Electrostatic Potential and Capacitance - Class 12

A semi-infinite solid rod of constant cross-sectional area $A$ extends along the x-axis from $x = 0$ to $x \to \infty$. The thermal conductivity of the material varies spatially as $k(x) = k_0 \left(1 + \frac{x}{L}\right)^2$, where $k_0$ and $L$ are positive constants. Heat is generated internally throughout the rod at a steady volumetric rate (power per unit volume) given by $q(x) = \frac{q_0}{\left(1 + \frac{x}{L}\right)^2}$, where $q_0$ is a positive constant. The end of the rod at $x = 0$ is perfectly insulated. If the temperature as $x \to \infty$ approaches a constant reference temperature $T_\infty$, what is the steady-state temperature $T_0$ at the insulated end $x = 0$?

A thick non-conducting spherical shell has an inner radius $R$ and an outer radius $2R$. The shell carries a non-uniform volume charge density given by $\rho(r) = \alpha r$ for $R \le r \le 2R$, where $\alpha$ is a positive constant with units of $\text{C m}^{-4}$, and $r$ is the radial distance from the center of the shell. The regions $r < R$ and $r > 2R$ contain no charge. Assuming the electric potential at infinity is zero, what is the electric potential at the center of the shell ($r = 0$)?

A particle of mass $m$ and charge $+q$ is placed in a one-dimensional region where the electric potential is given by $V(x,t) = \frac{1}{2} k (x - ut)^2$, with $k$ and $u$ being positive constants. The reference potential $V=0$ is defined at the moving boundary $x = ut$. At $t=0$, the particle is released from rest at $x = x_0$ (where $x_0 > 0$). Defining $\omega = \sqrt{\frac{qk}{m}}$, what is the kinetic energy of the particle in the laboratory frame at time $t = \frac{\pi}{2\omega}$?

An ideal current source supplies a total current $I_{\text{total}}(t) = I_{DC} + I_m \sin(\omega t)$ to a circuit consisting of a p-n junction diode in parallel with an AC-coupled variable load resistor $R_L$. The coupling capacitor acts as an ideal short circuit for the AC signal and an ideal open circuit for the DC signal. The diode's forward current is dominated by the diffusion of carriers over the potential barrier, following the relationship $I = I_0 \exp\left(\frac{eV}{kT}\right)$. The DC operating point is established at a forward voltage $V_{DC} = 0.60\text{ V}$ and a DC current $I_{DC} = 2.0\text{ mA}$. The thermal voltage is $\frac{kT}{e} = 25\text{ mV}$, and the AC signal amplitude is $I_m = 40\ \mu\text{A}$. To extract the maximum average AC power into $R_L$, what must be the value of $R_L$, and what is the corresponding maximum average AC power $P_{\text{max}}$ dissipated in $R_L$?

A uniform time-varying magnetic field $\vec{B} = \alpha t \hat{k}$ exists in a cylindrical region $x^2 + y^2 \le R^2$, where $\alpha$ is a positive constant in $\text{T/s}$, $t$ is time in $\text{s}$, and $R$ is the radius in $\text{m}$. Outside this region, the magnetic field is zero. Consider two points $P(R, 0, 0)$ and $Q(0, R, 0)$ with coordinates given in meters. Let $W_1$ be the work done (in Joules) by the induced electric field on a point charge $+q$ (in Coulombs) as it is moved from $P$ to $Q$ along the path $P \to O \to Q$ (where $O$ is the origin). Let $W_2$ be the work done by the induced electric field on the same charge as it is moved from $P$ to $Q$ along the straight line segment connecting them. Which of the following gives the correct values for $W_1$ and $W_2$?

A trolley moves along the horizontal $+x$-axis with a constant acceleration $a = \frac{3g}{4}$. Inside the trolley, a uniform electric field $\vec{E} = \frac{mg}{q} \hat{i} + \frac{mg}{2q} \hat{j}$ is maintained. A straight, rigid, frictionless dielectric wire of length $L$ is fixed inside the trolley. The ends of the wire are at coordinates $A \left(0, \frac{4L}{5}\right)$ and $B \left(\frac{3L}{5}, 0\right)$ with respect to a coordinate system fixed to the trolley, where the $y$-axis is vertically upwards. A small bead of mass $m$ and charge $+q$ is released from rest at point $A$. What is the kinetic energy of the bead when it reaches point $B$, as measured in the trolley's reference frame?

