Alternating Current - Class 12

An ideal $L-C$ circuit consists of a capacitor of capacitance $C$ and a solenoid of initial inductance $L_0$. The solenoid is tightly wound on a solid cylindrical non-magnetic rod having a coefficient of linear thermal expansion $\alpha$. The circuit is set into electromagnetic oscillations with an initial peak current $I_0$. The rod is then heated very slowly (adiabatically with respect to the oscillation period) such that its temperature increases by $\Delta T$. Assuming the total number of turns of the solenoid remains constant and the rod expands isotropically, what is the new peak current $I$ in the circuit?

A series $R-L-C$ circuit has a resistance $R = 12\ \Omega$. At an angular frequency $\omega_0$, the inductive reactance is $X_L = \frac{41}{3}\ \Omega$ and the capacitive reactance is $X_C = \frac{68}{3}\ \Omega$. The circuit is driven by a composite voltage source given by $V(t) = V_0 \sin(\omega_0 t) + V_0 \sin(2\omega_0 t)$. Calculate the effective power factor of the circuit, defined as $pf = \frac{P_{avg}}{V_{rms} I_{rms}}$, where $P_{avg}$ is the average power dissipated, and $V_{rms}$ and $I_{rms}$ are the root-mean-square voltage and current of the source.

An AC voltage source of RMS value $V$ and angular frequency $\omega$ is connected in series with an ideal inductor of inductance $L$ and a resistor. At the ambient temperature $T_0$, the resistor has a resistance $R_0$ (where $R_0 < \omega L$). The resistance varies with temperature $T$ as $R(T) = R_0[1 + \alpha(T - T_0)]$, where $\alpha > 0$ is the temperature coefficient of resistance. The resistor dissipates heat to the surroundings according to Newton's law of cooling with a cooling constant $K$. If the circuit reaches a steady thermal state where the power factor of the circuit is exactly $\frac{1}{\sqrt{2}}$, what is the required RMS voltage $V$ of the source?

A voltage source $V(t) = V_0 (1 + \cos \omega t)$ is connected across a parallel combination of an ideal capacitor of capacitance $C$ and a real inductor having inductance $L$ and resistance $R$. The circuit has been operating for a long time such that steady-state conditions are established. Given that $\omega L = R$ and $\omega C = 1/R$, find the total heat dissipated in the circuit during the time interval from $t=0$ to $t=\pi/\omega$.

A uniform plane electromagnetic wave of angular frequency $\omega$ propagates along the $+z$-direction in a non-magnetic ($\mu = \mu_0$) lossy dielectric medium. The medium is characterized by a complex permittivity $\epsilon^* = \epsilon_0 (4 - 3j)$, where $j = \sqrt{-1}$. In electromagnetic theory, the complex Poynting vector is defined as $\vec{S} = \frac{1}{2} \vec{E} \times \vec{H}^*$, where $\vec{E}$ and $\vec{H}$ are the complex phasor representations of the electric and magnetic fields. The real part of $\vec{S}$ represents the time-averaged active (real) power density, while its imaginary part represents the reactive (wattless) power density. What is the ratio of the magnitude of the reactive power density to the time-averaged active power density at any point in the medium?

A uniform, purely resistive metallic rod of Young's modulus $Y$ (in $\text{N/m}^2$) and coefficient of linear thermal expansion $\alpha$ (in $\text{K}^{-1}$) is rigidly clamped between two fixed, perfectly insulating walls at an initial temperature $T_0$. The rod has a temperature-dependent electrical resistance $R(T) = R_0[1 + \beta(T - T_0)]$, where $R_0$ is the resistance at $T_0$ (in $\Omega$) and $\beta$ is the temperature coefficient of resistivity (in $\text{K}^{-1}$). The rod is connected in parallel with a pure inductor, and this parallel combination is driven by an AC source.

In the steady state, the rod reaches a constant temperature $T > T_0$. At this steady state, the total RMS current drawn from the AC source is $I_0$ (in $\text{A}$), and the overall power factor of the parallel circuit is $\cos\phi$. The rod loses heat to the surroundings (which remain at $T_0$) at a rate $K(T - T_0)$, where $K$ is a positive constant (in $\text{W/K}$).

Assuming the walls do not yield and the rod does not buckle, which of the following expressions gives the compressive thermal stress $\sigma$ (in $\text{N/m}^2$) developed in the rod in the steady state?

