located at a distance \( \frac{4r}{3\pi} \) from the central axis of the semicylinder. The gravitational acceleration is denoted by \( g \).
Now, we assume the cylinder is free to move on a horizontal surface under gravity. The static friction coefficient between the cylinder and the surface is sufficiently large to ensure pure rolling without slipping. At time \( t = 0 \), the cylinder is in its equilibrium position and has an initial angular velocity \( \omega_0 \).
If \( \omega_0 \) is sufficiently small, the cylinder will undergo periodic motion around its stable equilibrium position. For small oscillations, what is the period of oscillation? Express your answer in terms of the given parameters.
| ||
323 | MECHANICS | A thin steel wire is bent into an ellipse with the polar equation $r={\frac{p}{1+\varepsilon\cos\theta}}$, where $p\cdot\varepsilon$ is a positive constant and $0<\varepsilon<1$. The major axis of the ellipse intersects the wire at points $A$ and $B$, with $A$ close to the right focus and $B$ close to the left focus. Point charges with charges ${Q}_{1}$ and ${Q}_{2}\left({Q}_{1}>0,{Q}_{2}>0\right)$ are fixed at the two foci of the ellipse. Another small ball with mass $m$ and charge $q$ $(q>0)$ is threaded on the wire, insulated from it, and can slide along the wire without friction. Gravity and other resistances are not considered.
Find the angular frequency of small oscillations of the ball around its stable equilibrium position on the wire between $A$ and $B$ (excluding points $A$ and $B$). | ||
681 | MODERN | Consider the following acceleration method: An object with total mass (including the mass of the mirror) \(m_0\) and a mirror of area \(S\) is placed facing the mirror direction with a nuclear bomb, which is detonated to produce photons that are perfectly reflected on the mirror surface, thereby propelling the mirror forward. Considering that a single nuclear bomb is insufficient to achieve the goal, a series of nuclear bombs are arranged in a straight line within the solar system, and the object is launched along the line. As it passes near each nuclear bomb, the bomb is detonated, further accelerating the object. Assume all nuclear bombs are identical and stationary before explosion, generating a flux density \(J\) of energy in the vicinity of the mirror at the moment of explosion (using the stationary bomb as a reference frame). Assume the entire flux consists of photons, which are incident perpendicular to the mirror and last for a duration of \(t_0\), with subsequent radiation from the explosion being negligible. Hint: The Doppler effect indicates that when moving away from a light source at speed \(v\), the observed frequency ratio compared to the original frequency is \(\sqrt{\frac{c-v}{c+v}}\). When the mirror velocity is \(v\), what is the momentum obtained by the mirror after one nuclear explosion? Express it in terms of \(J, v, c, t_0\) (assuming the change in \(v/c\) after one explosion is very small). | ||
396 | MECHANICS | A smooth bowl with a radius $R$ is fixed, and the opening plane is horizontal. A smooth, uniform thin rod $AB$ with length $L=4\sqrt{3}R/3$. End B is located outside the bowl, while end A presses against a point inside the bowl, achieving static equilibrium in the plane passing through the sphere center $O$. The points $D$ and $D^{\prime}$ on the rod nearly coincide with the support points on the bowl's edge, but $D$ is slightly to the lower left, and $D^{\prime}$ is slightly to the upper right. Let the angle between the rod and the horizontal plane be $\theta$.
Suddenly cut the rod at $D^{\prime}$. Note that after this point, $D^{\prime}$ will lightly rest on the edge of the bowl. Find the instantaneous angular acceleration of the rod $\beta^{\prime}=\ddot{\theta}$. | ||
301 | MECHANICS | Assume the Earth is a sphere with radius $R$, mass $M$, and uniformly distributed mass. A small object is launched from a space station at a height $h$ above the Earth's surface. The object moves with an initial velocity relative to the Earth, where the direction of the initial velocity is perpendicular to the line connecting the object to the Earth's center. The gravitational constant is denoted as $G$. If the magnitude of the initial velocity of the object is $v_{0}$, and the object is able to land on the Earth's surface, find the total time it takes from launch to landing. | ||
745 | ELECTRICITY | In the polar coordinate system, there is a logarithmic spiral $r = r_0 \mathrm{e}^\theta$. On this trajectory, a point charge is placed starting at $\theta = 0$, and for each angle $\theta_0$ turned, another point charge is placed. Their magnitude of the charge is sequentially $-q (q > 0), -2q, 3q, -4q, \cdots$ and so on to infinity. (The electrostatic constant is known to be $k$.) Find the magnitude of the electric potential $\varphi$ at the origin. | ||
788 | THERMODYNAMICS | The mechanical properties of the rubber membrane are similar to the surface layer in thermodynamics. In the first approximation, the tension inside the rubber membrane can be described by the surface tension coefficient. A layer of rubber membrane is equivalent to a surface layer (the refractive index of the membrane is the same as that of water, and the membrane is extremely thin). Tension formula: \[ F = \sigma l \] A hollow cylinder, with height \(L \), radius \(R \), has its two ends covered with a transparent rubber membrane with the same surface tension coefficient, and contains an ideal gas inside at pressure \(p_0 \). The cylinder is submerged in a pool to a depth \(H \) (assume \(L \ll H \), and the liquid-induced pressure on the upper and lower membranes is the same). When the cylinder is at a distance of \(H/3 \) from the water surface, the bottom of the pool is imaged exactly at the water surface. Assume the density of water is \( \rho \), refractive index is \(n \), gravitational acceleration is \(g \), and atmospheric pressure is \(p_0 \). The temperature of the water is independent of depth, and the cylinder has good thermal conductivity. If the cylinder can be treated as a thin lens when submerged in water, but the change in internal gas pressure cannot be ignored, and since the deformation of the membrane is still relatively small (the spherical radius of the membrane \(r \gg \dfrac{R^2}{L} \)), only first-order small quantities need to be retained. Find the surface tension coefficient \( \sigma \) of the rubber membrane. | ||
399 | ELECTRICITY | Consider a cylindrical dielectric with a radius of $a$ and a length of $l$, where it is known that $l \gg a$. First, permanently polarize the dielectric so that the direction of the polarization vector at every point is radially outward, and its magnitude is proportional to the distance from the axis. We establish a cylindrical coordinate system $(r, \theta, z)$, with the cylinder's axis coinciding with the $z$-axis, and appropriately choose the origin so that the cylinder is symmetric with respect to the $xOy$ plane. Thus, the polarization vector is given by $\vec{P} = \frac{\rho_0}{2} r \hat{r}$. Next, use an external force to rotate the cylinder about the $z$-axis with a constant angular velocity $\overrightarrow{\omega} = \omega \hat{z}$, while maintaining the polarization strength unchanged. Find the electric field $\overrightarrow{E}_{f}(r, \theta, z)$ at a point far from the cylinder $(\sqrt{r^{2} + z^{2}} \gg l)$, retaining terms up to the $-4$ power of $r$ and $z$. Possible mathematical knowledge you may use: the expression for divergence in cylindrical coordinates is $\nabla \cdot \vec{A} = \frac{1}{r} \frac{\partial}{\partial r} (r A_{r}) + \frac{1}{r} \frac{\partial A_{\theta}}{\partial \theta} + \frac{\partial A_{z}}{\partial z}$. | ||
232 | MECHANICS | Two balls, 1 and 2, each with mass $m$, are connected by a massless, rigid, light rod of length $\ell$, and are resting vertically in the corner of a wall, with ball 1 at the upper end of the rod. Assume both the wall and the ground are frictionless. Initially, give ball 2 a small initial velocity away from the wall. During the motion of the system, at what angle between the rod and the vertical wall does ball 1 begin to detach from the vertical wall? | ||
254 | THERMODYNAMICS | In a vacuum, there exists an electrically neutral plasma cloud composed of electrons and protons. The plasma cloud is in thermal equilibrium within a non-uniform external magnetic field, with an equilibrium temperature of $T$. It is known that the mass of an electron is $m_{e}$, the charge of an electron is $-e$, the mass of a proton is $m_{p}$, the charge of a proton is $e$, the Boltzmann constant is $k_{B}$, and the vacuum permeability is $\mu_{0}$. The electrostatic effects of the plasma are neglected.
