Electric Charges and Fields - Class 12

A uniformly charged fixed ring of radius $R$ and total charge $Q$ lies in the $y-z$ plane with its center at the origin. A particle of mass $m$ and charge $q$ (where $qQ > 0$) is constrained to move along the $x$-axis. The particle is attached to a light ideal spring of spring constant $k$, whose other end is fixed at the origin. If $k = \frac{q Q}{32 \pi \epsilon_0 R^3}$, the particle is displaced slightly from its stable equilibrium position $x_0 > 0$. What is the angular frequency $\omega$ of its small oscillations?

A spherical cloud of dust of radius $R$ (in $\text{m}$) and total mass $M$ (in $\text{kg}$) is held together by its own gravity. The mass density of the cloud varies radially as $\rho(r) = c r$, where $c$ is a positive constant in $\text{kg}\cdot\text{m}^{-4}$ and $r$ is the distance from the center in $\text{m}$. Let $V_c$ be the gravitational potential at the center of the cloud in $\text{J}\cdot\text{kg}^{-1}$ (taking the potential at infinity to be zero), and $U$ be the total gravitational self-energy of the cloud in $\text{J}$. Which of the following expressions correctly relates $U$, $M$, and $V_c$?

A fixed non-conducting sphere of radius $R$ has a spherically symmetric, variable volume charge density given by $\rho(r) = \frac{\rho_0 R}{r}$ for $0 < r \le R$, where $\rho_0$ is a positive constant and $r$ is the distance from the center. A narrow, smooth diametric tunnel is drilled through the sphere. A particle of mass $m$ and charge $-q$ ($q > 0$) is released from rest from the surface of the sphere into the tunnel. Let $T_1$ be the time period of its resulting oscillatory motion. Another identical particle of mass $m$ and charge $-q$ is placed in a circular orbit of radius $R$ around the sphere, grazing its surface. Let $T_2$ be the time period of this circular orbit. Assuming no gravitational forces are present, what is the ratio $\frac{T_1}{T_2}$?

A solid non-conducting sphere of radius $R$ has a total charge $Q$ uniformly distributed throughout its volume. A thin, uncharged conducting spherical shell of radius $R/2$ is embedded concentrically within the non-conducting sphere (assume the shell is infinitesimally thin and does not alter the uniform volume charge distribution). This embedded shell is connected via a long, thin, insulated conducting wire to a distant, initially uncharged solid conducting sphere of radius $R$. The wire contains a switch which is initially open. When the switch is closed, charge flows until electrostatic equilibrium is reached. Calculate the total heat dissipated in this process. (Assume the permittivity of free space is $\epsilon_0$ and the distant sphere is far enough that proximity effects are negligible).

A solid non-conducting sphere of radius $R$ has a uniform volume charge density $\rho$. A small electric dipole with dipole moment $\vec{p} = p_0 \hat{j}$ is placed at a point $A$ inside the sphere. The position vector of point $A$ from the center of the sphere is $\vec{r} = \frac{R}{2} \hat{i}$, where $\hat{i}, \hat{j}, \hat{k}$ are standard Cartesian unit vectors. What is the net electric force $\vec{F}$ and the net torque $\vec{\tau}$ acting on the dipole? (Assume the presence of the dipole does not alter the charge distribution of the sphere, and $\varepsilon_0$ is the permittivity of free space).

A solid non-conducting sphere of mass $M$, radius $R$, and total positive charge $Q$ is placed on a rough inclined plane that makes an angle $heta = 37^\circ$ with the horizontal. The incline slopes upwards to the right. The sphere has a non-uniform internal charge distribution such that its Effective Center of Charge (ECC) is located at a distance $R/2$ vertically below its Center of Mass (CM). A uniform horizontal electric field of magnitude $E$ is applied in the rightward direction. The incline is sufficiently rough to prevent slipping. If the sphere remains in both translational and rotational equilibrium, what is the required magnitude of the electric field $E$? (Take $\sin 37^\circ = 3/5$ and $\cos 37^\circ = 4/5$)

Two point charges, each of magnitude $+Q$, are fixed at $x = 0$ and $x = L$ in a vacuum. A solid isotropic dielectric slab of thickness $t = L/4$ is placed such that its left face is fixed at $x = 0$. The initial relative permittivity of the slab is $\epsilon_r = 4$. A small test charge $+q$ is placed at a point $x = x_0$ ($t < x_0 < L$) such that it is in electrostatic equilibrium. (Assume the effective distance in vacuum for a thickness $t$ of dielectric $\epsilon_r$ is $t\sqrt{\epsilon_r}$).\The temperature of the slab is then increased by $\Delta T$. The slab has a coefficient of linear thermal expansion $\alpha$. The relative permittivity $\epsilon_r$ of the slab depends on its density $\rho$ according to the relation $\epsilon_r - 1 = K \rho$, where $K$ is a positive constant. Assuming the slab expands freely, find the shift in the equilibrium position of the test charge, $\Delta x = x_0' - x_0$. (Assume $\alpha \Delta T \ll 1$).