A mechanical oscillator consists of a block of mass $m = 3 \text{ kg}$ attached to a spring of stiffness $k = 4 \text{ N/m}$ and a damper with damping coefficient $b = 3 \text{ N s/m}$ on a smooth horizontal surface. It is driven by a horizontal force $F(t) = 25 \sin(2t) \text{ N}$ (where $t$ is in seconds) and reaches a steady state. The instantaneous power $P(t)$ delivered by the driving force can be uniquely decomposed as $P(t) = P_{\text{active}}(t) + P_{\text{reactive}}(t)$, where $P_{\text{active}}(t)$ has a non-zero time average and $P_{\text{reactive}}(t)$ is a purely sinusoidal function with a zero time average. At the first instant $t > 0$ when the magnitude of the reactive power is maximum, a lump of putty of mass $M = 5 \text{ kg}$ is dropped vertically onto the block and sticks to it. What is the loss in the total mechanical energy of the oscillator during this collision?

An AC voltage source $v(t) = 200 \sin(\omega t)\text{ V}$ is connected in series with a resistor $R = 100\ \Omega$ and a symmetric Zener clipper. The clipper consists of two identical ideal Zener diodes connected back-to-back. Each Zener diode has a breakdown voltage of $100\text{ V}$ and a forward voltage drop of $0\text{ V}$. What is the average power supplied by the AC source to the circuit?

An isolated LCR circuit consists of an inductor of inductance $L$, a capacitor of capacitance $C$, and a resistor of mass $m$, specific heat capacity $s$, and initial resistance $R_0$ at ambient temperature $T_0$. The resistance varies with temperature $T$ as $R(T) = R_0 [1 + \alpha (T - T_0)]$, where $\alpha$ is the temperature coefficient of resistance. At $t=0$, the capacitor is charged to a voltage $V_0$ (giving an initial total energy $E_0 = \frac{1}{2} C V_0^2$) and the circuit is closed. The resistor is perfectly thermally insulated from its surroundings. Assume the circuit is highly underdamped ($Q \gg 1$) so that the fractional energy loss per cycle is very small, allowing the cycle-averaged power dissipation to be approximated as $\langle P \rangle = \frac{E}{L} R(T)$, where $E$ is the instantaneous total electrical energy of the circuit. Which of the following expressions gives the time $t$ at which the total electrical energy of the circuit reduces to $E_0 / 2$? (Let $\gamma = \frac{\alpha E_0}{ms}$)

A uniform conducting rod of mass $m$ and length $l$ is pivoted at one of its ends and can rotate freely in a horizontal plane. A uniform vertical magnetic field $B$ exists in the region. The pivot and the free end of the rod are connected via frictionless sliding contacts to a stationary circuit consisting of an inductor of inductance $L$, a capacitor of capacitance $C$, and a resistor of resistance $R$ in series. Initially, the rod is at rest and the capacitor is uncharged. At $t = 0$, an impulsive torque $\int \tau dt = J$ is applied to the rod. Assuming the resistance of the rod and the contacts is negligible, what is the Quality Factor ($Q$) of the subsequent transient electrical oscillations in the circuit?

A series RLC circuit is driven by an AC voltage source $V(t) = V_0 \sin(\omega t)$. The circuit consists of a capacitor $C$, an ideal resistor $R_0$, and an inductor with self-inductance $L$. The inductor is formed by winding $N$ turns of wire around a toroidal magnetic core of mean radius $r$, cross-sectional area $A$, and relative permeability $\mu_r$. The core material exhibits magnetic hysteresis, and the energy dissipated per unit volume per cycle is given by $w_h = \beta B_m^2$, where $B_m$ is the peak magnetic field in the core and $\beta$ is a material constant. Assuming the magnetic field is uniform across the core's cross-section and ignoring eddy current losses, which of the following expressions represents the quality factor $Q$ of the circuit at its resonant frequency $\omega_0$?

A block of mass $m = 0.8 \text{ kg}$ is attached to a spring of force constant $k = 80 \text{ N/m}$ and undergoes damped oscillations on a smooth horizontal surface. The damping force is non-linear and given by $F_d = -c v |v|$, where $v$ is the velocity of the block and $c = 0.01 \text{ kg/m}$. At $t = 0$, the amplitude of oscillation is $A_0 = 0.3 \text{ m}$. Assuming the damping is weak enough that the motion remains approximately sinusoidal over any single cycle, find the approximate number of complete oscillations $N$ after which the amplitude reduces to $A_N = 0.15 \text{ m}$.