The distribution function of electrons and protons (distinguished using the index $\alpha$, which can be used to simplify calculations) in six-dimensional phase space $(x, y, z; v_{x}, v_{y}, v_{z})$ is defined as:
$$
f(x, y, z; v_{x}, v_{y}, v_{z}):={\frac{d{\mathrm{P}}}{d x d y d z d v_{x} d v_{y} d v_{z}}}
$$
where $\mathrm{P}$ is the probability density of finding a particle at position $(x, y, z)$ with velocity $(v_{x}, v_{y}, v_{z})$. Statistical physics indicates that if there exist additive conserved quantities in the single-particle motion of electrons or protons, such as energy $E$ and momentum $p$, then the thermal equilibrium distribution function $f_{\alpha}$ is:
$$
f_{\alpha} \propto \exp\left(-{\frac{E_{\alpha} + V_{\alpha} p_{\alpha} + \dots}{k_{B} T}}\right)
$$
Here, $V$ is a velocity-related constant determined by the specific situation.
If the magnetic field in space can be neglected, and the plasma cloud as a whole begins to rotate about an axis with angular velocity $\Omega$,
If the electron density at the center of the rotation axis is $n$, and in the rotating frame relative to the plasma itself, the axial angular velocity of the plasma is $\Omega_{c}$. The boundary radius of the plasma cloud is $R$ ($R \ll \sqrt{k_{B}T / m_{e}(\Omega + \Omega_{c})^{2}}$), and the boundary magnetic field is zero. Find the magnetic field distribution $\boldsymbol{B}(\boldsymbol{r})$ generated by the plasma in the rotating frame. The answer should be expressed as the product of various factors. | ||
289 | OPTICS | By utilizing crystal materials with non-uniform thickness or non-uniform refractive indices, humans invented convex lenses. Now, let's reverse the thought process: use an ideal convex lens model combination to reconstruct a crystal material with a varying refractive index!
Consider infinitely many ideal semi-convex lenses, each placed equidistantly with an angular spacing of $\alpha$ between them. Each semi-convex lens is infinitesimally thin, allowing them to be arranged without mutual interference. These lenses are halved along their optical centers, and all optical centers are overlapped at the origin $O$ without crossing one another. Although there are infinitely many semi-convex lenses, $\alpha$ is an infinitesimal quantity, so the total angular range remains finite and does not extend infinitely in rotation. Finally, the focal length of each convex lens is $f$, which is an infinitely large quantity. However, the quantity $f\alpha = r_{0}$ is a finite, non-zero characteristic length in the limit. Below, all answers to the subsequent questions will be expressed in terms of $r_{0}$.
To solve the problem of determining the position and direction of a light ray perpendicular to the first lens upon exiting the last lens, we first perform an equivalence transformation: in fact, by tracing the trajectory of the light ray as it propagates between the lenses, it can be found that this problem is completely equivalent to the refraction of a light ray inside a spherical medium with a refractive index that varies with radius. Determine the refractive index distribution $n(r)$ of the sphere. It is required that $n(r_{0}) = n_{0}$, and it is known that at $r = r_{0}$, the angle between the propagation direction of the light ray and the lens orientation is $\psi_0$. | ||
56 | MECHANICS | First, establish a polar coordinate system on a plane. Suppose the rabbit starts at the origin, and there is a safe hole at the coordinates \((\pi, S)\). When the rabbit runs into the hole without being caught by the fox, it is considered "safe." Now, the fox starts chasing the rabbit from any starting point \((\varphi, r)\) and always keeps its speed direction along the line connecting it to the rabbit. At the same time, the rabbit starts moving in a straight line towards the safe hole. Assume the speed of the fox is \(v_2\), and the speed of the rabbit is \(v_1 (v_1 < v_2)\). Suppose the fox is just able to catch the rabbit before it reaches safety. Find the envelope of all starting points of the fox that satisfies the condition of "catching the rabbit" as described above; that is, provide the polar equation \(r = r(\varphi)\) for the critical state (just able to catch the rabbit before it is "safe"). | ||
665 | MECHANICS | A homogeneous cylinder is skewered at an angle on a rigid thin rod (considered as fixed), which passes through the center of the cylinder and intersects perfectly with the circumferences of its two bases. It is constrained by two very small parts, A and B, which do not provide power but only serve to secure the cylinder, and whose inner diameters are very close to the outer diameter of the metal rod. Parts A and B themselves are fixed. Given: the cylinder's mass is $m$, the radius of its base is $R$, and the cylinder's height is $4R$. The mass of the thin rod is negligible. If there is friction at parts A and B, the coefficient of friction is $\mu$. Initially, the cylinder has an angular velocity $\omega_{0}$ along its axis. Gravity is not considered. The radius of the thin rod is $r$. Without approximation, solve for the relationship between the magnitude of the cylinder's angular velocity $\omega$ and time $\omega(t)$.
| ||
429 | MECHANICS | There is a non-stretchable, lightweight rope with a length of $2l$, and its two ends are fixed at two points on the ceiling that are $2c$ apart. A smooth small ring with mass $m$ is placed on the rope. The gravitational acceleration is known to be $g$.
