Current Electricity - Class 12

A long straight wire carrying a steady current $I_0$ is placed along the $x$-axis. A long rectangular conducting slab of width $w$ and thickness $t$ (where $t \ll a$) is placed in the $xy$-plane such that its length is parallel to the $x$-axis, and its two longitudinal edges are located at $y = a$ and $y = a + w$. The slab carries a steady current $I$ in the $+x$-direction. The number density of free electrons in the slab is $n$, and the elementary charge is $e$. Assuming steady-state conditions, what is the magnitude of the Hall potential difference $\Delta V$ developed across the width of the slab?

A uniform conducting wire of constant cross-sectional area $A$ and free electron density $n$ is bent into a semi-circle of radius $R$. A time-varying voltage is applied across its ends such that the tangential electric field inside the wire is $E(t) = E_0 \sin(\omega t)$ for $0 \leq t \leq \pi/\omega$, where $E_0$ is a positive constant. An average conduction electron enters one end of the semi-circle at $t=0$ and exactly reaches the other end at $t = \pi/\omega$. Let $\mu$ be the electron mobility and $e$ be the elementary charge. Which of the following gives the magnitude of the average drift velocity vector $|\langle \vec{v}_d \rangle|$ of this electron during this time interval, and the total charge $Q$ that flows through any cross-section of the wire?

A parallel plate capacitor with circular plates of radius $R$ is being charged by a long cylindrical copper wire of radius $r$ (where $r < R$). The free electron density in the copper wire is $n$, and the magnitude of the elementary charge is $e$. At a given instant, the electrons in the wire are moving with a drift velocity of magnitude $v_d$. Assuming the electric field $E$ between the capacitor plates is uniform and neglecting fringe effects, what is the rate of change of the electric field $\frac{\partial E}{\partial t}$ in the region between the plates at this instant? Let $\epsilon_0$ be the permittivity of free space.

A straight conducting wire is uniformly accelerated along its length (the $+x$ direction) with a constant acceleration $a_0$. A uniform constant electric field of magnitude $E$ is applied along the $+x$ direction within the wire. The conduction electrons have mass $m$, charge $-e$, and an average relaxation time $\tau$ between successive collisions with the lattice. Assume the standard Drude model where electrons lose their entire drift momentum relative to the lattice upon each collision, and the steady-state drift velocity is given by $\vec{v}_d = \vec{F}_{\text{net}}\tau/m$. In the non-inertial frame of the accelerating wire, let $P_{\text{heat}}$ be the average rate of heat dissipation per electron, and $P_E$ be the average rate at which the electric field does work on a single electron. Which of the following expressions correctly represents the difference $(P_{\text{heat}} - P_E)$? (All given quantities are in standard SI units)

A uniform cylindrical metallic rod of length $L$ and cross-sectional area $A$ is placed between two thermal reservoirs. The ends at $x=0$ and $x=L$ are maintained at steady temperatures $T_0$ and $4T_0$, respectively. Assume steady-state heat conduction and that the thermal conductivity of the metal is constant. According to the classical Drude model, the mean free path of conduction electrons is independent of temperature, making the relaxation time $\tau$ inversely proportional to the root-mean-square speed of the electrons. If the electrical resistivity of the metal at temperature $T_0$ is $\rho_0$, what is the total electrical resistance $R$ of the rod?

In a modified Drude model for a metal, the instantaneous collision frequency $\nu$ (in $s^{-1}$) of electrons with the lattice is a stochastic variable due to local thermal fluctuations. The collision frequency has a mean value $\nu_0$ and a standard deviation $\Delta \nu$, such that the relative fluctuation is $\delta = \frac{\Delta \nu}{\nu_0} \ll 1$. Assume the fluctuations are slow enough that the electron drift velocity continuously reaches its steady-state value $v_d = \frac{eE}{m\nu}$ under a constant applied electric field $E$ (in $V \cdot m^{-1}$). The macroscopic effective conductivity $\sigma_{\text{eff}}$ is defined as the ensemble average of the instantaneous conductivity $\sigma$. Let $\sigma_0 = \frac{ne^2}{m\nu_0}$, where $n$ is the electron number density (in $m^{-3}$), $e$ is the elementary charge (in $C$), and $m$ is the electron mass (in $kg$). Which of the following expressions correctly gives the effective conductivity $\sigma_{\text{eff}}$ (up to order $\delta^2$) and the relative standard deviation of the conductivity $\frac{\Delta \sigma}{\sigma_{\text{eff}}}$ (to leading order in $\delta$)?