A parallel plate capacitor consists of two rectangular plates of length $l_0$ and width $w$, separated by a distance $d$. A solid dielectric slab of the same dimensions ($l_0 \times w \times d$) and dielectric constant $K$ is inserted a distance $x$ ($0 < x < l_0$) between the plates. The capacitor is connected in series with an inductor of inductance $L$, a resistor of resistance $R$, and an AC voltage source of RMS value $V_{\text{rms}}$ and variable angular frequency $\omega$. For any given position $x$ of the slab, the AC source is tuned to the instantaneous resonant frequency $\omega_0$ of the circuit. Assuming fringing fields are negligible, which of the following expressions correctly gives the time-averaged electrostatic force $\langle F \rangle$ pulling the slab into the capacitor, and the half-power bandwidth $\Delta\omega$ of the circuit at this position $x$? (All quantities are in SI units)

A long single-layer solenoid is used in a high-frequency circuit where its continuous charge distribution cannot be ignored. It is modeled as an ideal lossless distributed-parameter transmission line of length $l$, with a total low-frequency inductance $L_0$ and a total distributed capacitance to ground $C_0$. One end of the solenoid is directly grounded. An ideal AC voltage source of variable angular frequency $\omega$ is connected between the other end of the solenoid and ground. What is the lowest non-zero angular frequency $\omega_1$ at which the amplitude of the current drawn from the AC source becomes zero?

An ideal capacitor of capacitance $C$ is initially given a charge $Q_0$. At time $t=0$, it is connected in a closed series circuit with a real coil having inductance $L$ and internal resistance $r$. The resistance is small enough ($r \ll 2\sqrt{\frac{L}{C}}$) that the circuit undergoes weakly damped oscillations. The total electromagnetic energy of the circuit decays exponentially over time. What is the total electrical energy dissipated as heat in the coil's resistance from $t=0$ to $t = \frac{L}{r} \ln 4$?

An ideal LC circuit consists of an inductor and a parallel-plate capacitor with circular plates of radius $R$ and separation $d$. The circuit oscillates with an angular frequency $\omega$. The current in the circuit is given by $i(t) = I_0 \cos(\omega t)$, where $t$ is time. Neglecting edge effects, what is the magnitude of the Poynting vector at the radial edge ($r=R$) of the region between the capacitor plates at time $t = \frac{\pi}{6\omega}$? (Here $\epsilon_0$ is the vacuum permittivity)

An ideal inductor of inductance $L$ is connected in series with a non-linear capacitor. The voltage $V$ across the capacitor depends on the charge $Q$ on its plates as $V = kQ^3$, where $k$ is a positive constant. The circuit has no resistance. At $t=0$, the current in the circuit is zero and the charge on the capacitor is $Q_0$. The current $I$ and charge $Q$ form a closed trajectory in the phase space. Given the definite integral $\int_{0}^{1} \frac{dx}{\sqrt{1-x^4}} = c$, which of the following expressions correctly represents the time period $T$ of the oscillations in the circuit? (Assume all quantities are in SI units)

An ideal LC circuit consists of a capacitor of capacitance $C = 6 \text{ mF}$ and a non-linear inductor. The capacitor is initially charged to a voltage $V_0 = 6 \text{ V}$, and the initial current in the circuit is zero. Due to magnetic core saturation, the total magnetic flux linkage $\Phi$ of the inductor varies non-linearly with the current $I$ as $\Phi(I) = \alpha I + \beta I^3$, where $\alpha = 30 \times 10^{-3} \text{ H}$ and $\beta = 4 \times 10^{-3} \text{ H/A}^2$. Assuming no resistance in the circuit, what is the maximum current $I_{max}$ that will flow through the inductor?

A series LCR circuit with resistance $R$ (in $\Omega$), inductance $L$ (in $\text{H}$), and capacitance $C$ (in $\text{F}$) is driven by an AC voltage source $V(t) = V_0 \sin(\omega t)$ (in $\text{V}$). The circuit operates in a steady state with an impedance $Z$ (in $\Omega$) and a phase angle $\phi$ (in $\text{rad}$) between the voltage and the current. Let $P_S(t)$ be the instantaneous power delivered by the source, and $P_R(t)$ be the instantaneous power dissipated in the resistor. If $\Delta P_{max}$ is the maximum value of the difference $P_S(t) - P_R(t)$ during one complete cycle, which of the following expressions correctly represents $\Delta P_{max}$?