Determine the period of oscillation of the small ring near its equilibrium position as it moves left and right. | ||
552 | MECHANICS | On a slope with an incline angle \(\alpha\), a projectile is located above the slope. The launch point of the projectile is at a height \(H\) from the slope directly below it. A projectile is launched with an initial velocity \(v_0\) that is much less than the first cosmic velocity, ignoring air resistance, with gravitational acceleration \(g\). Find the area of the region on the slope that can be controlled by the projectile. | ||
383 | THERMODYNAMICS | Most of us are familiar with sound waves propagating in the Earth's atmosphere, which are pressure waves. However, within the Sun, we need to consider that the gas density decreases with height due to gravity. We simulate the Sun as an isothermal atmospheric model, where the density $\rho$ decreases exponentially with height $z$ due to gravity. Let the mass of the atmospheric particles be $m$, the gravitational acceleration at the Sun's surface be $g$ (neglecting its variation with height $z$), Boltzmann constant be $k_B$, and the temperature of the solar atmosphere be $T$. Its adiabatic index (ratio of specific heat at constant pressure to specific heat at constant volume) is $\gamma$.
When sound waves propagate vertically within the atmosphere, particles will undergo small vertical displacements. Let $u(z,t)$ be the vertical displacement of gas particles at time $t$, where $z$ is their equilibrium position. Next, we study a sinusoidal wave with angular frequency $\omega$. When a wave propagates vertically upwards at a constant speed in the direction of decreasing density $\rho(z)$, we expect the wave's energy density $w=\rho \omega^2 u^2$ to remain unchanged. Thus, we can set $u(z,t)=\frac{f(z)}{\sqrt{\rho(z)}} \exp(-i\omega t)$, where $f(z)$ is an unknown function to be determined. When the frequency of sound waves at the Sun's surface is lower than the critical frequency $\omega_c$, they will be trapped inside the Sun and unable to propagate vertically upwards. Find this frequency $\omega_c$. | ||
695 | MODERN | As is well known, an ideal black body can be well approximated by a small opening on a cavity. Let the temperature inside the cavity be $T$, the volume be $V$, the Stefan-Boltzmann constant be $\sigma$, and the black body move along the x-axis with a velocity $v=\beta c$ relative to a certain inertial frame S, while the small opening of the cavity faces exactly along the x-axis. Try to find in frame S: the black body radiation power density $J$ Express the answer in terms of $\beta$, $c$, $\sigma$, $T$, and other basic physical constants. | ||
590 | ELECTRICITY | In a uniformly positively charged large sphere, a small spherical cavity is excavated (the cavity radius is \(R/2\), and the distance from the large sphere's center is \(R/2\)), such that the total positive charge of the system is \(Q\). Now, electrons are emitted from the cavity center \(o\) in every direction with the same initial speed \(v\), but their kinetic energy is insufficient to reach the opposite end of the cavity diameter. Ignoring gravitational forces, find the envelope equation of all electron trajectories. The final answer should be an explicit formula in terms of \(x\) and \(y\), where the effective acceleration acting on the electrons is \[ g=\frac{eQ}{7\pi\varepsilon_0 m R^2} \] acting to the left. | ||
548 | THERMODYNAMICS | Consider a capillary tube with an inner radius $R$ fixed, vertically placed in a certain liquid, with a surface tension coefficient $\sigma$, and a contact angle that is an acute angle $\theta$. The liquid density is $\rho$. The height the liquid rises in the capillary tube is $H$. In the 17th century, the mathematician Torricelli proposed a model known historically as Torricelli's trumpet, which refers to the following capillary tube: the part above the water surface within a height $H^{\prime}$ has a radius maintained as $R^{\prime}$, but at a height $h > H^{\prime}$, its radius $r$ is inversely proportional to its height, that is, $h r = H^{\prime} R^{\prime}$. Find the expression for $H^{\prime}$ such that the height $h$ to which the liquid rises in the capillary tube exactly equals $H$, assuming the capillary tube is thin enough, approximating the tube wall to be nearly vertical at $h = H$. Express the answer in terms of $R^{\prime}$, $\sigma$, $\theta$, $\rho$, and the gravitational acceleration $g$. | ||
329 | ELECTRICITY | In a certain type of semiconductor, a class of carriers with charge $q$ and mass $m$ collide with the lattice. Prior to the collision, the carrier moves freely. After the carrier collides with the lattice, the probability $P$ of the next collision occuring and the uncollided motion time $\Delta t$ satisfy the relation:
$$
P(\Delta t)=1-e^{-\Delta t/\tau}
$$
$\tau$ is a known constant determined by the thermodynamic average. After each collision, the carrier's momentum is reset by the lattice. The gas composed of these carriers can be approximately regarded as a monoatomic ideal gas. Unless otherwise stated, the number density of carriers is uniform and of magnitude $n$, and the thermodynamic temperature of the system is $T$. The Boltzmann constant is known as $k_{B}$.
Assume the average collision time of carriers is $t_{1}$ (unknown), and the characteristic vibration time of the lattice is $t_{2}$. If $t_{1}\sim t_{2}$, there is a correlation in the impulse components $i, j$ provided by two consecutive collisions, which depends on the time difference $\Delta t$ of the two impulse actions:
$$
\langle p_{0,i}(0)\cdot p_{0,j}(\Delta t)\rangle=
\begin{cases}
\displaystyle\frac{\alpha}{2}\left[1+\cos\left(\frac{\Delta t}{t_{2}}\right)\right]&{(i=j)}\\
0&{(i\neq j)}
\end{cases}
$$
where $t_{2}$ is a given constant, and the same collision can be regarded as the process $\Delta t=0$.
When there is no electromagnetic field in the semiconductor, the spatial number density has a gradient distribution:
$$
n(x,y,z)=n_{0}-\Lambda x
$$
where $\Lambda$ is a given constant, and $|\Lambda x|\ll n_{0}$.