A non-linear electronic device $D$ has a voltage-current characteristic given by $I = k V^3$, where $k = 1.0 \times 10^{-3} \text{ A/V}^3$. The device is connected in series with a linear resistor $R$ and a DC voltage source of EMF $E = 10.0 \text{ V}$ (with negligible internal resistance). Under DC steady-state conditions, the voltage drop across the non-linear device is measured to be $V_D = 2.0 \text{ V}$. A small AC voltage signal $v(t) = v_0 \sin(\omega t)$ with amplitude $v_0 = 16.25 \text{ mV}$ is then added in series with the DC source. Assuming the AC signal is small enough that the dynamic resistance of the device remains constant, what is the amplitude of the resulting AC current in the circuit?

An n-type semiconductor bar is positioned along the $x$-axis. The donor impurity concentration is non-uniform, resulting in a steady-state electron concentration profile $n(x) = n_0 \left(1 + \frac{x}{L}\right)^{-2}$ for $x \ge 0$, where $n_0$ and $L$ are positive constants. The semiconductor is in thermal equilibrium at an absolute temperature $T$. Assuming the Einstein relation $D_e = \frac{\mu_e k_B T}{e}$ holds (where $D_e$ is the diffusion coefficient, $\mu_e$ is the electron mobility, $k_B$ is the Boltzmann constant, and $e$ is the elementary charge), what is the built-in potential difference $\Delta V = V(L) - V(0)$ between $x = L$ and $x = 0$?

A resistor has a resistance $R_0 = 120 \, \Omega$ at an ambient temperature $T_0 = 300 \, \text{K}$. Its temperature coefficient of resistance is $\alpha = 5 \times 10^{-3} \, \text{K}^{-1}$, such that its resistance varies with temperature $T$ as $R(T) = R_0[1 + \alpha(T - T_0)]$. The resistor is connected to a constant DC voltage source of $V = 60 \, \text{V}$. It dissipates heat to the surroundings according to Newton's law of cooling, with a thermal conductance $K = 0.2 \, \text{W/K}$ (i.e., the rate of heat loss is $K(T - T_0)$). What is the steady-state electrical power dissipated by the resistor?

An ideal battery of EMF $E = 12 \text{ V}$ is connected in series with a switch $S$ and a resistor $R_1 = 2 \text{ }\Omega$. The circuit then splits into two parallel branches that recombine and return to the battery. Branch 1 contains a resistor $R_2 = 4 \text{ }\Omega$ and an inductor $L = 4 \text{ H}$ in series. Branch 2 contains a single resistor $R_3 = 4 \text{ }\Omega$. The switch $S$ is closed at $t = 0$. At the specific instant when the total current drawn from the battery is $I = 2.5 \text{ A}$, what is the rate of change of current $\frac{di_1}{dt}$ in Branch 1?

A long coaxial cable consists of a solid inner cylindrical conductor of radius $a$ and a thick outer coaxial conducting cylindrical shell of inner radius $b$ and outer radius $c$. Both conductors are made of a non-magnetic metal with resistivity $\rho$ and permeability $\mu_0$. A high-frequency alternating current $I(t) = I_0 \cos(\omega t)$ flows down the inner conductor and returns via the outer shell. The angular frequency $\omega$ is sufficiently high such that the skin depth $\delta = \sqrt{\frac{2\rho}{\omega \mu_0}}$ is much smaller than $a$, $b$, and $(c-b)$. What is the time-averaged power dissipated as heat per unit length of the cable?

A spherical capacitor consists of two concentric conducting shells of inner radius $a$ and outer radius $b$. The space between the shells ($a < r < b$) is filled with a leaky, non-uniform dielectric material. The permittivity of the dielectric varies radially as $\epsilon(r) = \epsilon_1 \left(\frac{r}{a}\right)^2$, and its electrical conductivity varies as $\sigma(r) = \sigma_1 \left(\frac{a}{r}\right)^2$, where $\epsilon_1$ and $\sigma_1$ are positive constants. A steady DC voltage $V_0$ is applied across the capacitor, maintaining the inner shell at a higher potential than the outer shell. In the steady state, what is the total free charge accumulated strictly within the volume of the dielectric material (excluding the surfaces of the conducting shells)?

A cylindrical container of length $L$ (in m) and cross-sectional area $A$ (in m$^2$) is filled with a liquid metal. The uncompressed mass density of the liquid is $\rho_m$ (in kg/m$^3$), its bulk modulus is $K$ (in Pa), and its uncompressed electrical conductivity is $\sigma_0$ (in $\Omega^{-1}$m$^{-1}$). The container is accelerated uniformly with acceleration $a$ (in m/s$^2$) along its central axis (the $+z$ direction, from $z=0$ to $z=L$). A steady electrical current $I$ (in A) flows through the liquid metal in the $+z$ direction. Assume the mean free time $\tau$ (in s) of the conduction electrons remains constant, and the local conductivity is strictly proportional to the local mass density. The mass of an electron is $m_e$ (in kg) and its charge is $-e$ (in C). The pressure at the front face ($z=L$) is maintained at the uncompressed pressure $P_0$ (in Pa). Assuming $\rho_m a L \ll K$, calculate the total flux of the Poynting vector (in W) into the lateral surface of the liquid metal cylinder. Let $R_0 = \frac{L}{\sigma_0 A}$ be the unaccelerated resistance of the cylinder.