Two identical point charges, each of charge $+Q$, are fixed at positions $(d, 0, 0)$ and $(-d, 0, 0)$ in a Cartesian coordinate system. A small ideal electric dipole of dipole moment $\vec{p} = p \hat{j}$ (where $p > 0$) is constrained to move only along the $y$-axis with its orientation fixed. The dipole is placed near its stable equilibrium position $y_0 > 0$ on the $y$-axis. If the system is in thermal equilibrium at absolute temperature $T$, find the root-mean-square (RMS) displacement of the dipole from its equilibrium position. Assume the displacement is small enough that the restoring force is linear. Let $k_B$ be the Boltzmann constant and $\epsilon_0$ be the vacuum permittivity.

A small bead of mass $m$ (in $\text{kg}$) and charge $+q$ (in $\text{C}$) is threaded on a smooth, rigid, insulating wire shaped as a parabola $y = \alpha x^2$ (where $\alpha$ is a positive constant in $\text{m}^{-1}$) in the $x$-$y$ plane. A uniform electric field $\vec{E} = E_0 \hat{i}$ (in $\text{V/m}$) exists throughout the region. The bead is released from rest at the origin $(0,0)$. Assuming the entire setup is in a gravity-free space, find the magnitude of the normal force (in $\text{N}$) exerted by the wire on the bead when it reaches the point where the tangent to the wire makes an angle of $60^\circ$ with the positive $x$-axis.

A point charge $Q$ (in $\text{C}$) is placed at the origin. It is surrounded by a thick spherical dielectric shell of inner radius $R$ and outer radius $2R$ (both in $\text{m}$). The relative permittivity (dielectric constant) of the shell varies with the radial distance $r$ from the origin as $K(r) = \alpha r$, where $\alpha$ is a positive constant with units of $\text{m}^{-1}$. Which of the following expressions correctly represents the volume bound charge density $\rho_b$ inside the dielectric shell at a distance $r$ ($R < r < 2R$) from the origin?

A parallel plate capacitor with plates of length $L$, width $w$, and separation $d$ is fixed horizontally. It is connected to a constant voltage source $V$. A dielectric slab of mass $M$, dielectric constant $K$, and dimensions $L \times w \times d$ is initially placed such that its front edge just enters the capacitor ($x=0$). The slab is connected to a massless, inextensible string that passes over a rough solid cylindrical pulley of mass $m_p$ and radius $R$. The other end of the string is attached to a hanging block of mass $m$. The system is released from rest. Assuming no friction between the slab and the capacitor plates, which of the following expressions gives the speed $v$ of the hanging block after the slab has been inserted a distance $x$ into the capacitor?

In a Young's Double Slit Experiment (YDSE), a white light source emitting wavelengths from $\lambda_V = 400 \text{ nm}$ to $\lambda_R = 700 \text{ nm}$ is used. The slit separation is $d = 1.0 \text{ mm}$ and the screen is at a distance $D = 1.0 \text{ m}$ from the slits. A transparent dielectric slab of thickness $t = 10 \text{ \mu m}$ is placed directly in front of the upper slit. The refractive index of the slab varies with wavelength $\lambda$ due to dispersion, having values $n_V = 1.52$ for violet light and $n_R = 1.50$ for red light. Determine the spatial spread $\Delta y$ of the zero-order maxima on the screen for the visible spectrum, and identify which color's zero-order maximum is closest to the geometric center of the screen.

An infinite homogeneous dielectric medium of relative permittivity $K$ contains a spherical vacuum cavity of radius $R$. A point charge $q$ is fixed at the center of the cavity. Another point charge $Q$ is embedded in the dielectric medium at a distance $d$ from the center of the cavity. Assuming $d \gg R$, what is the magnitude of the net electrostatic force experienced by the charge $q$?