On this basis, if a uniform magnetic field $\vec{B}=B\hat{z}$ is applied in space, the diffusion Hall effect can be observed in the semiconductor. Find the diffusion Hall conductivity $R_{H~D}=j_{D,y}/\Lambda$, where $j_{D,y}$ is the diffusion current in the unit $y$ direction. | ||
179 | ELECTRICITY | The coaxial cylindrical capacitor with length $l$ consists of a metal rod with radius $a$ and a metal cylindrical shell with an inner diameter of $b$. One end of the capacitor is vertically inserted into a liquid medium with relative permittivity $\varepsilon_{r}$ and density $\rho$. Assume the liquid surface is very large, and the cylindrical capacitor is inserted into the liquid to a depth of $h_{0}$ (with $h_{0}\gg b$, and all edge effects are neglected). A voltage ${V}_{0}$ is applied to the capacitor. Assume the liquid is an insulator, there is no conductive current in the liquid, and the surface tension of the liquid, the adhesive force, and the viscous force at the liquid-metal interface are not considered. The acceleration due to gravity is $g$. Neglect higher-order effects from the liquid flow in the tank and find the angular frequency $\Omega$ of the micro-vibrations of the liquid inside the cylinder (assuming the liquid surface in the cylinder remains horizontal during the vibrations). | ||
278 | MODERN | The mass of particle X is $m$, and it can decay into two photons with their emission directions uniformly distributed in the center-of-mass system. A large number of X particles are emitted from the origin at high speed $v$ along the positive $z$ axis, and the average lifetime of X particles observed in the laboratory is $\tau$. Establish a cylindrical coordinate system $\rho, z, \varphi$. Construct a photon detector on a cylindrical surface with radius $\rho=R$. Find the probability distribution function $f(z)$ of the $z$ coordinates of photons detected on the detector after a sufficiently long time. In this problem, it is allowed to express the final result using an integral over angles, without needing to evaluate the integral. Consider the theory of special relativity, and use $c$ to represent the speed of light. | ||
583 | ELECTRICITY | In a sufficiently long cylindrical region with a radius of $a$, the outer side is a conductor with zero resistance, and the interior is vacuum.
Inside the cylindrical region, there is an infinitely long straight wire (compared to $a$), with the wire's center at a distance $b$ from the cylinder's axis. The current gradually increases from $0$ to $I$. The radius of the straight wire satisfies $r \ll a$.
Find the work done by the power source per unit length, $\begin{array}{r}{\omega^{\prime}=\frac{W^{\prime}}{L}}\end{array}$. | ||
655 | MECHANICS | A container filled with water rotates around a vertical axis $OB$ at a constant angular velocity $\omega$. Inside the container, there is a thin rod $OA$ fixed at an angle $\theta$ to the horizontal plane. A small ball is threaded onto the rod $OA$ and can slide without friction along the rod. Assume the volume of the ball is $V$, its density is $\rho$, and the density of the water is $\rho_{0}(\rho_{0}>\rho)$, with gravitational acceleration $g$. Initially, the ball is located at the bottom end $O$ of the rod, and then it starts sliding along the rod from rest. Find: the magnitude of the force exerted by the ball on the rod when the ball's speed in the rotating frame of the container is at its maximum. Assumption: Compared to the displacement caused by the ball, its radius can be neglected; the ball will not touch the walls of the container; the force exerted by the water on the moving ball relative to the water is approximately equal to the force exerted when the ball is at rest relative to the water.
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57 | MECHANICS | There is a planet with a constant volume \( V \) and a constant mass density \( \rho \). What shape should the planet take so that the gravitational force at a specific point on its surface is maximized? Provide the equation for the shape of its surface. | ||
308 | MECHANICS | When a lightweight ball is placed on a horizontally rotating disk, an interesting phenomenon occurs: the object can move on the disk without leaving it. If the ball performs pure rolling without slipping on the disk, then when the ball is placed at a certain position on the disk, the ball exhibits a special motion state. To facilitate the analysis of the ball's motion, a rectangular coordinate system is established that is stationary relative to the experimental frame.
Given that the angular velocity of the disk is $\Omega$, the radius of the ball is $a$, the moment of inertia is $kma^2$, point $C$ is the center of mass of the ball, and point $P$ is the position of contact between the ball and the disk in space, and further assuming that the ball does not leave the disk during motion, determine the motion period $T$ of the ball for this special motion. | ||
230 | THERMODYNAMICS | A uniform, infinitely long isothermal pipe is placed vertically in a gravitational field. The gas molecules inside are monoatomic, with a mass of $m$. The number density at the bottom is $n_{0}$, and the area is $S$. At the bottom, there is a piston that does not leak. The gravitational acceleration is known to be $g$, and the temperature is $T$. All the questions below consider local equilibrium, assuming that the relaxation time is extremely short and the local gas is adiabatic when vibrating. Find the change in air vibration over time at height $h$ when the piston vibrates as $x = A \cos{\omega t}$. That is, determine the propagation of the sound field, where $A$ is a small quantity. It is known that the (additional) force applied at the piston is $F = f \cos{\omega t}$, the adiabatic index of the gas is $\gamma$, and $4k_{B}T m\omega^{2}/\gamma - m^{2}g^{2} < 0$. The Boltzmann constant is $k_{B}$. Find the change in air vibration over time at height $h$ when vibrating. | ||
189 | MODERN | Considering the effects of special relativity, establish a rectangular coordinate system \((xOy)\) in space. In a given inertial frame, a particle with mass moves along the \(x\)-axis with velocity \(v_0\) at time \(t = 0\). From \(t = 0\), the particle is subjected to a force only in the direction of the \(y\)-axis, causing the particle to undergo uniformly accelerated linear motion along the \(y\)-axis relative to this reference frame, where the acceleration of the particle in the \(y\)-direction is \(a\). Before the particle's velocity reaches the velocity of light \(c\), provide the trajectory equation \(x(y)\) of the particle. There is no need to specify the range of values for \(x\) and \(y\), with the stipulation that at time \(t = 0\), \(x = y = 0\). | ||
370 | ELECTRICITY | An electric quadrupole has two forms, one of which is a linear electric quadrupole: it consists of two point charges with charge $+q$ and one point charge with charge $-2q$. The three point charges are arranged in a straight line, with the charge $-2q$ located at the midpoint of the two charges and equidistant from the two $+q$ charges by a distance $l$. All scenarios below are placed in vacuum, with the known electrostatic constant in vacuum being $k$. Place a linear electric quadrupole on the left side outside a grounded conductive sphere with a radius $R$. The center of the electric quadrupole is at a distance $r$ (r > R) from the center of the conductive sphere. The orientation of the electric quadrupole is parallel to the radial vector $r$.Find the magnitude of the force exerted by the conductive sphere on the electric quadrupole.