A uniform heavy wire of un-stretched length $L$, mass $M$, initial cross-sectional area $A_0$, and resistivity $\rho$ hangs vertically. It is initially supported along its entire length such that its strain is zero everywhere. The support is suddenly removed, allowing the wire to fall and stretch under its own weight. A constant direct current $I$ flows downwards through the wire. The wire is also immersed in an external radial magnetic field $\vec{B} = B_0 \hat{r}$, which exerts an azimuthal torque on the wire; assume the resulting torsional deformation is negligible and does not affect the wire's dimensions. Assuming the volume of each infinitesimal element of the wire remains constant during elastic deformation, and the material obeys Hooke's law with Young's modulus $Y$, what is the exact instantaneous electrical resistance of the wire at the instant of its maximum downward displacement?

A thin insulating disk of uniform mass density, total mass $M$, and radius $R$ lies in the $xy$-plane. It is rotating with a constant angular velocity $\vec{\omega} = \omega \hat{k}$ about its central axis (the $z$-axis). The disk has a non-uniform surface charge density given by $\sigma(r) = \sigma_0 \frac{r}{R}$, where $r$ is the radial distance from the center and $\sigma_0$ is a positive constant. A uniform magnetic field $\vec{B} = B_0 \hat{i}$ is applied across the region. Assume the setup is in a gravity-free space and the disk is pivoted at its center. The magnetic torque causes the rotation axis of the disk to undergo gyroscopic precession. Assuming the precession angular velocity $\Omega_p$ is much smaller than $\omega$, what is the magnitude of the precession angular velocity $\Omega_p$?

A perfectly insulated cylindrical conductor of length $L$ (m), cross-sectional area $A$ (m$^2$), and density $D$ (kg/m$^3$) is initially solid at a temperature $T_0$ (K). A constant total direct current $I$ (A) is passed through it. The material has a melting point $T_m$ (K) (where $T_m > T_0$), specific heat capacity $c$ (J/kg$\cdot$K) in the solid phase, and latent heat of fusion $L_f$ (J/kg). The electrical resistivities of the solid and liquid phases are $\rho_s$ ($\Omega\cdot$m) and $\rho_l$ ($\Omega\cdot$m) respectively, assumed independent of temperature. Assume infinite radial thermal conductivity (uniform temperature across any cross-section) and zero longitudinal heat conduction. As the conductor melts, the solid and liquid phases form parallel cylindrical channels extending the full length $L$. What is the total time $\tau$ (s) required for the conductor to completely melt?

A cylindrical metallic rod of initial length $L_0$, uniform cross-sectional area $A_0$, and initial electrical resistance $R_0$ is rigidly fixed between two unyielding massive walls. The material of the rod has a Young's modulus $Y$, Poisson's ratio $\nu$, coefficient of linear thermal expansion $\alpha$, and temperature coefficient of resistivity $\alpha_R$. The temperature of the rod is uniformly increased by a small amount $\Delta T$. Assuming the walls do not deform and neglecting any buckling of the rod, which of the following expressions correctly represents the fractional change in the electrical resistance of the rod, $\frac{\Delta R}{R_0}$, to the first order in $\Delta T$?

A thermistor is made of an intrinsic semiconductor whose resistance varies with absolute temperature $T$ as $R(T) = A \exp\left(\frac{E_g}{2k_B T}\right)$, where $E_g$ is the energy band gap, $k_B$ is the Boltzmann constant, and $A$ is a constant. The thermistor is connected in series with a temperature-independent resistor of resistance $R_0$ and an ideal battery of electromotive force $V$. At a specific temperature $T_0$, the potential difference across the thermistor is observed to be $\frac{V}{2}$. What is the rate of change of the potential difference across the thermistor with respect to temperature, $\frac{dV_{th}}{dT}$, at $T = T_0$?

A constant potential difference $V$ is applied across the ends of a long straight cylindrical solid wire of radius $R$ and length $L$. The resistivity of the material of the wire varies with the radial distance $r$ from its central axis as $\rho(r) = \rho_0 \frac{R}{r}$, where $\rho_0$ is a positive constant. What is the magnitude of the magnetic field $B$ at a radial distance $r = \frac{R}{2}$ from the axis of the wire?

A parallel plate capacitor has rectangular plates of length $L$, width $w$, and separation $d$. It is connected across an ideal DC voltage source of EMF $\varepsilon$. A dielectric slab of the same dimensions ($L \times w \times d$), with dielectric constant $K$ and finite resistivity $\rho$, initially fills the entire space between the plates. At time $t=0$, the slab starts being pulled out along its length at a constant velocity $v$. Edge effects are negligible, and the space vacated by the slab becomes vacuum (permittivity $\epsilon_0$). What is the total current $I$ drawn from the voltage source at the instant when a length $x$ of the slab remains inside the capacitor?