A long straight wire carries an alternating current $I(t) = I_0 \sin(\omega t)$, where $I_0$ is the current amplitude in Amperes and $\omega$ is the angular frequency in rad/s. A rigid rectangular conducting loop of side lengths $a$ and $b$ (in meters) is placed in the same plane as the wire. The side of length $b$ is parallel to the wire and is at a distance $d$ (in meters) from it. The loop has a constant self-inductance $L$ (in Henries). Due to high-frequency skin effects, the effective resistance of the loop is strictly proportional to the angular frequency, given by $R = \alpha \omega$, where $\alpha$ is a positive constant in Henries. Let $M = \frac{\mu_0 b}{2\pi} \ln\left(1 + \frac{a}{d}\right)$ be the mutual inductance (in Henries) between the wire and the loop. What is the total electrical energy $E$ dissipated as heat in the loop during exactly one complete cycle of the alternating current?

A cylindrical region of space $x^2 + y^2 < R_s^2$ contains a uniform time-varying magnetic field $\vec{B} = \alpha t \hat{k}$ (where $\alpha > 0$ is a constant). Outside this region, the magnetic field is zero. A non-conducting solid sphere of radius $R$ (where $R \ll R_s$) has a uniform volume charge density and a total charge $Q$. The sphere is centered at coordinates $(d, 0, 0)$ such that $d + R < R_s$. Find the interaction electric energy $U_{int} = \int \epsilon_0 (\vec{E}_{stat} \cdot \vec{E}_{ind}) dV$ stored within the hemisphere $y > 0$ of the charged sphere, where $\vec{E}_{stat}$ is the static electric field of the sphere and $\vec{E}_{ind}$ is the induced electric field.

A real transformer is modeled as an ideal transformer with a primary-to-secondary turns ratio of $N_1 : N_2 = 1 : 2$, in series with a secondary leakage inductance $L_l = 20 \text{ mH}$. The primary winding is driven by an AC voltage source $V_1(t) = 100 \sin(\omega t) \text{ V}$. The secondary terminals of the transformer are connected to a load consisting of a resistor $R = 50 \ \Omega$ and a capacitor $C = 50 \ \mu\text{F}$ in series. The angular frequency $\omega$ of the source is adjusted such that the secondary current $I_2(t)$ is exactly $180^\circ$ out of phase with the primary voltage $V_1(t)$. Calculate the peak voltage across the secondary terminals of the transformer at this frequency.

A small magnetic dipole of constant magnitude $\mu$ and mass $m$ is constrained to move along the $z$-axis in a viscous medium with a linear drag coefficient $b$. An external mechanism forces the dipole to rotate uniformly in the $x$-$z$ plane such that its magnetic moment vector is given by $\vec{\mu}(t) = \mu \sin(\omega t) \hat{i} + \mu \cos(\omega t) \hat{k}$. The system is permeated by an AC magnetic field with a spatial gradient, given by $\vec{B}(x, z, t) = (-\beta x \hat{i} + \beta z \hat{k}) \cos(\omega t)$, where $\beta$ is a positive constant. Calculate the average power delivered by the magnetic force (given by $\vec{F} = \nabla(\vec{\mu} \cdot \vec{B})$) to the dipole in the steady state.

A parallel plate capacitor consists of two rectangular plates of length $L$, width $w$, and separation $d$. A dielectric slab of identical dimensions is partially inserted between the plates to a depth $x$. The dielectric constant of the slab varies along its length as $\kappa(y) = 1 + \alpha y$, where $y$ is the distance measured from its leading edge (the edge deepest inside the capacitor). An alternating voltage $V(t) = V_0 \sin(\omega t)$ is applied across the capacitor plates.

An external mechanical agent forces the slab to oscillate such that its insertion depth is given by $x(t) = x_0 + A \sin(2\omega t)$, where $A < x_0$ and $x_0 + A < L$. During its motion, the slab experiences a viscous drag force $F_{\text{drag}} = -b v$, where $v$ is its instantaneous velocity and $b$ is a positive constant. Assuming edge effects are negligible, which of the following expressions represents the average mechanical power $\langle P_{\text{ext}} \rangle$ that the external agent must supply to maintain this steady oscillation over one complete cycle?