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29 | MECHANICS | On a horizontal plane, there is a large homogeneous wooden plank with a surface mass density of $\sigma$. Above the plank, there exists a static magnetic field $\begin{array}{r}{\vec{B}(r)=\frac{3m v_{0}r}{q r_{0}^{2}}\vec{e_{z}}}\end{array}$. A particle with a charge-to-mass ratio of $\frac{q}{m}$ is launched horizontally from the origin with an initial velocity of $v_{0}$. The particle leaves a dumbbell-shaped trajectory on the wooden plank. Symmetrically cutting out the "water-droplet-shaped" half from this trajectory as its boundary, we take this portion as a rigid planar body. Placing this rigid body on a rough horizontal plane and considering the gravitational acceleration $g$, certain configurations of the rigid body can maintain stable equilibrium. Derive the small oscillation period at the point of stable equilibrium. | ||
666 | OPTICS | A student accidentally smeared out one slit on the grating while conducting a grating diffraction experiment and discovered a different \"landscape.\" The following research is based on this phenomenon. It is known that the wavelength of light used in the experiment is $\lambda$, the grating constant is ${d}$, and the slit width is $a$. The missing orders phenomenon is not considered. Assume that there are a total of $2N+1$ slits on the grating and due to a manufacturing oversight, one slit is missing in the center. Discuss the distribution of light intensity. Assume the central light intensity of diffraction from one slit of this grating is $I_{0}$ | ||
200 | OPTICS | This question explores time coherence from the perspective of randomly emitted wave packets: a light source always emits waves in the form of one wave packet after another. Within the same wave packet, there is a regular variation in the phase between two points, whereas the phase relationship between different wave packets is completely random. At this point (at a given location), time coherence can be understood as
$$
\gamma(t_{0})=\left|\left\langle\frac{E^{*}(t+t_{0})E(t)}{\left|E(t+t_{0})E(t)\right|}\right\rangle\right|
$$
where $<>$ denotes averaging or expectation over $t$. Coherence can be seen as a measure of the correlation between the phase of the electric field at time $t+t_{0}$ and at time $t$. Thus, the electric field magnitude is not discussed here. In this expression, the quantity being averaged is the electric field $E$ divided by its own magnitude. As a result, only the phase information at different times is retained, yielding a dimensionless constant. Additionally, note that averaging is performed first, followed by taking the modulus.
If there is a light source that emits random wave packets of different lengths, and the number of wave packets of different lengths is distributed according to a probability density, the number of wave packets with a length distributed between $\tau$ and $\tau+\mathrm{d}\tau$ is
$$\mathrm{d}n=\frac{1}{\tau_{0}}e^{-\frac{\tau}{\tau_{0}}}d\tau,$$
where $\tau_{0}$ is a known constant. Determine the time coherence of these randomly emitted wave packets. | ||
434 | MODERN | Action integrals are not only of great significance in quantum theory; they also appear as adiabatic invariants in classical physics. Consider the following problem: A negative charge $-q$ initially moves in a circular orbit around a stationary positive charge $Q$, with angular momentum, radius, radial momentum, and angle denoted as $L$, $r$, $p_{r}$, and $\theta$ respectively. Now, for some reason, the charge of the positive charge $Q$ changes very slowly to $Q^{\prime} = \lambda Q$. This change only affects the electrostatic force on the negative charge without causing other effects. Then, theoretical mechanics can prove that $J_{1} = \oint p_{r}\mathrm{d}r$ and $J_{2} = \oint L\mathrm{d}\theta = \int_{0}^{2\pi}L\mathrm{d}\theta$ both remain unchanged over a long period of time, meaning they have the same value in the initial and final states, which is called an adiabatic invariant. Find the final state radius $r^{\prime}$ (express it as a power of $\lambda$ to indicate the growth factor). | ||
348 | MECHANICS | There is now an elastic beam AB fixed to a vertical wall, where at the contact point A with the wall, it is tangentially horizontal. The length of the elastic beam is $L$, and the bending stiffness is $EI$ (meaning that the restoring moment of the elastic beam is $M=EI \frac{d^2y}{dx^2}$). The coefficient of friction between the block and the beam is $\mu$. There is now a slider C, whose dimensions relative to $L$ can be neglected, with a weight of $G$. Assume the block is at point D on the beam, and set the initial sliding position of the block to be $s$, where it just begins to slide. Given this setup, find the velocity $v$ at the end point B as the block slides along the beam and falls off.
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214 | MODERN | The uncertainty principle in quantum mechanics refers to the fact that two non-commuting quantities cannot be simultaneously measured with perfect accuracy. Suppose we observe two mechanical operators $A$ and $B$, which satisfy the commutation relation:
$$
[A,B]=A B-B A=C
$$
Then, when we simultaneously measure $A$ and $B$, their measurement uncertainties satisfy:
$$
\langle(\Delta A)^{2}\rangle\langle(\Delta B)^{2}\rangle\geq\frac{1}{4}\langle-i C\rangle^{2}
$$
In classical mechanics, the ground state energy of a harmonic oscillator is zero. However, in quantum mechanics, due to the influence of the uncertainty principle, the harmonic oscillator’s ground state energy is not zero but rather $\begin{array}{r}{\cdot\frac{1}{2}\hbar\omega=\frac{1}{2}h\nu.}\end{array}$ The ground state energy of a harmonic oscillator, $\cdot{\frac{1}{2}}\hbar\omega$, is also called the vacuum zero-point energy. Consider the electromagnetic field can also be quantized as infinite harmonic modes. The electric field strength of the standing wave field between two parallel conducting plates can be decomposed as:
$$
\overrightarrow{E}_{(z,t)}=\sum_{j}\widehat{\epsilon}_{j}A_{j}q_{j}(t)\sin{(k_{j}z)}
$$
Substituting each harmonic oscillator into the equation satisfied by electromagnetic waves in a vacuum:
$$
\nabla^{2}\vec{E}-\frac{1}{c^{2}}\frac{\partial^{2}\vec{E}}{\partial t^{2}}=0
$$
where $c$ is the speed of light, and $\nabla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}$ is the scalar operator. We obtain:
$$
\ddot{q}_{j}+c^{2}k_{j}^{2}q_{j}=0
$$
which is the harmonic oscillator equation, where $k_{j}$ represents the wavevector of the propagating electromagnetic wave. Therefore, the vacuum also possesses the ground state energy of the electromagnetic field (for each harmonic oscillator):
$$
E_{0}=\frac{1}{2}h\nu
$$
This energy is referred to as the vacuum zero-point energy. As a result, even in the absence of photons, there exists a Casimir force between two parallel conducting plates in a vacuum.