A carbon resistor has color bands: Red, Black, Red, Silver, indicating its resistance at a reference temperature $T_0 = 300 \, \text{K}$. It is connected across a parallel combination of two non-ideal cells. Cell 1 has an EMF $E_1 = 12 \, \text{V}$ and Cell 2 has an EMF $E_2 = 6 \, \text{V}$. At $T_0$, the internal resistances of both cells are $r_{01} = r_{02} = 1000 \, \Omega$.

According to Drude's model, the electrical resistance is inversely proportional to the relaxation time $\tau$ of the charge carriers. For the external resistor, $\tau$ varies with temperature $T$ as $\tau_R \propto T^{-2}$. For the internal resistance material of the cells, the relaxation time varies as $\tau_r \propto T^{-1}$.

If the entire circuit is operated at a steady temperature of $T = 600 \, \text{K}$, what is the steady current flowing through the external resistor? (Assume the dimensions of the resistors and the EMFs of the cells remain strictly independent of temperature).

Four identical cylindrical resistors are connected to form a Wheatstone bridge $ABCD$, with the resistors occupying arms $AB$, $BC$, $CD$, and $DA$. Each resistor has the color band sequence: Brown, Black, Brown, and Gold. A DC voltage source of $V = 50\text{ V}$ is connected across terminals $A$ and $C$ (current enters at $A$ and leaves at $C$). At a uniform ambient temperature of $20^\circ\text{C}$, the bridge is perfectly balanced.

The resistor in arm $AB$ is then uniformly heated to $120^\circ\text{C}$, while the other three resistors are maintained at $20^\circ\text{C}$. For the resistor material, the temperature coefficient of resistivity is $\alpha_\rho = 4.0 \times 10^{-3}\ ^\circ\text{C}^{-1}$ and the coefficient of linear thermal expansion is $\alpha_L = 1.0 \times 10^{-3}\ ^\circ\text{C}^{-1}$. Assuming isotropic thermal expansion and that the exact relations for temperature dependence apply (do not use small-parameter binomial approximations), calculate the magnitude of the open-circuit voltage across terminals $B$ and $D$. Assume nominal resistance values for all calculations.

Consider a Cartesian coordinate system where the $z$-axis is vertically upwards. A fixed, infinitely long straight wire lies along the $x$-axis and carries a constant current $I_1 = 5000 \text{ A}$ in the $+x$ direction. A straight conducting rod of mass $m = 10 \text{ g}$ and length $L = 1.0 \text{ m}$ is suspended by two massless, rigid, non-conducting rods of length $R = 0.2 \text{ m}$ from the $x$-axis. The rod is always parallel to the $x$-axis and is free to swing as a pendulum in the $yz$-plane. A uniform external magnetic field $\vec{B} = 1.0 \text{ T} \hat{k}$ is present in the region. The swinging rod is part of a closed electrical circuit containing a resistor, with the rest of the circuit arranged such that it does not experience any magnetic force or contribute to the motional EMF. The resistor has a color code: Brown, Black, Silver, Gold. At a certain instant, the pendulum passes through its lowest point (on the negative $z$-axis) with an angular velocity $\omega = 5 \text{ rad/s}$, moving in the $+y$ direction. Calculate the total tension $T$ in the two suspending rods at this instant. (Take $g = 10 \text{ m/s}^2$ and $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$. Ignore the self-inductance of the circuit.)

An ideal capacitor of capacitance $C = 40 \, \mu\text{F}$ is initially charged to a potential difference $V_0 = 100 \, \text{V}$. At $t = 0$, it is connected in series with an inductor $L = 1.0 \, \text{mH}$ and a resistor $R$. The resistor has color bands in the sequence: Blue, Black, Gold, Silver. Assuming the nominal value of the resistance, calculate the exact root-mean-square (RMS) current $I_{\text{rms}}$ in the circuit over the first complete period $T$ of its damped oscillation. (Given: The period $T$ is defined as the time taken for the oscillatory part of the current to complete one full cycle).