A particle of mass $m$ and initial speed $v_0$ approaches a fixed repulsive scattering center from infinity. The potential energy of the interaction is given by $V(r) = \frac{\alpha}{r^2}$, where $\alpha > 0$ is a constant and $r$ is the radial distance from the center. If the impact parameter of the particle is $b$, what is the scattering angle $\theta$ of the particle?

A solid non-conducting sphere of radius $R$ has a radially varying volume charge density given by $\rho(r) = \rho_0 \frac{r}{R}$ for $0 \le r \le R$, where $\rho_0$ is a positive constant and $r$ is the radial distance from the center. If the total charge of the sphere is $Q$, what is the total electrostatic self-energy of this charge configuration? (Assume the potential at infinity is zero, and $\epsilon_0$ is the permittivity of free space).

A solid non-conducting sphere of radius $R$ is centered at the origin. The volume charge density within the sphere varies with the radial distance $r$ as $\rho(r) = \rho_0 \left(\frac{r}{R}\right)^2$, where $\rho_0$ is a positive constant. Assuming the electric potential at infinity is zero, what is the electric potential at a distance $r = \frac{R}{2}$ from the center of the sphere?

A smooth, rigid, insulating circular ring of radius $a$ is fixed in the $xy$-plane, with its equation given by $(x-a)^2 + y^2 = a^2$. A point charge $+Q$ is fixed at the point $(-a, 0)$. A uniform but time-varying magnetic field $\vec{B} = \alpha t \hat{k}$ exists in a cylindrical region $(x-a)^2 + y^2 \le R^2$ (where $R > 2a$), where $\alpha$ is a positive constant and $t$ is time. A small bead of mass $m$ and charge $+q$ is released from rest at the origin $(0,0)$. Due to the induced electric field, the bead begins to slide along the upper semicircle ($y \ge 0$) of the ring. What is the kinetic energy of the bead when it reaches the point $(2a, 0)$ for the first time? (Ignore gravity)

A cart of initial mass $m_0 = 2 \text{ kg}$ is moving with a velocity $v_0 = 4 \text{ m/s}$ along a smooth horizontal surface. At $x = 0$, it enters a region where it experiences a conservative restoring force $F = -kx$, where $k = 4 \text{ N/m}$. The region is filled with a stationary dust cloud of uniform linear mass density $\lambda = 1.5 \text{ kg/m}$. As the cart moves, it sweeps up all the dust it encounters, which sticks to the cart. What is the maximum distance $x_{max}$ the cart travels into the dust cloud before coming to rest momentarily?

A uniform electric field $\vec{E}_1$ exists in a vacuum region ($y > 0$). The electric field lines make an angle $\theta_1 = 60^\circ$ with the normal to the $x$-$z$ plane (the $y$-axis). The region $y < 0$ is filled with a linear, isotropic dielectric material of dielectric constant $K = 3$. There is no free charge on the boundary interface at $y = 0$. Let $d_1$ (in meters) and $d_2$ (in meters) be the perpendicular distances between two adjacent equipotential surfaces separated by the same potential difference $\Delta V$ (in volts) in the vacuum and the dielectric, respectively. Due to the surface polarization charge density at the interface, the equipotential surfaces exhibit a geometric 'kink'. Calculate the ratio $\frac{d_2}{d_1}$.

A rigid, non-conducting circular loop of radius $R$ is free to rotate in three dimensions about its center, which is fixed at the origin. A thin wire carrying a steady current $I$ is wound around the loop. Two small point charges, $+q$ and $-q$, are fixed at diametrically opposite points on the loop's circumference. The system is placed in a region with a uniform magnetic field $\vec{B} = B_0 \hat{k}$ and an electric field described by the equipotential surfaces $V(x,y,z) = -E_0 z$. What is the minimum work required by an external agent to rotate the loop slowly from its most stable equilibrium orientation to its most unstable equilibrium orientation?