Next, we use a simplified model to theoretically calculate the magnitude of the force arising from the Casimir effect. Assume our universe is a one-dimensional ring of length $L$, and the two conducting plates are placed parallel to each other at a distance $x$. In this system, a series of electromagnetic wave modes satisfying the boundary conditions contribute to the vacuum zero-point energy of the system. In reality, metallic plates cannot reflect arbitrarily high-frequency radiation: the highest energy modes will escape. To account for this effect, an exponential term is introduced to truncate the high-energy modes. The truncation position is arbitrary, so we expect it will not affect any measurable quantities. Assume that when summing over different electromagnetic wave modes to obtain the total energy, a Boltzmann factor is introduced, i.e.,
$$
E=\sum_{j}e_{j}^{-\frac{E_{j}}{\Lambda}}E_{j}
$$
where $E_{j}$ is the zero-point energy for each mode, and $\Lambda$ is a constant.
Find the Casimir force $F_{(x)}$ in the lowest-order approximation, assuming $L \gg x$ and $h c \ll x\Lambda$. | ||
602 | ELECTRICITY | In a vertical plane, there is a circular coil with a radius of $a$ carrying a current $I$. OP is perpendicular to the plane of the circle and has a length of $h$. Point O is the center of the circular coil. At point P, a very small metal ring is placed, with its center at P and a radius of $b$, where $b$ is much smaller than $a$ and $h$. At $t=0$, the plane of the small ring is also perpendicular to OP. The current in the large ring remains constant, and the small ring rotates uniformly with an angular velocity $\omega$. The small ring has a resistance of $R$, with its self-inductance neglected. The angular velocity direction of the small ring is vertically upward, and the direction of the current $I$ in the large coil, when viewed from along the PO direction, is counterclockwise. To maintain uniform angular velocity of the small ring, find the magnitude of the torque required on the small ring at time $t$. Given the vacuum permeability $\mu_0$. | ||
638 | OPTICS | Laser has been widely used due to its high power; however, various dissipations in reality can reduce the laser power. It is known that due to the gain of the medium, light satisfies $$ \frac{dA}{dx}=\alpha A $$ where \(A\) is the light intensity, and \(\alpha\) is a constant. It is also known that the length of the laser \(L\). Assume that the light will not be completely reflected at the front and back surfaces of the laser but is reflected in the opposite direction with a light intensity reflection rate \( r < 1 \). In fact, the medium not only brings gain but also causes some reflection. Define \(k\) as the ratio of the reflected light intensity to the original light intensity when advancing a unit distance in the medium. In order to enhance the laser, try to express the minimum value of \(L\) using \(r\), \(k\), and \(\alpha\). Known \(\alpha > k\), directly perform light intensity superposition without considering the phase. | ||
210 | OPTICS | Let a beam of monochromatic parallel light with wavelength $\lambda$ and amplitude $A$ be irradiated from left to right onto a biconvex lens with a radius $R$, curvature radii of the left and right surfaces $\rho_{1}, \rho_{2}(\rho_{1,2} \gg R)$, and refractive index $n$. The direction of the light beam is parallel to the optical axis of the lens.
Let the position on the axis and the optical center distance be $z$. Calculate the intensity distribution $I(z)$ on the optical axis behind the lens. | ||
642 | MECHANICS | On a free homogeneous disk A with a radius $R$, there is an inextensible light string of length $L$ wound around it (which does not affect the disk's diameter). The free end of the string is connected to object B, and then passes over a frictionless fixed pulley to connect to object C. Initially, A and B are at rest, with disk A exactly situated in the vertical plane, and C is stationary on the ground. The segments of the string between A and B, between B and the fixed pulley, and between the fixed pulley and C all remain vertical. It is known that the string length $L$ is much greater than the radius $R$ of the disk, and that after the entire string on disk A is unwound, disk A will fall off the string; during motion, the string remains in the vertical direction, and disk A will not collide with the ground. The masses of objects A, B, and C are $M_A$, $M_B$, and $M_C$, respectively. If the pulley is smooth, and disk $A$ is released from rest, $A, B, C$ all start moving simultaneously. Try to find the acceleration of $B$ at the moment of release.
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589 | MECHANICS | A ladder of length \(2l\) and mass \(m\) is placed against a vertical wall, initially making an angle \(\alpha\) with the horizontal. It starts to slide under frictionless conditions. While the ladder is still in contact with the wall, its total mechanical energy (taking potential energy at \(y=0\) as zero) can be expressed as \[ E(\theta,\dot{\theta})=\frac{2}{3}ml^2\dot{\theta}^2+mgl\sin\theta, \] where \(\theta(t)\) is the angle between the ladder and the horizontal. Derive the relationship between the critical angle \(\theta_c\), at which the ladder loses contact with the wall, and the initial angle \(\alpha\). Provide the final answer in a clear expression.",
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489 | MODERN | The Bohr model holds historical significance as the first semi-classical quantum model of the hydrogen atom. It explained the challenges of classical electrodynamics regarding electron radiation and atomic stable states, as well as the results of spectroscopic observations at the time. Due to the limitations of quantum theory during its early development, it underwent modifications by Sommerfeld and others, eventually leading directly to the establishment of the Schrödinger equation by Schrödinger and others, which serves as the fundamental equation of quantum mechanics.
In the Bohr model, the principal quantum number $n$ simultaneously determines the energy and angular momentum of the electron state, which is actually incorrect. Through observation of hydrogen atoms in electric fields and magnetic fields, we point out that degenerate states under the same principal quantum number $n$ with different angular quantum numbers $l$, magnetic quantum numbers $m$, and even spin quantum numbers $s$ will experience energy level splitting. Specifically, some states' energy levels increase, while others decrease, and spectroscopic observations can reveal distinct spectral lines resulting from electronic level transitions between different states. The Stark effect is a typical effect when the hydrogen atom couples with an electric field. This problem attempts to estimate of the magnitude of energy level splitting due to the Stark effect using a semi-classical approach.