A cell of electromotive force $E$ and temperature-dependent internal resistance $r(T)$ is connected across a variable external resistor $R$. The internal resistance varies with the cell's temperature $T$ according to the relation $r(T) = \frac{r_0}{1 + \alpha(T - T_0)}$, where $r_0$ is the internal resistance at the constant ambient temperature $T_0$, and $\alpha$ is a positive constant. The cell loses heat to the surroundings at a rate $P_{loss} = k(T - T_0)$, where $k$ is the heat transfer coefficient. The external resistance $R$ is continuously adjusted so that the electrical power delivered to it is always maximum. If the cell reaches a steady-state temperature $T_{ss}$, what is the steady-state temperature difference $\Delta T = T_{ss} - T_0$? (Assume $4kr_0 > E^2 \alpha$)

A spherical battery consists of an inner conducting electrode of radius $a$ and an outer conducting electrode of radius $b$. The space between the electrodes ($a < r < b$) is filled with a leaky dielectric material characterized by a constant relaxation time $\tau$. The relative permittivity of the material varies radially as $\epsilon_r(r) = \frac{\lambda}{r^2}$, where $\lambda$ is a positive constant with units of $\text{m}^2$ and $r$ is the radial distance from the center. The battery has an ideal electromotive force $\mathcal{E}$ generated at the inner electrode-dielectric interface. If the battery is short-circuited by connecting the inner and outer electrodes with an ideal zero-resistance wire, what is the magnitude of the steady-state electric field $E(r)$ at a distance $r$ inside the dielectric?

A specialized liquid-electrolyte battery of EMF $E = 12 \text{ V}$ is connected across an external resistor $R = 4 \ \Omega$. Due to pressure-induced viscosity changes, the average ion mobility in the electrolyte is inversely proportional to the hydrostatic pressure difference $\Delta P$ between the bottom and top of the electrolyte column. Consequently, the internal resistance $r$ of this battery depends on $\Delta P$. When the battery is at rest on the ground, its internal resistance is $r_0 = 2 \ \Omega$. The entire setup is placed inside an elevator. If the elevator accelerates upwards with a constant acceleration $a = 5 \text{ m/s}^2$, calculate the terminal voltage across the battery. (Assume the density of the electrolyte and the height of the column remain constant, and take $g = 10 \text{ m/s}^2$).

A rigid rectangular wire loop is placed in a uniform magnetic field $\vec{B}$ perpendicular to its plane, pointing into the page. The loop is divided into two identical adjacent square loops, Loop 1 (left) and Loop 2 (right), by a central straight wire. The side length of each square is $a$ (in $\text{m}$). The magnetic field magnitude increases with time as $B(t) = \beta t$, where $\beta > 0$ is a constant in $\text{T/s}$. The left-most vertical branch has a resistance $R$ (in $\Omega$), and the right-most vertical branch has a resistance $2R$ (in $\Omega$). The top and bottom horizontal branches have negligible resistance. The central vertical branch contains an ideal inductor of inductance $L$ (in $\text{H}$) and has negligible resistance. Assume the self-inductance of the square loops is negligible compared to $L$. At $t=0$, the magnetic field starts increasing. Find the magnitude of the current flowing through the central branch as a function of time $t$ (in $\text{s}$).

A spherical capacitor consists of an inner conducting sphere of radius $a$ and an outer conducting shell of inner radius $c$. The space between them is filled with two concentric layers of weakly conducting solid dielectrics. The inner layer ($a < r < b$) has electrical conductivity $\sigma_1$ and permittivity $\epsilon_1$. The outer layer ($b < r < c$) has electrical conductivity $\sigma_2$ and permittivity $\epsilon_2$. A constant potential difference is maintained across the electrodes such that a steady radial current $I$ flows outward from the inner sphere to the outer shell. What is the total free charge $Q$ accumulated at the dielectric interface $r = b$ in the steady state?

An ideal diode and a sinusoidal AC voltage source are connected in series to a junction $J$. From junction $J$, two parallel branches connect to the ground, containing ideal resistors $R_1 = 30 \ \Omega$ and $R_2 = 60 \ \Omega$ respectively. A moving-coil ammeter (which measures the average current) is connected in the main branch just before junction $J$ and reads a steady value of $2.0 \text{ A}$. Assuming the diode operates as a perfect half-wave rectifier, what is the total average electrical power dissipated in the two resistors?

A resistor of initial resistance $R_0$ at temperature $T_0$ is connected across an ideal DC voltage source of EMF $V$. The resistor is thermally insulated from its surroundings and has a constant heat capacity $C$. Its resistance varies with temperature as $R(T) = R_0[1 + \alpha(T - T_0)]$, where $\alpha$ is the temperature coefficient of resistance. Find the time $t$ required for the current in the circuit to drop to exactly half of its initial value.

Two identical coils, each having self-inductance $L$ and resistance $R$, are placed near each other such that their mutual inductance is $M$ ($M > 0$). At $t=0$, a constant DC voltage source of EMF $V_0$ is connected across the first coil, while the second coil is maintained as a closed short-circuited loop. Which of the following expressions gives the time $t_m$ at which the induced current in the second coil reaches its maximum magnitude?

A stationary circular loop of radius $a$ is formed by joining two semicircular conducting wires of resistances $R_1$ and $R_2$. The loop is placed in a uniform magnetic field $\vec{B}$ perpendicular to its plane. The magnitude of the magnetic field varies with time as $B(t) = \alpha t$, where $\alpha$ is a positive constant. An ideal voltmeter is connected between the two junction points $P$ and $Q$ of the semicircles. If the connecting wires of the voltmeter lie exactly along the straight diameter joining $P$ and $Q$, what is the magnitude of the reading of the voltmeter? (Assume all given parameters are in SI units, and the options represent the reading in Volts).