A series circuit consists of an ideal capacitor of capacitance $C$, an inductor of inductance $L$, and a radiation resistance $R_{rad}$ (accounting for electromagnetic energy radiated by the system). The circuit is driven by an alternating voltage source $V(t) = V_0 \sin(\omega t)$. The driving frequency is set to $\omega = 2\omega_0$, where $\omega_0 = \frac{1}{\sqrt{LC}}$ is the natural undamped frequency of the $LC$ combination. At this driving frequency, the steady-state current lags the source voltage by a phase angle of $\frac{\pi}{3}$ rad. If the peak voltage across the capacitor is measured to be $V_C$, what is the average power $P_{avg}$ dissipated as radiation?

A series LCR circuit consists of a resistor $R = 12 \, \Omega$, an inductor $L = 90 \, \text{mH}$, and a capacitor $C = 2.5 \, \text{mF}$. The circuit is driven by an alternating voltage source given by $v(t) = 260 \sin(100 t) + 100 \sin(200 t)$, where $v(t)$ is in volts and $t$ is in seconds. Calculate the total average power dissipated in the circuit.

A parallel plate capacitor has a plate area $A$ and an initial plate separation $d_0$ at $t=0$. An external agent pulls one of the plates outwards at a constant velocity $v$, such that the separation becomes $x(t) = d_0 + vt$. A time-varying voltage source $V(t) = \alpha t$ (where $\alpha$ is a positive constant) is connected across the capacitor plates. What is the average electrical power delivered by the voltage source during the time interval from $t=0$ to $t=d_0/v$? (Assume ideal conditions and let $\epsilon_0$ be the permittivity of free space).

A toroidal solenoid of rectangular cross-section has an inner radius $a$, an outer radius $b$, and consists of $N$ tightly wound turns carrying a steady current $I$. A particle of mass $m$ and positive charge $q$ is projected from the inner edge ($r=a$) radially outwards with an initial speed $v_0$ in the mid-plane of the toroid. Assume the axial height of the toroid is sufficiently large so the particle does not strike the top or bottom walls, and the magnetic field outside the toroid is zero. Which of the following expressions gives the minimum initial speed $v_0$ required for the particle to just reach the outer edge ($r=b$)?

A bimetallic strip with an effective elastic spring constant $k$ acts as a movable armature for an electromagnet. The electromagnet has $N$ turns, a cross-sectional area $A$, and carries a constant DC current $I$. Due to a constant thermal power input, the unconstrained equilibrium position of the strip (the position it would take if $I=0$) moves towards the electromagnet at a constant speed $u$. Let $x$ be the instantaneous air gap between the strip and the electromagnet. Assuming the motion is quasi-static (the strip is always in mechanical equilibrium), what is the rate of change of the magnetic flux linkage $\frac{d\lambda}{dt}$ of the electromagnet at the instant the air gap is $x$? (Assume the magnetic flux is entirely confined to the air gap, fringing effects are negligible, and the permeability of free space is $\mu_0$).

An LCR series circuit contains an inductor $L$, a capacitor $C$, and a non-linear resistor across which the voltage drop is given by $V_R = \beta i^3$, where $i$ is the current and $\beta$ is a positive constant. The circuit is driven by an AC voltage source $V(t) = V_0 \cos(\omega_0 t)$, where $\omega_0 = 1/\sqrt{LC}$ is the resonant frequency. Assume that the steady-state current can be approximated as purely sinusoidal at the fundamental frequency $\omega_0$.

A separate mechanical damped harmonic oscillator of mass $m$ and spring constant $k$ is subjected to a linear damping force $F_d = -b v$, where $v$ is the velocity and $b$ is the damping constant. It is driven by an external force $F(t) = F_0 \cos(\omega_0 t)$ at its resonant frequency $\omega_0 = \sqrt{k/m}$.

If the average power dissipated over one complete cycle in both systems is equal, which of the following expressions gives the equivalent linear damping constant $b$?