A gas of Hydrogen atoms is placed in a uniform external electric field $\vec{E} = E_0 \hat{k}$. The first-order Stark effect perturbs the first excited state ($n=2$), splitting the degenerate manifold into three distinct energy levels: $E_2^{(0)}$, $E_2^{(0)} + 3ea_0E_0$, and $E_2^{(0)} - 3ea_0E_0$, where $E_2^{(0)}$ is the unperturbed energy, $e$ is the elementary charge, and $a_0$ is the Bohr radius. The ground state ($1s$) energy $E_1^{(0)}$ remains unshifted. The states corresponding to the shifted energies $E_2^{(0)} \pm 3ea_0E_0$ are linear combinations of the $2s$ and $2p_z$ orbitals, while the unshifted energy $E_2^{(0)}$ corresponds to the $2p_x$ and $2p_y$ orbitals. Let $\lambda_0$ be the wavelength of the Lyman-$\alpha$ line in the absence of the electric field. Assuming $3ea_0E_0 \ll (E_2^{(0)} - E_1^{(0)})$, what is the number of allowed distinct spectral lines observed for transitions from the $n=2$ manifold to the ground state, and what is the maximum difference in wavelength $\Delta \lambda$ between the highest and lowest energy photons emitted?

A simple pendulum consists of a bob of mass $m$ and positive charge $q$ suspended by a massless, inextensible string of length $L$ from a fixed pivot. A fixed point charge $+Q$ is located exactly at the pivot. The entire setup is placed in a uniform horizontal electric field of magnitude $E$. The acceleration due to gravity is $g$ downwards. The bob is allowed to reach its stable equilibrium position, where the string makes an angle $\theta_0$ with the downward vertical. If the bob is given a small angular displacement from this equilibrium position within the vertical plane, what is the time period $T$ of its small oscillations? (Assume $\epsilon_0$ is the permittivity of free space).

A region of space contains a net electric field $\vec{E}$ composed of a static conservative component and a non-conservative induced component due to a localized time-varying magnetic field. In cylindrical coordinates $(r, \theta, z)$, the net electric field in the $z=0$ plane is given by: $\vec{E} = E_0 \frac{a}{r} \hat{r} + E_0 \frac{r}{a} \hat{\theta}$ where $E_0$ is a positive constant with units of $\text{V}\cdot\text{m}^{-1}$ and $a$ is a positive constant with units of $\text{m}$. A point charge $q$ (in Coulombs) is moved from point $P_1$ at $(r=a, \theta=0)$ to point $P_2$ at $(r=a e^{\pi}, \theta=\pi)$ along a logarithmic spiral path defined by $r(\theta) = a e^{\theta}$. What is the total work done by the net electric field on the point charge $q$ during this motion?

A particle of mass $m$ and charge $+q$ is constrained to move along a frictionless, non-conducting straight wire aligned with the $x$-axis. The setup is permeated by an external electric field given by $\vec{E} = E_0 \left( \frac{2x}{L} - \frac{x^2}{L^2} \right) \hat{i}$, where $E_0$ and $L$ are positive constants. The entire wire assembly is placed inside a rocket that accelerates along the positive $x$-axis with a constant acceleration $a_0 = \frac{3 q E_0}{4 m}$. In the reference frame of the rocket, the particle is released from rest at a position slightly displaced from its stable equilibrium point. Determine the position of the stable equilibrium point $x_{eq}$ and the angular frequency $\omega$ of its small oscillations.

An electric dipole of dipole moment $p$ (in $\text{C}\cdot\text{m}$) is constrained to rotate in the $xy$-plane about a fixed pivot at the origin. A frictionless mechanism alters the mass distribution of the dipole assembly such that its moment of inertia about the pivot depends on its angular position $\theta$ (the angle between the dipole moment and the positive $x$-axis) as $I(\theta) = I_0 (1 + \alpha \sin^2\theta)$, where $I_0$ (in $\text{kg}\cdot\text{m}^2$) and dimensionless $\alpha$ are positive constants. The mechanism is fixed to the ground and does no net work, ensuring the mechanical energy of the assembly is conserved. A uniform electric field $E$ (in $\text{N/C}$) exists along the positive $x$-axis. If the assembly is released from rest at $\theta = \pi/2 \text{ rad}$, what is the magnitude of its angular acceleration (in $\text{rad/s}^2$) when it passes through $\theta = \pi/4 \text{ rad}$?