A modified metre bridge is constructed with a uniform wire AB of length $L = 100 \text{ cm}$ and total resistance $5 R_0$. Two resistors $R_1 = 4 R_0$ and $R_2 = R_0$ are connected in the left gap (between A and D) and right gap (between D and B) respectively. An ideal battery of emf $E$ is connected across the ends A and B (with the positive terminal at A). A non-ideal galvanometer is connected between the junction D and a jockey J that slides on the wire. The galvanometer has a resistance $R_g = 0.4 R_0$ and a mechanical zero error, such that it shows zero deflection only when a constant current $i_0 = \frac{E}{40 R_0}$ flows through it from D to J. Due to the mechanical design of the track, the jockey exerts a variable pressure on the wire, creating a position-dependent contact resistance $R_c$ in series with the galvanometer. For a jockey position $x$ (measured from A in cm), this contact resistance is given by $R_c(x) = R_0 \left[ 1.05 + 13 \left( \frac{x - 80}{100} \right) \right]$. Find the apparent null point position $x$ where the galvanometer shows zero deflection.

An AC Wheatstone bridge ABCD is connected to an ideal AC voltage source $V(t) = V_0 \sin(\omega t)$ across nodes A and C. The arms of the bridge contain the following components: Arm AB consists of a resistor $R$ in series with a capacitor $C$. Arm BC consists of a resistor $R$ in parallel with a capacitor $C$. Arm AD is a pure resistor $R_1$, and Arm DC is a pure resistor $R_2$. An ideal AC voltmeter is connected between nodes B and D. It is observed that the RMS reading of the voltmeter remains strictly constant and non-zero for all possible values of the source angular frequency $\omega \in (0, \infty)$. What is the ratio $\frac{R_1}{R_2}$?

A straight conducting rod of length $L$ has the shape of a truncated cone with circular end faces of radii $a$ and $b$ ($b > a$). The rod is made of an isotropic material of uniform electrical resistivity $\rho$. A constant potential difference $V$ is applied across its end faces. Assuming that the current density vector is uniform across any cross-section and parallel to the axis of the rod, which of the following expressions gives the magnitude of the electric field at the exact midpoint of the rod's axis?

A dielectric medium contains $n$ polarizable molecules per unit volume, each modeled as a pair of bound charges $+q$ and $-q$. An alternating electric field $E(t) = E_0 \cos(\omega t)$ is applied to the medium. Due to internal viscous drag opposing the rotation and stretching of the molecular dipoles, the macroscopic polarization $P(t)$ lags the electric field by a phase angle $\delta$, such that $P(t) = P_0 \cos(\omega t - \delta)$. Which of the following expressions correctly gives the average thermal power dissipated per unit volume $\langle P_{th} \rangle$ due to the work done against the internal torque, and the maximum relative drift velocity $v_{d0}$ between the bound charges of a single molecule?

A circuit consists of a battery of constant EMF $\mathcal{E}$, an ideal inductor of inductance $L$, and a resistor of resistance $R$ connected in series. The resistor is made of a uniform wire of cross-sectional area $A$ and free electron density $n$. The switch is closed at $t=0$. The electric field signal propagates through the circuit almost instantaneously, but the macroscopic current is governed by the circuit's inductance. Assuming the drift velocity of electrons is directly proportional to the instantaneous current, what is the net displacement of an average free electron in the resistor during the time interval from $t=0$ to $t = L/R$? (Here $q$ is the elementary charge and $e$ is the base of the natural logarithm).

Consider a classical electron gas of mass $m$ and charge $-e$ at temperature $T$ in a uniform electric field $E$. In the microscopic Drude-Lorentz model, the drift velocity of an electron with thermal speed $v$ is given by $v_d(v) = \frac{eE}{m}\tau(v)$, where $\tau(v)$ is the relaxation time. Assume the electrons scatter off neutral impurities such that their mean free path $\lambda$ is constant and independent of $v$. The macroscopic drift velocity $\langle v_d \rangle$ is defined as the average of $v_d(v)$ over the 3D Maxwell-Boltzmann speed distribution $f(v)$. Which of the following expressions gives the correct macroscopic drift velocity? (Here $k_B$ is the Boltzmann constant).

An electron of mass $m$ and charge $-e$ is injected at $x=0$ with zero initial velocity into a semiconductor region of length $L$. A non-uniform electric field $\vec{E} = -E_0 \left(1 - \frac{x}{L}\right) \hat{i}$ is applied across the region ($0 \le x \le L$). Due to collisions with the lattice, the electron experiences a drag force. In the 'hot electron' regime, the mean free path $\lambda$ is constant, and the momentum relaxation time is given by $\tau = \frac{\lambda}{v}$, where $v$ is the instantaneous speed of the electron. Which of the following expressions gives the square of the electron's velocity, $v^2$, at $x=L$?