A hypothetical atomic nucleus of mass number $A$ and atomic number $Z$ is modeled as a sphere of radius $R = R_0 A^{1/3}$, where $R_0$ is a constant. The electric charge is distributed within the nucleus such that the volume charge density $\rho(r)$ at a distance $r$ from the center is given by $\rho(r) = \rho_0 \left(\frac{r}{R}\right)^2$ for $r \le R$, where $\rho_0$ is a constant. Let $U_E$ be the electrostatic self-energy of this nucleus. If $U_0$ is the electrostatic self-energy of a nucleus with the same total charge $Ze$ and radius $R$, but with a uniform volume charge distribution, find the ratio $\frac{U_E}{U_0}$ and the proportional dependence of $U_E$ on $Z$ and $A$.

A bead of mass $m$ and charge $+q$ is constrained to slide without friction along a vertical non-conducting wire aligned with the $y$-axis. Two fixed point charges, each of charge $+Q$, are located at coordinates $(d, 0)$ and $(-d, 0)$ in the $xy$-plane. The bead is released from rest at $y = h$. A programmable external mechanism applies a vertical force to the bead such that it undergoes exact vertical free fall kinematics, maintaining a constant downward acceleration $g$. Simultaneously, a time-varying uniform electric field $\vec{E}(t) = E_0 t \hat{j}$ is active in the region. What is the total work done by the external mechanism from the moment of release until the bead reaches the origin ($y=0$)? (Assume $k = \frac{1}{4\pi\epsilon_0}$)

A smooth horizontal table has a fixed, rough, vertical circular wall of outer radius $R$. A small block of mass $m$ and charge $-q$ ($q > 0$) is placed on the table, in contact with the outer surface of the wall. Two point charges, each of charge $+Q$ ($Q > 0$), are fixed on the vertical axis passing through the center of the circular wall, at heights $H = \sqrt{3}R$ above and below the table. The coefficient of kinetic friction between the block and the wall is $\mu$. The block is given an initial tangential velocity $v_0 = \sqrt{\frac{qQ}{32 \pi \epsilon_0 m R}}$. What is the total distance $s$ traveled by the block along the wall before coming to rest?

A long cylindrical conducting shell of inner radius $R_1$ and outer radius $R_2$ is placed in a uniform magnetic field $\vec{B}$ pointing in the $+z$ direction. The magnetic field exists everywhere in space and its magnitude varies with time as $B(t) = \alpha t$, where $\alpha$ is a positive constant. A point charge $q$ is moved slowly by an external agent from point $P$ to point $Q$ through the conducting material. In cylindrical coordinates $(r, \phi, z)$ with the z-axis along the cylinder's axis, the coordinates of $P$ and $Q$ are $(R_2, 0, 0)$ and $(R_2, \pi, 0)$ respectively.

Consider two paths for moving the charge: Path A: A semi-circular arc along the outer surface of the shell ($r = R_2$). Path B: A path consisting of three segments: radially inward from $P$ to $(R_1, 0, 0)$, then a semi-circular arc along the inner surface ($r = R_1$) to $(R_1, \pi, 0)$, and finally radially outward to $Q$.

Assuming the charge is moved slowly enough that radiation and self-magnetic field effects are negligible, what is the difference in the work done by the external agent along Path A and Path B, i.e., $\Delta W = W_A - W_B$?

A spherical capacitor consists of two concentric perfectly conducting shells of inner radius $a$ and outer radius $b$. The space between the shells ($a < r < b$) is completely filled with a leaky dielectric material of uniform resistivity $\rho$ and permittivity $\epsilon$. A battery of constant electromotive force $\mathcal{E}$ and finite internal resistance $R_0$ is connected across the two shells via thin insulated wires. After a long time, steady-state conditions are reached. Which of the following expressions gives the magnitude of the electric field $E(r)$ at a radial distance $r$ ($a < r < b$) from the center? (Assume all quantities are in standard SI units)

An isolated, uncharged solid conducting sphere of radius $R$ is placed with its center at the origin $(0,0,0)$. Two point charges, $q_1 = +2Q$ and $q_2 = +Q$, are initially fixed at positions $(-d, 0, 0)$ and $(d, 0, 0)$ respectively, where $d \gg R$. The charge $q_2$ is then displaced by a small distance $\delta$ ($\delta \ll d$) along the positive x-axis to a new position $(d+\delta, 0, 0)$, while $q_1$ remains fixed. What is the new electrostatic potential at a point $P$ located inside the conducting sphere at $(R/2, 0, 0)$? (Assume all given variables are in SI units and $\epsilon_0$ is the permittivity of free space).