A rigid electric dipole of dipole moment magnitude $p$ is moved slowly along the x-axis from the origin $(0,0)$ to the point $(L,0)$. The region contains a static electric field given by $\vec{E} = (E_0 + \alpha x)\hat{i} - \alpha y \hat{j}$, where $E_0$ and $\alpha$ are positive constants. During the motion, an external mechanism continuously rotates the dipole such that its angle $\theta$ with the positive x-axis changes linearly with its position $x$, according to the relation $\theta(x) = \frac{\pi}{2} \left(1 - \frac{x}{L}\right)$. What is the work done strictly by the translational electric force on the dipole during this entire process?

A rigid insulating rod of length $2a$ has point charges $+q$ and $-q$ fixed at its ends, forming an electric dipole of moment $\vec{p}$ (where $p = 2qa$). The dipole is placed such that its center is fixed on the axis of a circular loop of radius $R$ ($R \gg a$) carrying a steady counter-clockwise current $I$. The loop lies in the $xy$-plane with its center at the origin $(0,0,0)$. The center of the dipole is located at coordinates $(0, 0, R)$. The dipole is constrained to rotate in the plane $z = R$ with a constant angular velocity $\vec{\omega} = \omega \hat{k}$. A uniform electric field $\vec{E} = E_0 \hat{i}$ is also present in the entire region. At the instant the dipole moment vector is aligned with the positive $y$-axis ($\vec{p} = p \hat{j}$), what is the net torque $\vec{\tau}$ acting on the dipole about its center? (Assume the magnetic field variation over the length of the dipole is small but non-zero, and account for all spatial components of the magnetic field).

A thin non-conducting ring of mass $M$ (in $\text{kg}$), radius $R$ (in $\text{m}$), and uniformly distributed charge $Q$ (in $\text{C}$) is pivoted at its center on a horizontal surface. The surface exerts a viscous frictional torque $\tau_v = -b \omega$ on the ring, where $\omega$ is its angular velocity (in $\text{rad/s}$) and $b$ is a positive constant (in $\text{kg}\cdot\text{m}^2/\text{s}$). A uniform magnetic field $\vec{B}$ is applied perpendicular to the plane of the ring. The magnetic field magnitude varies with time as $B(t) = B_0 (1 - e^{-t/t_0})$ for $t \ge 0$, where $B_0$ is in Tesla and $t_0$ is a time constant in seconds. The ring is initially at rest. What is the total angular displacement of the ring from $t=0$ until it finally comes to rest?

A deep-space rocket is traveling at a constant velocity $v = \frac{\sqrt{3}}{2}c$ relative to an inertial space station. At $t=0$, the rocket begins to accelerate with a constant proper acceleration $a$ along its direction of motion ($+x$-axis). Inside the rocket, a rigid electric dipole is pivoted frictionlessly at its negative charge $-q$. The positive charge $+q$ has a rest mass $m_0$ and is at a distance $2l$ from the pivot. A uniform static electric field $\vec{E} = E_0 \hat{i}$ and a uniform photon beam of intensity $I$ propagating along the $+y$-axis exist inside the rocket (both measured in the rocket's frame). A perfectly absorbing flat disk of area $A$ is attached to the $+q$ charge, rigidly fixed such that its surface normal is always parallel to the dipole axis (directed from $-q$ to $+q$). The $-q$ charge and the connecting rod do not interact with the radiation. Assuming the dipole reaches a steady-state equilibrium angle $\theta \in (0, \pi/2)$ with respect to the $+x$-axis, which of the following expressions correctly gives $\cos\theta$?

A non-conducting solid sphere of initial radius $R_0$ has a radially varying mass density $\rho_m(r)$. The moment of inertia of the spherical region enclosed within any radius $r$ (about a diametric axis) is given by $I(r) = I_0 \left(\frac{r}{R_0}\right)^6$, where $I_0$ is the total initial moment of inertia of the sphere. The sphere carries a total charge $Q$, distributed such that the volume charge density is $\rho_e(r) = \beta \rho_m(r) r^2$, where $\beta$ is a positive constant. The sphere is then heated, increasing its temperature by $\Delta T$. The coefficient of linear thermal expansion of the material is $\alpha$. Assuming the charge is bound to the material and $\alpha \Delta T \ll 1$, what is the magnitude of the electric field at a fixed spatial distance $r = R_0/2$ from the center of the sphere?