An n-type semiconductor at a constant temperature $T$ has a non-uniform donor impurity profile. Assuming complete ionization, the electron concentration is given by $n(x) = n_0 e^{-x/L}$ for $x \ge 0$, where $n_0$ and $L$ are positive constants. The electron mobility is dominated by ionized impurity scattering, such that the momentum relaxation time $\tau(x)$ is inversely proportional to the local impurity concentration. At $x = 0$, the electron diffusion coefficient is $D_0$. Which of the following expressions gives the magnitude of the electron diffusion current density $J_{\text{diff}}$ at $x = 2L$? (Let $e$ be the elementary charge).

A magnetic Wheatstone bridge is constructed to measure small fluctuations in magnetomotive force. Three of the arms are made of a linear magnetic material, each having a constant magnetic reluctance $\mathcal{R}_0$. The fourth arm is a non-linear ferromagnetic core of length $l$ and uniform cross-sectional area $A$. The material of the fourth arm follows the Fröhlich-Kennelly relation $B = \frac{\alpha H}{1 + \beta H}$, where $B$ is the magnetic flux density, $H$ is the magnetic field intensity, and $\alpha$ and $\beta$ are positive constants. The fourth arm is biased by a steady DC magnetic field intensity $H_0$. If the bridge is perfectly balanced for an infinitesimally small superimposed AC magnetomotive force, what is the required value of $\mathcal{R}_0$?

Consider a Wheatstone bridge circuit powered by an ideal DC voltage source $V_0$. The left branch consists of a temperature-independent resistor $R$ in series with a p-n junction diode $D$ (connected to ground). The right branch consists of two temperature-independent resistors, $R$ and $R_v$. The output voltage is measured between the midpoints: $V_{out} = V_{right} - V_{left}$, where $V_{left}$ is the voltage across the diode.

The diode operates in the forward-bias region with current $I = I_s(T) e^{qV_D/kT}$, where $V_D$ is the diode voltage, $q$ is the elementary charge, and $k$ is the Boltzmann constant. The reverse saturation current depends on absolute temperature $T$ as $I_s(T) = C T^3 e^{-E_g/kT}$, where $C$ is a material constant and $E_g$ is the bandgap energy.

At a specific temperature $T$, the bridge is balanced ($V_{out} = 0$) by adjusting $R_v$. At this operating point, the diode has a dynamic resistance $r_d = \frac{kT}{qI}$. Determine the temperature sensitivity of the bridge, defined as $S = \frac{dV_{out}}{dT}$, at this balanced state.

A potentiometer wire of initial length $L_0 = 100\text{ cm}$ and resistance $R_{w0} = 100\ \Omega$ is rigidly clamped at both ends so that its length cannot change. It is connected in series with a resistance $R_s = 105\ \Omega$ and a primary cell of EMF $E_0 = 10\text{ V}$ (with negligible internal resistance). A secondary cell of EMF $E_1 = 2\text{ V}$ is balanced on this wire. The temperature of the potentiometer wire is then uniformly increased by $\Delta T = 50^\circ\text{C}$. Given the following properties for the wire material: temperature coefficient of resistivity $\gamma = 1.6 \times 10^{-3}\ ^\circ\text{C}^{-1}$, coefficient of linear thermal expansion $\alpha = 2.0 \times 10^{-4}\ ^\circ\text{C}^{-1}$, and Poisson's ratio $\nu = 0.5$. Assuming the clamps and the series resistor $R_s$ do not change temperature, what is the magnitude of the shift in the null point?

A potentiometer wire $AB$ of length $L = 10 \text{ m}$ and resistance $R_w = 10 \ \Omega$ is connected in a primary circuit with a DC battery of EMF $V_0 = 20 \text{ V}$ and an external resistance $R_0 = 10 \ \Omega$. The primary circuit also contains an ideal coil (zero resistance and self-inductance) of $N = 100$ turns and area $A = 0.1 \text{ m}^2$. A uniform magnetic field perpendicular to the coil's plane varies with time as $B(t) = \alpha t^2$, where $\alpha = 0.5 \text{ T s}^{-2}$. By Lenz's law, the induced EMF in the coil opposes the battery $V_0$.

A secondary cell of constant EMF $E_s = 2 \text{ V}$ is connected to find the null point $x(t)$ from end $A$. Due to electromagnetic interference, a time-varying zero error $e(t)$ appears at end $A$, such that the actual potential difference balanced by the secondary cell is $V = \lambda(t) x(t) + e(t)$, where $\lambda(t)$ is the instantaneous potential gradient of the wire. If $e(t) = \gamma |\varepsilon_{ind}|$, where $\varepsilon_{ind}$ is the induced EMF in the coil and $\gamma = 0.05$, find the velocity of the null point $\frac{dx}{dt}$ at $t = 1 \text{ s}$.