Three concentric thin conducting spherical shells A, B, and C have radii $R$, $2R$, and $3R$ respectively. Shells A and C are connected by a thin conducting wire and together carry a net charge of $-Q$. Shell B is given a charge of $+Q$. The entire system is placed in a region where the reference point for zero electric potential is chosen to be at a radial distance $r = 2.5R$ from the common center. The apparent self-capacitance of shell B in this configuration is defined as $C_B = \frac{Q}{V_B}$, where $V_B$ is the absolute electric potential of shell B. What is the value of $C_B$?

A parallel plate capacitor consists of a fixed left plate and a movable right plate, each of area $A$. The initial separation between the plates is $d$. The right plate has mass $m$, rests on a rough horizontal surface with a coefficient of friction $\mu$, and is connected to a relaxed ideal spring of spring constant $k$. The left plate is rigidly attached to a wall via an insulating cylindrical rod of cross-sectional area $S$ and breaking stress $\sigma_b$. A constant voltage $V$ is suddenly applied across the plates. The right plate moves towards the left plate and momentarily comes to rest at a separation of $d/2$. At this exact instant, the insulating rod holding the left plate breaks. Assuming the coefficient of static friction is equal to the kinetic friction, find the breaking stress $\sigma_b$ of the rod.

A uniform conducting rod of mass $m$ and length $L$ is pivoted at one of its ends and rotates in a horizontal plane with an initial angular velocity $\omega_0$. A uniform vertical magnetic field $B$ exists throughout the region. The free end of the rod slides without friction on a fixed, horizontal, circular conducting ring of radius $L$ centered at the pivot. An uncharged capacitor of capacitance $C$ is connected between the pivot and the circular ring. The total resistance of the circuit (rod, ring, and wires) is $R$. What is the steady-state charge $q_f$ on the capacitor after a long time?

A perfectly rigid, sealed parallel-plate capacitor has a plate area $A$ (in $\text{m}^2$) and a plate separation $d$ (in $\text{m}$). The space between the plates is completely filled with an incompressible solid dielectric of dimensionless dielectric constant $K$. The capacitor is placed horizontally at the bottom of a vessel whose cross-sectional area increases exponentially with height. The vessel is initially filled with a non-conducting liquid of density $\rho$ (in $\text{kg/m}^3$) such that the top of the capacitor is at a depth $h$ (in $\text{m}$) below the liquid's free surface. The atmospheric pressure is $P_0$ (in $\text{Pa}$). The capacitor is connected to an ideal battery that maintains a constant potential difference $V$ (in $\text{V}$) across its plates. If the liquid is slowly drained from the vessel until it is completely empty, what is the net electrical work done by the battery during this draining process? (Assume acceleration due to gravity is $g$ in $\text{m/s}^2$ and ignore fringing edge effects of the electric field).

In a Young's Double Slit Experiment (YDSE), the maximum intensity on the screen is $I_m$. A parallel plate capacitor of length $L$ and plate separation $d$ is placed immediately behind one of the slits. The space between the plates is filled with an electro-optic crystal. Light of wavelength $\lambda$ passes through the crystal parallel to the capacitor plates. The other slit has no medium in front of it. The refractive index of the crystal responds to the electric field $E$ between the plates as $n(E) = n_0 + \alpha E$, where $n_0$ and $\alpha$ are constants with appropriate SI units.

An alternating voltage $V(t) = V_0 \sin(\omega t)$ is applied across the capacitor. Assume the low-frequency capacitance remains constant, and the frequency $\omega$ is much higher than the response rate of the intensity detector. The static phase difference introduced by the crystal (when $V=0$) at the central point $O$ on the screen (where the geometric path difference is zero) is $\phi_0 = \frac{\pi}{3}$.

Given the integral identities $\frac{1}{2\pi} \int_0^{2\pi} \cos(z \sin \theta) d\theta = J_0(z)$ and $\frac{1}{2\pi} \int_0^{2\pi} \sin(z \sin \theta) d\theta = 0$, what is the time-averaged intensity $\langle I \rangle$ detected at point $O$?