Consider the slow evolution of the electron's orbital in space under the influence of a weak uniform electric field $\vec{E}$. First, consider a simpler situation where at time $t=0$, the normal vector $\vec{n}$ of the plane of the electron's orbit and the electric field strength $\vec{E}$, along with the axis pointing from the nucleus to the pericenter of the orbit (the $x$-axis), are all situated in the $xOz$ plane, with an angle $\theta$ between the latter two. Use the perturbation method to prove that the magnitude of the angular momentum of the electron's orbit does not change over time, and the orbit merely precesses around the direction of the electric field $\vec{E}$ (that is, rotates around the axis). Determine this angular velocity $\Omega$. Given the orbital energy $-U$, eccentricity $e$, electron mass $m$, and the magnitude of charge for both the atomic nucleus and electron as $q$.
Hint: The so-called perturbation method refers to calculating the instantaneous derivative of the dynamical conserved quantity of the original system under the action of the newly added perturbing force at each moment. Substitute the position and velocity of the periodic motion of the original system into this derivative and calculate the periodic average. Finally, this averaged quantity is interpreted as the time derivative of a dynamical conserved quantity along the instantaneous orbit.
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775 | THERMODYNAMICS | We study a specific thermodynamic system, often exploring it from two perspectives: constant volume and constant pressure.
Consider a system describable by pressure p and volume V.
Define its constant-pressure temperature scale t such that, when pressure is held constant, the temperature scale t has a linear relationship with the volume V.
Define its constant-volume temperature scale t
′
such that, when volume V is held constant, the temperature scale t
′
has a linear relationship with the pressure p.
It is also known that when the pressure is constant at p
0
, the temperature scale t is directly proportional to the volume V, with a proportionality constant of
V
0
1
. When the volume is constant at V
0
, the temperature scale t
′
is directly proportional to the pressure p, with a proportionality constant of
p
0
1
.
If, for a thermodynamic system, the values given by both temperature scales are always equal (i.e., t=t
′
, denoted as t
sys
), and it satisfies:
When the pressure p is constantly p
0
,
t
sys
=
V
0
V
.
When the volume V is constantly V
0
,
t
sys
=
p
0
p
.
When both pressure p and volume V are sufficiently large, the temperature
t
sys
satisfies the asymptotic relation
t
sys
+α∝pV
, where
α
is a known parameter (easily determined experimentally).
Derive the equation of state for this thermodynamic system, expressed as a relationship
t
sys
=t(p,V)
involving p,V, and
t
sys
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79 | ELECTRICITY | In vacuum, there is a plane wave with angular frequency $\omega$ propagating in the $\pmb{x}$ direction, given by $\begin{array}{r}{\vec{E}(x,y,z,t)=E_{0}\cos(\omega t-k x)\hat{y},}\end{array}$ $\begin{array}{r}{\vec{B}(x,y,z,t)=\frac{E_{0}}{c}\cos(\omega t-k x)\hat{z},k=\frac{\omega}{c}.}\end{array}$
The electromagnetic wave propagates in the $\pmb{x}$ direction with amplitude $E_0$. An ideal mirror has an angle $\pmb{\theta}$ with the direction $-\hat{\pmb{x}}$, and has velocity $\pmb{v}$, such that the wave is incident on the mirror in its reference frame. Find the radiation pressure $_{p_{2}}$, expressing the answer using $I_{0}={\textstyle\frac{1}{2}}\varepsilon_{0}E_{0}^{2}c$. The permittivity in vacuum is $\epsilon_0$, the speed of light in vacuum is $c$, and consider special relativity. | ||
520 | ELECTRICITY |
---
A ring with mass $m$, charge $q$, and radius $R$, which is uniformly charged, homogeneous, and made of insulating material, is located on a smooth horizontal surface. The plane of the ring coincides with the surface of the ground. Friction between the ground and the ring is negligible. In the space above, there exists a non-uniform magnetic field directed vertically upward (in the $-z$ direction), given by ${{B}}={{B_{0}}}+k{{x}}$. Initially, the center of the ring is located at the origin. Analyze the motion of the ring after being subjected to a small perturbation along the positive $x$ direction.
The ring will undergo small oscillations in the $x$ direction. Ignoring higher-order small terms of $x$, find the period of small oscillations in the $x$ direction.
--- | ||
447 | ELECTRICITY | A small permanent magnetic needle can be considered as a current loop with a very small radius, and its magnetic moment $\mu$ is given by $\mu=I S$, where $I$ is the constant current strength of the loop, $S=\pi R^{2}$, and $R$ is the radius of the current loop. The direction of the magnetic moment is perpendicular to the plane of the current loop and follows the right-hand rule with the direction of the current. Two small magnetic needles, A and B, have a constant magnetic moment of size $\mu$ and a mass of $m$. We establish a coordinate system by taking the downward vertical direction as the positive $z$ axis. Magnetic needle A is fixed at the origin $\mathrm{o}$ of this coordinate system, with its magnetic moment pointing in the positive $x$ direction; magnetic needle B is placed directly below A. The magnetic permeability of vacuum is given as $\mu_{\mathrm{0}}$, and the gravitational acceleration is $g$. Assuming the magnetic moment direction of B makes an angle $\theta$ with the $x$ axis, and its center of mass coordinates are $(0,0,z)$ ($z >> R$), find the angle $\theta_{\oplus}$ that B can maintain stable equilibrium with the $x$ axis. Stable equilibrium means that if B's magnetic moment direction deviates slightly from $\theta_{\oplus_{\tt G}^{\tt G}}$, there will still be a tendency to return to the original direction. | ||
604 | THERMODYNAMICS | Consider a material with emissivity $e$, which consists of two large parallel plates, $A$ and $B$, with constant temperatures $T_{A}$ and $T_{B}$, where $T_{A} > T_{B}$. The emissivities of $\boldsymbol{A}$ and $\boldsymbol{B}$ are $e_{1}$ and $e_{2}$, respectively. Find the net heat flow $J$ from $A$ to $B$. | ||
620 | OPTICS | As is well known, the human eye is most sensitive to green light around $532 \mathrm{nm}$, so a laser at this wavelength can cause the most damage if it directly hits the eye. Consider an optical system where the refractive index on the left side is $n_{1}$, and the refractive index on the right side is $n_{2}$. A point $A$ on the optical axis is perfectly imaged at point $A'$. It is known that the angle between the light emitted from $A$ and the optical axis is $u$, and the angle between the light received at $A'$ and the optical axis is $u'$. For another point $B$, which is very close to the optical axis and at a perpendicular distance $\delta y$ from $A$, it can also be perfectly imaged at point $B'$ on the image plane. Place a green laser pointer at point $O$, located at a distance $R$ from point $A$ along the line of the light emitted from $A$ at an angle $u$ to the optical axis, but extending in the opposite direction. The laser beam has a divergence angle of $\theta \ll 1$ and is incident at a small angle $u+\theta/2$ with the optical axis. Find the minimum distance that the eye needs to maintain from the optical axis in the original $A'B'$ image plane region below the optical axis to avoid being damaged by the green laser. | ||