A semiconductor wire of length $L$ is placed along the $x$-axis from $x=0$ to $x=L$. The wire is maintained at a constant temperature $T$. Due to a specific doping profile, the free electron concentration in the wire varies as $n(x) = n_0 e^{-\alpha x^2}$, where $n_0$ and $\alpha$ are positive constants. No external voltage source is connected across the wire.

A student uses this wire to find the null point for an ideal cell of electromotive force $\varepsilon$. The positive terminal of the cell is connected to the end $x=0$, and the negative terminal is connected to a jockey that can slide along the wire. Assuming the Einstein relation $D = \frac{kT}{e}\mu$ holds (where $D$ is the diffusion coefficient, $\mu$ is the mobility, $k$ is the Boltzmann constant, and $e$ is the elementary charge), at what distance $l$ from the end $x=0$ will a galvanometer connected in series with the cell show zero deflection?

Three concentric thin conducting spherical shells of radii $a$, $2a$, and $3a$ are arranged in a vacuum. The innermost shell is grounded. The middle shell is given a charge $+Q$ and the outermost shell is given a charge $-Q$. The potential difference between the middle and outermost shells is connected across a secondary circuit to find its null point on a potentiometer. The potentiometer consists of a uniform wire of length $L$ and resistance $R$, driven by a primary ideal battery of EMF $E_0$ and a series resistance of $R/2$. If the null point is obtained at a distance $x$ from the high-potential end of the potentiometer wire, what is the value of $x$? (Assume the secondary circuit draws no current at the null point and $x < L$).

A small coil of $N=50$ turns, area $A=200 \text{ cm}^2$, and internal resistance $r=2 \text{ }\Omega$ moves with a constant velocity $\vec{v} = 5 \hat{i} \text{ m/s}$ along the x-axis. The coil is in a region of non-uniform magnetic field $\vec{B} = \beta x^2 \hat{k}$, where $\beta = 2 \text{ T/m}^2$ and $\hat{k}$ is the unit vector pointing out of the page. Assume the coil dimensions are small enough that the magnetic field is uniform over its area at any instant. The coil is connected in series with a variable external resistor $R$. The potential difference across $R$ is fed to a potentiometer circuit. The potentiometer consists of a wire of length $L=1 \text{ m}$ connected across an ideal $10 \text{ V}$ DC source. The connections to the potentiometer are made such that a null point can only be obtained if the induced current in the moving coil is in the clockwise direction. If $R$ is set to extract maximum electrical power from the coil, at what position $x$ of the coil will a null point be observed on the potentiometer wire at a balancing length of $l = 50 \text{ cm}$?

A potentiometer wire of initial physical length $L_0$, uniform cross-sectional area $A_0$, and resistivity $\rho_0$ at ambient temperature $T_0$ is connected in series with an ideal constant current source $I$. A standard cell of EMF $E_s$ is balanced across a physical length $l_0$ of the wire at $T_0$. Due to Joule heating, the entire wire reaches a uniform steady-state temperature $T = T_0 + \Delta T$. The wire material is isotropic with a linear coefficient of thermal expansion $\alpha$ and a temperature coefficient of resistivity $\alpha_r$. Assuming $\alpha \Delta T \ll 1$ and $\alpha_r \Delta T \ll 1$, what is the new physical balancing length $l$ for the same standard cell at the steady-state temperature $T$?

Consider a rectangular primary loop $ACDB$ in the $xy$-plane with vertices $A(0,0)$, $B(L,0)$, $D(L,w)$, and $C(0,w)$. The side $AB$ is a uniform potentiometer wire of length $L$ and resistance $R$. The other sides are made of ideal resistanceless wires. An ideal primary cell of EMF $V_0$ is connected in the side $CD$ such that it drives a steady current in the counter-clockwise direction. A uniform magnetic field $\vec{B} = \alpha t \hat{k}$ exists everywhere, where $\hat{k}$ is the unit vector in the $+z$ direction (out of the page) and $\alpha > 0$ is a constant. A secondary circuit consists of an ideal cell of EMF $E_s$ and a galvanometer. It connects point $A$ to a movable jockey $J$ at position $(x,0)$ on the wire $AB$. The connecting wires of the secondary circuit form a smaller rectangular loop $A J J' C'$ with vertices $A(0,0)$, $J(x,0)$, $J'(x, w/2)$, and $C'(0, w/2)$. The cell $E_s$ and the galvanometer are located in the branch $C'A$, with the positive terminal of $E_s$ connected to $A$. Given $V_0 > \alpha L w$, find the balancing length $x$ for which the galvanometer shows zero deflection.