A rigid, thin rod of mass $M$ and length $L$ has a uniform positive charge $Q$. The rod is placed along the $y$-axis with its ends at $y = 0$ and $y = L$, and is free to rotate about a frictionless pivot at the origin $(0,0)$ of a laboratory frame. A positive point charge $q$ is rigidly fixed at the coordinates $(-d, 0)$ in this frame. The laboratory frame accelerates with a constant acceleration $\vec{a} = a_0 \hat{i}$ relative to an inertial frame. If the rod is released from rest relative to the laboratory frame, what must be the magnitude of the acceleration $a_0$ such that the initial angular acceleration of the rod is zero? (Ignore gravity and assume $k = \frac{1}{4\pi\epsilon_0}$)

A non-conducting thin disc of radius $R$ has a surface charge density $\sigma(r) = \sigma_0 \left(1 - \frac{r}{R}\right)$, where $r$ is the radial distance from the center and $\sigma_0$ is a positive constant. The disc rotates with a constant angular velocity $\omega$ about its central geometric axis. The disc is placed entirely inside a long ideal solenoid such that the axis of rotation of the disc is perpendicular to the central axis of the solenoid. The solenoid has $n$ turns per unit length and is wound using a wire of cross-sectional area $S$. The wire material has a free electron number density $\eta$. Due to Joule heating, the temperature $T$ of the solenoid wire increases, causing the drift velocity of the free electrons to vary as $v_d(T) = v_0 e^{-\alpha T}$, where $v_0$ and $\alpha$ are positive constants. Assuming the external magnetic field is uniform over the disc, what is the magnitude of the magnetic torque $\tau$ acting on the disc at temperature $T$?

A solid non-conducting sphere of radius $R$ has a spherically symmetric volume charge density $\rho(r) = kr$, where $k$ is a positive constant and $r$ is the radial distance from the center. A very narrow, smooth tunnel is drilled through the center of the sphere along a diameter. A point particle of mass $m$ and charge $-q$ ($q > 0$) is released from rest inside the tunnel at a distance $A$ ($A < R$) from the center. Let $T$ be the time period of the resulting oscillatory motion. Given that $\int_0^1 \frac{du}{\sqrt{1 - u^3}} = \beta$, which of the following expressions correctly represents $T$?

A solid non-conducting sphere of radius $R$ contains a uniform free volume charge density $\rho_0$. The sphere is filled with a non-uniform linear dielectric material whose relative permittivity varies with the radial distance $r$ from the center as $\epsilon_r(r) = \frac{R}{r}$ for $0 < r \le R$. Which of the following expressions correctly represents the volume bound charge density $\rho_b(r)$ inside the dielectric at a distance $r$ from the center?

An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho$. The axis of the cylinder coincides with the $z$-axis. A uniform time-varying magnetic field $\vec{B} = \alpha t \hat{k}$ is present throughout the region, where $\alpha$ is a positive constant and $t$ is time. What is the line integral of the net electric force $W = \int_A^B q\vec{E} \cdot d\vec{l}$ on a point charge $q$ along a straight line path from point $A(R, 0, 0)$ to point $B(0, R, 0)$ at a given instant $t$?

An infinitely long solid insulating cylinder of radius $R$ has a volume charge density $\rho(r) = \rho_0 \frac{r}{R}$, where $r$ is the radial distance from the cylinder's axis and $\rho_0$ is a positive constant. A thin, infinitely long straight wire lies along the axis of the cylinder and carries a steady current $I$. A particle of mass $m$ and charge $+q$ is projected from the surface of the cylinder ($r = R$) with an initial velocity $\vec{v}_0 = v_0 \hat{k}$ parallel to the axis. Assuming the initial velocity $v_0$ and current $I$ are sufficient for the particle to reach a radial distance of $r = R/2$, what is the total speed of the particle at this instant? (Ignore gravity and edge effects).

A solid non-conducting sphere of radius $R$ has a uniform volume charge density $\rho$. A spherical cavity of radius $r$ ($r < R$) is made inside the sphere. The center of the cavity is displaced from the center of the sphere by a constant vector $\vec{d}$. A particle of mass $m$ and charge $q$ is placed inside the cavity. The particle is attached to the center of the cavity by a light insulating spring of force constant $k$ and zero natural length. The entire sphere is then given a constant acceleration $\vec{a}$ in a direction perpendicular to $\vec{d}$. If the particle is released from rest relative to the cavity from the cavity's center, what is the maximum speed of the particle in the reference frame of the sphere? (Ignore gravity and assume the particle does not collide with the cavity walls).