An intrinsic semiconductor component has a temperature-dependent resistance given by $R(T) = R_0 \exp\left(\frac{E_g}{2k_B T}\right)$, where $E_g$ is the energy band gap, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature. The component is connected to an AC voltage source $V(t) = V_0 \cos(\omega t)$. The frequency $\omega$ is sufficiently high such that the temperature $T$ of the component remains effectively constant at a steady-state value. The component dissipates heat to the surroundings at a constant temperature $T_0$ according to Newton's law of cooling, with a heat transfer coefficient $K$ (in $\text{W/K}$). As the peak voltage $V_0$ is gradually increased, the steady-state temperature $T$ increases. However, beyond a critical voltage $V_{0,c}$, thermal runaway occurs and no steady state is possible. Assuming $E_g \gg k_B T_0$, which of the following expressions best approximates the critical peak voltage $V_{0,c}$? (Here $e$ is the base of the natural logarithm).

An electron of mass $m$ and charge $-e$ is initially moving in a circular orbit of radius $r$ with angular velocity $\omega_0$ in the $xy$-plane. At $t=0$, a uniform, time-varying magnetic field $\vec{B}(t) = B_0 \sin(\Omega t) \hat{k}$ is applied perpendicular to the orbital plane. The field increases from zero to $B_0$ over the time interval $0 \le t \le \frac{\pi}{2\Omega}$. Assuming the orbital radius $r$ remains constant due to a strong central restoring force, what is the average power delivered to the electron by the induced electric field during this time interval?

A solid body of heat capacity $C$ (in $\text{J/K}$) is placed in a room maintained at a constant temperature $T_0$. The body is heated by an internal electrical heater that supplies power according to the relation $P(t) = P_0(1 + \sin(\omega t))$, where $P_0$ and $\omega$ are positive constants. The body loses heat to the surroundings strictly according to Newton's law of cooling, with a cooling constant $K$ (in $\text{W/K}$). After a sufficiently long time, the temperature of the body reaches a steady oscillatory state. Which of the following expressions represents the maximum temperature reached by the body in this steady state?

A series LCR circuit consists of an inductor of inductance $L$, a capacitor of capacitance $C$, a metallic resistor of temperature-independent resistance $R_m$, and a semiconductor resistor of resistance $R_s(T)$. The semiconductor is intrinsic with a band gap energy $E_g$, and its resistance varies with absolute temperature $T$ strictly due to the intrinsic carrier concentration $n_i \propto \exp\left(-\frac{E_g}{2k_B T}\right)$, where $k_B$ is the Boltzmann constant. At a specific temperature $T_0$, the metallic resistance and the semiconductor resistance are equal, such that $R_m = R_s(T_0) = R_0$. If $Q_0$ is the quality factor of the circuit at temperature $T_0$, what is the rate of change of the quality factor with respect to temperature, $\left. \frac{dQ}{dT} \right|_{T=T_0}$?

An AC Wheatstone bridge ABCD is constructed with the following components in its four arms: Arm AB contains a resistor $R_1 = 10\ \Omega$ in series with an inductor $L = 20\ \text{mH}$. Arm BC contains a pure resistor $R_2 = 50\ \Omega$. Arm AD contains a pure resistor $R_3 = 40\ \Omega$. Arm DC contains a resistor $R_4$ in series with a capacitor $C = 50\ \mu\text{F}$. An AC voltage source $V = V_0 \sin(\omega t)$ is connected across terminals A and C. An ideal AC null detector is connected across terminals B and D. If the detector registers zero current, what are the required values of the resistance $R_4$ and the angular frequency $\omega$?

A non-ideal transformer has a primary-to-secondary turns ratio of $1:n$. The primary winding has a finite magnetizing inductance $L_m$ (in H) and negligible resistance. The secondary winding is connected to a series load consisting of a resistor $R$ (in $\Omega$), an inductor $L$ (in H), and a capacitor $C$ (in F). An AC voltage source of frequency $\omega$ (in rad/s) is applied to the primary. It is observed that the total primary current is exactly in phase with the applied primary voltage. Which of the following conditions must necessarily hold true at this frequency?

A series LCR circuit with resistance $R$ (in $\Omega$), inductance $L$ (in H), and capacitance $C$ (in F) is driven by a superposition of two AC voltage sources: $V(t) = V_0 \cos(\omega_1 t) + V_0 \cos(\omega_2 t)$, where $V_0$ is the voltage amplitude (in V) and $t$ is time (in s). The angular frequencies (in rad/s) are symmetrically offset from the circuit's resonant frequency $\omega_0 = 1/\sqrt{LC}$ such that $\omega_1 = \omega_0 + \Delta \omega$ and $\omega_2 = \omega_0 - \Delta \omega$, where $\Delta \omega \ll \omega_0$. The total voltage $V(t)$ exhibits a beat pattern with an envelope that periodically reaches a maximum. The resulting steady-state current $I(t)$ also exhibits a beat pattern. What is the time lag $\Delta t$ of the maximum of the current envelope relative to the nearest maximum of the voltage envelope?