(Define $\beta = \frac{2\pi \alpha V_0 L}{\lambda d}$)

A parallel plate capacitor is formed by two identical vertical triangular metal plates of base $b$ and height $L$. The bases of the triangular plates are horizontal and located at the bottom edge, while their apexes point vertically upwards. The separation between the plates is $d$. A rectangular dielectric slab of mass $M$, thickness $d$, width $b$, and length $L$ (with dielectric constant $K$) is inserted vertically upwards from the bottom into the space between the plates. The slab is constrained to move only in the vertical direction. The plates are maintained at a constant potential difference $V$ by an ideal battery. Let $x$ be the vertical depth of the slab inserted into the capacitor (measured from the bottom base). Assume $V$ is large enough such that an equilibrium position $x_0$ exists within $0 < x_0 < L$. Neglecting fringing fields, which of the following correctly gives the equilibrium insertion depth $x_0$ and the nature of this equilibrium?

A dielectric medium contains $N$ molecules per unit volume ($\text{m}^{-3}$). Each molecule is modeled as a classical Lorentz oscillator, where an electron of mass $m$ ($\text{kg}$) and charge $-e$ ($\text{C}$) is bound to a heavy nucleus by a harmonic restoring force with natural angular frequency $\omega_0$ ($\text{rad/s}$). The electron also experiences a damping force $\vec{F}_d = -b \vec{v}$, where $\vec{v}$ is its velocity ($\text{m/s}$) and $b$ ($\text{kg/s}$) is a positive constant. The medium is subjected to a uniform, time-varying external electric field $\vec{E}(t) = E_0 \cos(\omega t) \hat{i}$ ($\text{V/m}$). Assuming the system is in a steady state, what is the average power dissipated per unit volume ($\text{W/m}^3$) of the dielectric due to the external field?

An uncharged solid dielectric sphere of radius $R$ and relative permittivity $\epsilon_r = 2$ is placed in a non-uniform external electric field. The external electric potential is given by $V_{\text{ext}}(x,y,z) = -k(2z^2 - x^2 - y^2)$, where $k$ is a positive constant with units of $\text{V m}^{-2}$. The origin of the coordinate system coincides with the center of the sphere. What is the electric potential produced solely by the induced bound charges at a point on the positive $z$-axis at a distance $r = 2R$ from the center?

A parallel plate capacitor has plate area $A$ (in $\text{m}^2$) and separation $d$ (in $\text{m}$). The space between the plates, from $x = 0$ to $x = d$, is filled with a non-uniform dielectric whose relative permittivity varies as $\epsilon_r(x) = 1 + \frac{x}{d}$. The plate at $x = 0$ is given a free charge $+Q$ (in $\text{C}$) and the plate at $x = d$ is given a free charge $-Q$ (in $\text{C}$). Let $x_0$ be the distance from the positive plate such that the electrostatic energy stored in the region $0 \leq x \leq x_0$ is exactly half of the total electrostatic energy stored in the entire capacitor. Find the value of $x_0$.

An incompressible, non-viscous fluid of mass density $\rho$ ($\text{kg/m}^3$), absolute permittivity $\epsilon$ ($\text{F/m}$), and uniform free charge density $\rho_e$ ($\text{C/m}^3$) flows steadily and irrotationally through a horizontal pipe. The pipe has a wide section of cross-sectional area $2A$ ($\text{m}^2$) (Region 1) and a narrow section of area $A$ ($\text{m}^2$) (Region 2). Region 1 is in a field-free space. Region 2 passes through a parallel-plate capacitor of plate separation $d$ ($\text{m}$), which maintains a uniform transverse electric field $E$ ($\text{V/m}$) perpendicular to the flow direction. The volume flow rate of the fluid is $Q$ ($\text{m}^3\text{/s}$). Assuming the flow remains strictly one-dimensional and accounting for the electro-hydrodynamic energy balance, what is the pressure difference $P_1 - P_2$ ($\text{Pa}$) between Region 1 and Region 2 along a central streamline?