218 | OPTICS | Humans utilized the reflection of the ionosphere to achieve the first transatlantic wireless communication, and the ionosphere is also the most crucial auxiliary weapon in modern electronic warfare. Imagine ultra-high precision directional signal transmission (emitting signals only in one precise direction), using extremely high-frequency signals to eliminate diffraction, striving to achieve "imaging communication" (where both parties are precisely positioned at each other's image points), and converging the transmitted electromagnetic waves at the receiving site (thus achieving fixed-point signal reception). Here, "imaging communication" means that the two points of the object can achieve convergence under the third-order small quantity approximation. Since an ellipse can achieve ideal imaging, if the reflective surface is similar to an ellipse within certain precision and range, then ideal imaging can be realized. Only consider signal propagation within a plane. The angular separation between the signal source and the receiving station with respect to the Earth's center is $\varphi$. Consider ground-to-ground communication with a single reflection; the base height of the ionosphere is $z$, and the Earth's radius is $R$. $\varphi=2\operatorname{arccos}R/(R+z)$, if the maximum deviation angle of the transmitted signal from the set value is $\alpha(\alpha\ll1)$, calculate how far from the ground the receiving station can be to receive the signal, and assume $z\ll R$ in this question. | ||
641 | MECHANICS | A particle \( P \) with a mass of \( m \) is subjected to a constant force of magnitude \( F \) that always points towards point \( O \). Initially, \( P \) is at a distance \( r_{0} \) from \( O \) and has a velocity perpendicular to \( OP \) with a magnitude of \( u_{0} \) (\( u_{0} > \sqrt{\frac{F r_{0}}{m}}\)). At the moment when the particle \( P \) reaches its farthest point from \( O \), an impulse is applied to it so that \( P \) can perform uniform circular motion. At some point during its subsequent movement, a perturbation is applied to \( P \) along the radial direction. Find the period \( T \) of the radial oscillation of particle \( P \).
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753 | ELECTRICITY | In a certain space, there is a uniform magnetic field $ \vec{B} = B_0 \hat{z}(B_0 > 0)$ in the region where $y > 0$. A homogeneous ring with a mass $M$, radius $R$, insulated and with a charge per unit length of $ \lambda$, moves with a velocity $ \vec{v} = v_0 \hat{y}(v_0 > 0)$. At time $t = 0$, the ring is located in the $xy$ plane, and the center of the ring is at $(0, -R, 0)$. Ignore the effects of all non-electromagnetic forces. Find the minimum initial velocity $v_{1}$ required for the ring to completely enter the magnetic field. | ||
727 | MECHANICS | A bomb explodes at a height $H$, splitting into many small fragments. After the explosion, each fragment is projected outward with the same velocity $u$ in all directions (with uniform angular distribution). Afterwards, all fragments fall to the ground, experiencing completely inelastic collisions upon landing. Find the distribution radius \(R\) of the bomb debris.
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781 | ELECTRICITY | In a vacuum, there is an infinitely long thin-walled conductive cylinder with a cross-sectional radius of $R$, carrying a uniform and constant azimuthal current $I$. The cylinder is cut along the axial direction into two halves, separated by a very short distance, assuming the current distribution remains unchanged. Given that the permeability of the vacuum is $ \mu_0$, find the force per unit length between the two halves. | ||
623 | MODERN | Consider the decay reaction $A \rightarrow B+C$, where the rest masses of the particles are $m_{B}=m_{C}=m$ and $m_{A}=\alpha m(\alpha >2)$. In the ground frame, the initial velocity of $A$ is $v=\beta c$. Let $\beta_{0}=\sqrt{1-\frac{4}{\alpha^{2}}}$, and it is known that $\beta_{0}<\beta<\frac{\beta_{0}}{\sqrt{1-\beta_{0}^{2}}}$. Find the maximum value of the angle between the velocity directions of $B$ and $C$ in the ground frame. | ||
433 | ELECTRICITY | When a metal conductor is placed into a plasma, changes in the electron and ion number densities and potential will occur near the surface. There is a presheath layer where the electron and ion number densities are equal, and a sheath layer where they are unequal, creating a current between them. By measuring the $U-I$ characteristic curve, certain characteristics of the plasma can be obtained, known as a plasma probe.
The problem is simplified to a one-dimensional problem, with the one-dimensional coordinate $x$. For $x>0$, it's the conductor, and for $x<0$, it's the plasma. From right to left, there are the metal conductor, the sheath, and the presheath.
At $x \rightarrow \infty$, the potential is set as the reference potential $V \approx 0$, and the electron and ion number densities are $n_0$.
Assume the ion mass is $m$, and the kinetic energy at the top of the presheath moving towards the conductor is $W_0$. It is assumed that ions do not experience collisions during their motion.
Assume the potential at the interface between the presheath and sheath is 0, the electric field intensity is $E_d$, and the potential at the conductor's surface is $V_f < 0$. It is assumed that the electron number density follows the Boltzmann distribution, $n_e = n_0 \exp\left(\frac{eV(x)}{kT}\right)$, where $T$ is the plasma temperature.
The electron charge is known as $e$, and the vacuum permittivity is $\epsilon_0$.
Assume the surface area is $A$. If a small alternating voltage is superimposed on the conductor on the basis of the potential $V_f$, an alternating current will be generated between them. Find the dynamic capacitance $C$ of the conductor. (The dynamic capacitance is defined as $C = \frac{dQ}{dV}$, which is the derivative of charge with respect to potential.) | ||
351 | MECHANICS | The acceleration due to gravity $g$ is directed vertically downward. A light rope of length $2c$ symmetrically passes through a smooth fixed hole $O$ and suspends a rectangular thin plate with length $2a$ and width $2b$. The suspension points are the two vertices on one edge of length $2a$. The plate has a mass $m$ uniformly distributed over its surface. Initially, the plate is in a state of symmetric suspended equilibrium. Suitable choices of $a,b,c$ ensure that the plate is in stable equilibrium.
We restrict disturbances and motion to occur only within a plane parallel to the plate, so the number of possible vibration modes is $2$. Let $a=3l$, $b=4l$, and $c=5l$.
Please output the larger angular frequency, requiring an exact solution of the larger angular frequency rather than a numerical solution, using $g$ and $l$ for expression. |
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