A non-uniform, spherically symmetric charge distribution has a volume charge density $\rho(r) = \frac{\alpha}{r}$ for $r \le a$, where $r$ is the spherical radial distance from the origin and $\alpha$ is a positive constant. In the same region, there exists a magnetic field given in cylindrical coordinates $(s, \phi, z)$ as $\vec{B} = \frac{B_0 R}{s} \hat{s}$, where $B_0$ and $R$ are positive constants. A thin, straight conducting rod of length $L$ is placed parallel to the $z$-axis at a cylindrical radial distance $s = R$. The ends of the rod are at $z = 0$ and $z = L$. The rod rotates about the $z$-axis with a constant angular velocity $\vec{\omega} = \omega \hat{k}$. Assuming the rod is entirely within the charge distribution ($R^2 + L^2 < a^2$), what is the steady-state potential difference $V(L) - V(0)$ between the ends of the rod?

An isolated thin metallic spherical shell of initial radius $R_0 \text{ m}$ carries a constant total charge $Q \text{ C}$. The shell is in a vacuum. Initially, the electric field magnitude just outside the surface of the shell is $E_0 \text{ V/m}$ and the total electric flux through a spherical Gaussian surface infinitesimally larger than the shell is $\Phi_0 \text{ V}\cdot\text{m}$. The temperature of the shell is then uniformly increased by $\Delta T \text{ K}$. If the coefficient of linear thermal expansion of the shell material is $\alpha \text{ K}^{-1}$ (where $\alpha \Delta T \ll 1$), what are the fractional changes in the electric field magnitude ($\frac{\Delta E}{E_0}$) just outside the shell and the total electric flux ($\frac{\Delta \Phi}{\Phi_0}$) through the new infinitesimally larger Gaussian surface, respectively?

In a region of space, two independent static electric potentials are established, given by: $V_1(x,y,z) = E_0 x \exp\left(-\frac{z^2}{R^2}\right)$ $V_2(x,y,z) = E_0 y \exp\left(-\frac{z^2}{R^2}\right)$ where $E_0$ is a positive constant with units of $\text{V/m}$, and $R$ is a positive constant with units of meters. The corresponding electric fields are $\vec{E}_1 = -\nabla V_1$ and $\vec{E}_2 = -\nabla V_2$. A new vector field is defined as $\vec{G} = \vec{E}_1 \times \vec{E}_2$. What is the magnitude of the flux of $\vec{G}$ through the open hemispherical surface defined by $x^2 + y^2 + z^2 = R^2$ with $z \ge 0$?

An elevator is accelerating vertically upwards with a constant acceleration $a$. Inside the elevator, an infinite flat non-conducting sheet with a uniform positive surface charge density $\sigma$ is fixed vertically. A rigid, rough, non-conducting vertical rod is fixed parallel to the sheet at a distance $d$ from it. A bead of mass $m$ and positive charge $q$ is threaded on the rod. The coefficient of kinetic friction between the bead and the rod is $\mu$. If the bead is released from rest relative to the elevator, what is its kinetic energy relative to the elevator after it has slid a distance $L$ down the rod? (Assume $m(g+a) > \frac{\mu q \sigma}{2\varepsilon_0}$ and neglect edge effects).

A fixed infinite non-conducting flat sheet lies in the $y-z$ plane ($x=0$) and has a uniform surface charge density $+\sigma$ and a uniform surface mass density $\mu$. A fixed thin non-conducting spherical shell of radius $R$, total mass $M$, and total charge $+Q$ (uniformly distributed) is centered at the coordinates $(4R, 0, 0)$. A point particle of mass $m$ and charge $+q$ is released from rest at the coordinates $(4.5R, 0, 0)$. What is the magnitude of the initial acceleration of the particle? (Given: $G$ is the universal gravitational constant, $\varepsilon_0$ is the permittivity of free space)

An isolated, perfectly conducting sphere of radius $a$ in vacuum is initially charged to a potential $V_0$. At $t=0$, it is connected to ground through a series combination of a time-varying internal resistor and a variable external load resistor $R_L$. The internal resistor is a cylinder of length $L$ and cross-sectional area $A$, filled with a material of constant permittivity $\epsilon$. Due to heating, the dielectric relaxation time of this material decreases as $\tau(t) = \frac{\tau_0}{1 + \alpha t}$. A control system continuously adjusts $R_L$ such that the instantaneous power dissipated in $R_L$ is always maximized. Assuming the capacitance of the cylindrical resistor is negligible, which of the following gives the potential of the sphere $V(t)$?