A long cylindrical region of radius $R$ contains a time-varying magnetic field directed parallel to its central axis. The magnitude of the magnetic field is given by $B(r,t) = \beta r t$, where $\beta$ is a positive constant, $r$ is the radial distance from the axis, and $t$ is time. The magnetic field outside this region is zero. Calculate the total electrical energy stored in the induced electric field per unit length of the cylinder, within the region $0 \le r \le R$. (Assume $\epsilon_0$ is the permittivity of free space).

Two large parallel conducting plates of area $A$ are placed at $x = 0$ and $x = d$ in vacuum. Both plates are grounded ($V = 0$). The space between the plates is filled with a frozen-in, non-uniform volume charge density $\rho(x) = \rho_0 \frac{x}{d}$, where $\rho_0$ is a positive constant. Neglecting edge effects, what is the total electrostatic energy stored in the region between the plates?

A solid isotropic dielectric sphere of outer radius $R_2$ has a concentric spherical cavity of radius $R_1$. The inner surface of the cavity and the outer surface of the dielectric are coated with infinitesimally thin conducting layers to form a spherical capacitor. The dielectric has a relative permittivity $K$ (assumed independent of temperature) and a coefficient of linear thermal expansion $\alpha$. The capacitor is connected to a battery that maintains a constant voltage $V$ across the conducting layers. The temperature of the dielectric is then increased by a small amount $\Delta T$. Assuming the conducting layers expand with the dielectric, what is the mechanical work done by the electrostatic forces during this expansion? (Here $\epsilon_0$ is the permittivity of free space).

A solid conducting sphere $A$ of radius $R$ is initially uncharged. A separate conducting spherical shell $B$ has an outer radius of $3R$ and contains an off-center spherical cavity. A point charge $+Q$ is fixed inside this cavity. The shell $B$ is initially given a net charge of $+2Q$. Sphere $A$ is placed at a large distance from shell $B$. The two conductors are then connected by a long, thin conducting wire. After electrostatic equilibrium is reached, what is the final charge on sphere $A$ and the total electric flux passing through a spherical Gaussian surface of radius $4R$ concentric with shell $B$?

A capacitor of capacitance $C$ is initially charged to a potential difference $V_0$. It is connected in a series circuit containing an open switch $S$, an uncharged capacitor of capacitance $2C$, and an ideal solenoid of self-inductance $L$. The ohmic resistance of the circuit is negligible. If the switch $S$ is closed at $t = 0$, what is the maximum potential difference developed across the initially uncharged capacitor during the subsequent LC oscillations?

A parallel plate capacitor has rectangular plates of length $L$, width $w$, and separation $d$. It is connected to an ideal battery of electromotive force $V$. A solid dielectric slab of length $L$, width $w$, thickness $d/2$, and dielectric constant $K$ is initially outside the capacitor. The following sequence of operations is performed:

1. The slab is slowly inserted into the gap, parallel to the plates, until it completely covers the area $L \times w$.
2. The battery is disconnected.
3. The slab is slowly pulled completely out of the capacitor. Find the total work done by the external agent during this entire process. (Ignore edge effects, $\epsilon_0$ is the permittivity of free space)

A parallel plate capacitor with plate area $A$ and separation $d$ is charged to a total free charge $Q$ and then electrically isolated. A dielectric slab of the same area $A$ and thickness $d$ is slowly inserted between the plates. The dielectric constant of the slab varies with the distance $x$ from one of its faces as $K(x) = \frac{K_0}{1 + \alpha x}$, where $K_0$ and $\alpha$ are positive constants. Assume $K_0 > 1 + \frac{\alpha d}{2}$ and that the electric field is strictly normal to the plates. What is the magnitude of the work done by the external agent to completely insert the slab? (Here, $\epsilon_0$ is the vacuum permittivity).

A parallel plate capacitor with plates of length $L$, width $b$, and separation $d$ is fixed horizontally. The lower plate is rough with a coefficient of kinetic friction $\mu$. A dielectric slab of mass $m$, dielectric constant $\kappa$, and dimensions $L \times b \times d$ is placed such that it can slide between the plates. The slab is connected to a fixed wall by a horizontal ideal spring of spring constant $k$. Initially, the spring is relaxed and the slab is just at the edge of the capacitor ($x=0$). A constant voltage $V$ is applied across the plates. Assume the normal force exerted by the lower plate on the slab is $mg$, the electrostatic force is sufficient to pull the slab in, and the maximum displacement $x_{max} < L$. What is the total work done by the battery from the instant the slab is released until it reaches its maximum displacement?

A parallel plate capacitor is formed by two fixed semi-circular metal plates. A semi-circular dielectric slab is pivoted at the center of its straight edge and can rotate freely between the plates. The capacitance of the system as a function of the angle $\theta$ (in radians) of the slab inside the plates is given by $C(\theta) = C_0 (\pi + \beta \theta)$, where $C_0$ and $\beta$ are positive constants. The capacitor is given a charge $Q$ and then electrically isolated. A torsional spring of stiffness $k$ exerts a restoring torque $\tau_s = -k\theta$ on the slab. The system is found to be in stable equilibrium at an angle $\theta = \theta_0$ ($0 < \theta_0 < \pi$). Let $I$ be the moment of inertia of the slab about the pivot. Which of the following expressions represents the angular frequency $\omega$ of small oscillations of the slab about this equilibrium position?

A transparent, non-magnetic, non-polar dielectric slab of thickness $t = 2.4 \text{ \mu m}$ is placed directly in front of one of the slits in a Young's Double Slit Experiment (YDSE) setup. The experiment uses monochromatic light of wavelength $\lambda = 400 \text{ nm}$. To determine the optical properties of the slab, it is tested in an electrostatic setup: when the slab completely fills the space between the plates of a parallel plate capacitor connected to a constant DC voltage source, the energy stored in the capacitor is $U_1 = 90 \text{ \mu J}$. When the slab is removed, leaving air between the plates while still connected to the same voltage source, the stored energy becomes $U_2 = 10 \text{ \mu J}$. Assuming the relative permittivity of the dielectric material is frequency-independent, by how many fringes does the central maximum shift on the screen due to the insertion of the slab?

A semi-infinite ladder network is constructed with alternating series and shunt branches. The input terminals are at the left end. For all integers $n \ge 1$, the $n$-th series branch contains a capacitor of capacitance $C$ (in Farads) and a resistor of resistance $R_n$ (in Ohms) connected in series. The $n$-th shunt branch contains a capacitor of capacitance $2C$ (in Farads) and a resistor of resistance $r_n$ (in Ohms) connected in series. The resistance values $R_n$ and $r_n$ are arbitrary, strictly positive finite constants. At $t=0$, an ideal DC voltage source of EMF $V_0$ (in Volts) is connected across the input terminals. Assuming all capacitors are initially uncharged, what is the total energy dissipated as heat (in Joules) in all the resistors of the network as $t \to \infty$?

An infinite straight wire carrying a uniform linear charge density $\lambda$ is placed along the $z$-axis. A sequence of small conducting spheres, numbered $n = 0, 1, 2, \dots$, each of radius $R$, are placed along the $x$-axis at positions $x_n = a \cdot 2^n$ (where $R \ll a$). Adjacent spheres $n$ and $n+1$ are connected by a thin wire with a lumped capacitor of capacitance $C$. The self-capacitance of each sphere is $C_0 = 4\pi\epsilon_0 R$. Given that $C_0 = \frac{4}{3}C$, and the system is assembled such that the net charge on each isolated section (i.e., sphere $n$ and the two capacitor plates directly connected to it form an isolated neutral island) is zero, find the steady-state charge $Q_0$ on the sphere at $n=0$.

An ideal LC oscillator, acting as a Simple Harmonic Motion system for charge, generates an AC voltage $V(t) = V_0 \sin(\omega t)$, where $V_0$ is the peak voltage in Volts and $t$ is time in seconds. This voltage is applied across the input terminals of an infinite ladder network. The ladder consists of series capacitors, each of capacitance $C$ (in Farads), and shunt inductors, each of inductance $L$ (in Henrys). The angular frequency of the oscillator is tuned to $\omega = \frac{1}{\sqrt{LC}} \text{ rad/s}$. Find the peak instantaneous power $P_{peak}$ delivered by the source to the infinite ladder network.

A solid non-conducting sphere of radius $R$ has a spherically symmetric volume charge density given by $\rho(r) = kr$, where $k$ is a positive constant and $r$ is the radial distance from the center of the sphere. Assuming the electrostatic potential at infinity is zero, what is the electrostatic potential $V(r)$ at a point inside the sphere ($r < R$)?