A small conducting sphere of mass $m$, radius $R$, and net charge $Q$ is attached to one end of a rigid, massless rod of length $L$. The other end of the rod is pivoted at the origin $(0,0)$ and connected to a torsional spring of restoring constant $C$. The system is constrained to rotate in a smooth horizontal $xy$-plane. An external electric field is present in the region, given by $\vec{E} = \left(E_0 + E_1 \frac{x}{L}\right) \hat{i} - E_1 \frac{y}{L} \hat{j}$, where $E_0$ and $E_1$ are positive constants. The rod is initially in equilibrium along the positive $x$-axis. If the rod is given a small angular displacement $\theta$ and released, what is the angular frequency $\omega$ of the resulting simple harmonic motion? (Assume $R \ll L$ and neglect any radiation or damping effects. The permittivity of free space is $\epsilon_0$.)

Consider a macroscopic intrinsic semiconductor in thermal equilibrium. The average electron and hole concentrations are given by the Law of Mass Action as $\langle n \rangle = \langle p \rangle = n_i$. To analyze the noise floor in the drift-diffusion balance, we model the semiconductor as a series of adjacent microscopic cubic regions, each of volume $V = L^3$. Due to the discrete nature of charge, the actual number of electrons ($N_e$) and holes ($N_h$) in each region fluctuate independently according to Poisson statistics.

The instantaneous electron and hole diffusion currents from region 1 to an adjacent region 2 are modeled as: $I_e = \frac{e D_e}{L^2} (N_{e2} - N_{e1})$ $I_h = -\frac{e D_h}{L^2} (N_{h2} - N_{h1})$ where $N_{e1}, N_{e2}$ and $N_{h1}, N_{h2}$ are the instantaneous number of electrons and holes in regions 1 and 2, respectively. $D_e$ and $D_h$ are the diffusion coefficients, and $e$ is the elementary charge. What is the root-mean-square (RMS) value of the total instantaneous diffusion current, $I_{RMS} = \sqrt{\langle (I_e + I_h)^2 \rangle}$, between the two regions?

A neutral liquid droplet of mass $M$ is moving with an initial kinetic energy $K_0$ along the positive $x$-axis in a region of uniform electric field $\vec{E} = E_0 \hat{i}$. At $x = d$, it undergoes a perfectly inelastic collision with a stationary dust particle of mass $M$ and charge $+q$, absorbing it completely. At $x = 2d$, it undergoes another perfectly inelastic collision with a second identical stationary dust particle of mass $M$ and charge $+q$, absorbing it as well. What is the kinetic energy of the droplet when it reaches $x = 3d$?

A particle of mass $m$ and charge $q = Ne$ (where $N$ is a positive integer and $e$ is the elementary charge) is accelerated from rest through a potential difference $V$. It then enters a vacuum chamber containing a cylindrically symmetric magnetic field given by $\vec{B} = \frac{\alpha}{r} \hat{k}$, where $r$ is the radial distance from the $z$-axis and $\alpha$ is a positive constant. The particle moves in a stable circular orbit of radius $R$ in the $xy$-plane, centered at the origin.

Assume the particle's orbit satisfies the Bohr-Sommerfeld quantization condition, meaning the circumference of the orbit is an integer multiple $k$ ($k = 1, 2, 3, \dots$) of the particle's de Broglie wavelength. Which of the following pairs $(V, R)$ correctly gives the required accelerating potential and the allowed orbital radii?

A rough horizontal insulating rod is aligned along the $x$-axis. A small bead B of mass $m$ and charge $+q$ is initially at rest at the origin $(0,0)$ on the rod. Another small bead A of charge $+Q$ is slowly moved along the line $y = h$ in the vertical $xy$-plane (where the $y$-axis points vertically upwards). The coefficient of static friction between bead B and the rod is $\mu_s = \frac{1}{3}$. What is the maximum value of $Q$ such that bead B never slips on the rod during the entire motion of bead A from $x = -\infty$ to $x = +\infty$? (Assume acceleration due to gravity is $g$ and $k = \frac{1}{4\pi\epsilon_0}$)

A thin, fixed, non-conducting disk of radius $R \text{ m}$ has a uniform surface charge density $\sigma \text{ C/m}^2$. A point charge $q \text{ C}$ is placed at a point $P$ which is at a height $z \text{ m}$ above the center of the disk and at a very small radial distance $r \text{ m}$ ($r \ll R, z$) from the central axis of symmetry. Which of the following expressions correctly represents the approximate radial outward force $F_r$ acting on the point charge $q$? (Assume vacuum with permittivity $\epsilon_0 \text{ F/m}$)

A simple pendulum consists of a small bob of mass $m$ and positive charge $q$ suspended by a massless, inextensible string of length $L$. The point of suspension is at $O$. The equilibrium position of the bob is at $P$, where $OP = L$. A fixed point charge $+Q$ is placed at point $S$, such that $P$ is the midpoint of the vertical line segment $OS$ (with $S$ located directly below $P$). The pendulum is displaced by a small angle $\theta$ from its equilibrium position and released. Assuming $mg > \frac{Qq}{2\pi\epsilon_0 L^2}$, what is the angular frequency $\omega$ of the resulting simple harmonic motion?