A long cylindrical region of radius $R$ contains a time-varying uniform magnetic field $\vec{B} = \beta t \hat{k}$, where $\beta$ is a positive constant and $t$ is time. A uniform line charge of linear charge density $\lambda$ is fixed along the $z$-axis. A point charge $q$ is moved slowly by an external agent from point $P(a, 0, 0)$ to point $Q(b, \pi/2, 0)$ in cylindrical coordinates $(r, \theta, z)$, where $a < b < R$. The chosen path consists of a circular arc from $(a, 0, 0)$ to $(a, \pi/2, 0)$ in the $z=0$ plane, followed by a radial line from $(a, \pi/2, 0)$ to $(b, \pi/2, 0)$. What is the total work done by the external agent? (Assume $\epsilon_0$ is the permittivity of free space).

Two concentric conducting spherical shells of radii $a$ and $b$ ($a < b$) form a spherical capacitor. The inner shell carries a free charge $+Q$ and the outer shell carries a free charge $-Q$. The space between the shells ($a < r < b$) is filled with a non-uniform isotropic dielectric material whose relative permittivity varies with the radial distance $r$ as $\epsilon_r(r) = \frac{\beta}{r}$, where $\beta$ is a positive constant with dimensions of length. Which of the following expressions correctly represents the bound volume charge density $\rho_b(r)$ within the dielectric region?

A particle of rest mass $m_0$ and charge $-q$ is projected tangentially with speed $v_0$ from a radial distance $r = 2R$ in the vacuum space between two fixed concentric non-conducting spherical shells of radii $R$ and $4R$. The inner shell carries a uniformly distributed charge $+Q$, while the outer shell carries a uniformly distributed charge $-Q$. The particle's trajectory just grazes the inner shell. It is given that the characteristic electrostatic energy parameter $U_0 = \frac{qQ}{4 \pi \epsilon_0 R}$ is equal to $10^{-6} m_0 c^2$, where $c$ is the speed of light in vacuum. Which of the following is the correct expression for the required projection speed $v_0$?

A total charge $Q$ is distributed within a non-conducting solid sphere of radius $R$ situated in a vacuum. The volume charge density $\rho(r)$ is not uniform but varies with the radial distance $r$ from the center as $\rho(r) = kr$, where $k$ is a positive constant. What is the total electrostatic self-energy of this charge configuration? (Here, $\epsilon_0$ is the permittivity of free space).

A uniform solid sphere of radius $R$ is assembled from a dust of infinitesimal particles. The sum of the rest masses of these particles when infinitely separated is $M_0$. The sphere also carries a total charge $Q$ uniformly distributed throughout its volume. According to the mass-energy equivalence principle ($E=mc^2$), the total energy of the assembled stationary system determines its effective inertial mass $M_{eff}$. Assuming the charge and mass distributions remain perfectly uniform, which of the following expressions correctly represents the effective mass $M_{eff}$ of the sphere? (Here $G$ is the universal gravitational constant, $\epsilon_0$ is the permittivity of free space, and $c$ is the speed of light in vacuum).

A thermally insulated spherical soap bubble of initial radius $r_0$ contains $n$ moles of an ideal monatomic gas at an initial temperature $T_0$. The bubble is given a uniformly distributed total charge $Q$. The system is placed in a vacuum and allowed to expand quasi-statically to a new radius $2r_0$. Assuming the capacitance of the bubble is that of a thin spherical shell and the surface tension of the soap solution is $S$, what is the final temperature $T_f$ of the gas? (Let $R$ be the universal gas constant and $\epsilon_0$ be the vacuum permittivity).

A solid conducting sphere of radius $R$ is given an initial charge $Q$ and is immersed in an infinite, homogeneous liquid dielectric medium of relative permittivity $\epsilon_r = 3$. The sphere is then connected via a long, thin conducting wire to an initially uncharged parallel-plate capacitor of capacitance $C = 8\pi\epsilon_0 R$ located in a vacuum far away. After the switch is closed, charge redistributes until electrostatic equilibrium is reached. What is the total heat dissipated in the wire during this process? (Assume the wire has negligible capacitance and the dielectric medium remains unperturbed).