A spherical capacitor consists of an inner conducting sphere of radius $a$ and an outer conducting spherical shell of inner radius $b$. The space between the conductors ($a \le r \le b$) is filled with a compressible dielectric fluid. Initially uniform, the system is subjected to a steady-state radial temperature gradient. Due to non-uniform thermal expansion, the local particle density of the fluid varies. According to the material's volume-polarizability model, this density gradient results in a spatially varying dielectric constant given by $K(r) = K_0 \frac{b}{r}$, where $r$ is the radial distance from the center and $K_0$ is the dielectric constant at the outer shell. What is the effective capacitance of this system?

A plane electromagnetic wave with a time-averaged total energy density $u_0$ propagates in a vacuum ($x < 0$) along the $+x$-direction. At $x = 0$, it enters a non-magnetic, lossless dielectric medium extending through $x \ge 0$. The relative permittivity of the medium, $K(x)$, increases continuously and very slowly from $K(0) = 1$ to $K(L) = 4$ at a depth $x = L$. The spatial gradient of $K(x)$ is sufficiently small such that any reflection of the wave can be neglected. Which of the following represents the time-averaged total energy density of the wave at $x = L$?

A spherical cloud of dust of total mass $M$ and radius $R$ has a non-uniform mass density given by $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ for $0 \le r \le R$, where $r$ is the radial distance from the center and $\rho_0$ is a positive constant. What is the magnitude of the gravitational self-energy of this dust cloud? (Here $G$ is the universal gravitational constant)

Three point charges are fixed on a circle of radius $R$ in the $xy$-plane centered at the origin. Charge $q_1 = +q$ is located at $\theta = 0^\circ$, $q_2 = -q$ is at $\theta = 60^\circ$, and $q_3 = +q$ is at $\theta = 120^\circ$. An ideal electric dipole with dipole moment $\vec{p} = p_0 \hat{i}$ is placed at the origin. What is the magnitude of the net electric force experienced by the dipole?

In a non-degenerate n-type semiconductor, the electron mobility $\mu_e$ depends on the absolute temperature $T$ as $\mu_e(T) = \mu_0 (T/T_0)^{-\alpha}$, where $\mu_0$ and $\alpha$ are positive constants ($\alpha$ is dimensionless), and $T_0$ is a reference temperature. The semiconductor is subjected to an electric field with an $x$-component $E$ and maintains a constant electron concentration gradient $\frac{dn}{dx} > 0$ along the $x$-axis. The electron concentration at the point of observation is $n$. Assume the Einstein relation $D_e = \frac{k_B T}{e} \mu_e$ holds valid, where $k_B$ is the Boltzmann constant and $e$ is the elementary charge. If the total electron current density $J_n$ along the $x$-axis is independent of small temperature variations around $T = T_0$ (i.e., $\left. \frac{dJ_n}{dT} \right|_{T=T_0} = 0$), what is the required value of $E$?

A stationary circular ring of radius $a$ (in meters) consists of two ideal semicircular conducting wires connected at diametrically opposite nodes P and Q. A lumped resistor of resistance $R$ is inserted in one semicircle, and a lumped ideal inductor of self-inductance $L$ is inserted in the other semicircle. The ring is placed in a uniform magnetic field perpendicular to its plane. The magnetic field is present everywhere inside the ring and its magnitude varies with time as $B(t) = \alpha t$, where $\alpha > 0$ is a constant in $\text{T s}^{-1}$. At $t=0$, the current in the circuit is zero. What is the magnitude of the scalar potential difference $|V_P - V_Q|$ between the nodes P and Q at time $t = \frac{L}{R} \ln 3$?

A continuous rectangular slit of width $a$ is illuminated by a coherent monochromatic plane wave of wavelength $\lambda$ at normal incidence. The intensity of the central maximum on a distant screen is $I_0$. A thin transparent wedge of refractive index $\mu = \frac{4}{3}$ is placed immediately behind the slit, covering it completely. The thickness of the wedge varies linearly from $t = 0$ at one edge of the slit to $t = \frac{9}{2}\lambda$ at the opposite edge. What is the new intensity at the original central point on the screen?