An AC voltage source $V(t) = V_0 \cos(\omega t)$ with a constant internal resistance $R_i$ is connected in series with a capacitor of capacitance $C$ and a special load wire. The load wire is a cylindrical conductor of length $l$, cross-sectional area $A$, electron number density $n$, and electron relaxation time $\tau$. The effective mass of an electron is $m$ and its charge is $-e$. According to the Drude model, the drift velocity $v_d$ of electrons under an electric field $E$ obeys $m \frac{dv_d}{dt} + \frac{m}{\tau} v_d = -e E$. At what angular frequency $\omega$ is the average power dissipated in the load wire maximized?

An engineer designs a circuit to deliver power to an ideal p-n junction diode from an AC voltage source $V(t) = V_0 \sin(\omega t)$ with amplitude $V_0 = 50 \text{ mV}$. Due to thermal drift, the diode stabilizes at $T = 300 \text{ K}$, where the thermal voltage is $V_T = \frac{kT}{q} = 25 \text{ mV}$. At this temperature, governed by the Law of Mass Action, the reverse saturation current is $I_s = 2.0 \mu\text{A}$.

Applying the Maximum Power Transfer principle linearly, the engineer assumes the diode acts as an Ohmic load with a constant resistance equal to its zero-bias dynamic resistance $r_d = \frac{V_T}{I_s}$, and calculates the expected average power as $P_{linear} = \frac{V_{rms}^2}{r_d}$.

However, the true average power $P_{true}$ is governed by the non-linear ideal diode equation $I(t) = I_s \left( e^{\frac{V(t)}{V_T}} - 1 \right)$.

Calculate the absolute error $\Delta P = |P_{true} - P_{linear}|$ in the engineer's calculation.

(To find $P_{true}$, approximate the exponential up to the cubic term: $e^x - 1 \approx x + \frac{x^2}{2} + \frac{x^3}{6}$. Use standard time averages $\langle \sin^2(\omega t) \rangle = \frac{1}{2}$ and $\langle \sin^4(\omega t) \rangle = \frac{3}{8}$.)

An ideal gas with adiabatic exponent $\gamma$ (dimensionless) is confined in a horizontal cylinder of cross-sectional area $A \text{ (m}^2\text{)}$. The gas is initially at equilibrium with volume $V_0 \text{ (m}^3\text{)}$ and pressure $P_0 \text{ (Pa)}$. A frictionless piston of mass $m \text{ (kg)}$ seals the gas. An external driving force $F(t) = F_0 \cos(\omega t) \text{ (N)}$ is applied to the piston, where $\omega \text{ (rad/s)}$ is the angular frequency. Assuming the process is strictly adiabatic and displacements are small, what is the root-mean-square (RMS) value of the fluctuating component of the gas pressure, $\Delta P_{rms} \text{ (Pa)}$, in the steady state?

A plane electromagnetic wave propagating in a vacuum has a wave vector $\vec{k}$ lying in the $xy$-plane. The wave vector makes a small angle $\alpha$ with the positive $x$-axis. The wave is linearly polarized with its electric field vector parallel to the $z$-axis, and its root-mean-square (RMS) electric field magnitude is $E_{\text{rms}} = 10\pi \text{ V/m}$. The wave is incident upon an opaque screen at $x = 0$ containing a single narrow slit of width $b$, extending from $y = -b/2$ to $y = b/2$. According to Huygens' principle, the slit acts as a continuous array of secondary wavelet sources. Assuming the small angle approximation ($\sin\alpha \approx \alpha$) is valid and $\alpha = \frac{\lambda}{2b}$ (where $\lambda$ is the wavelength), calculate the time-averaged intensity $I$ of the electromagnetic wave at a point $P$ located on a distant screen at $x = D$ ($D \gg b$) exactly on the $x$-axis ($y = 0$). Let $c$ be the speed of light and $\epsilon_0$ be the vacuum permittivity. Assume standard Fraunhofer diffraction conditions where the secondary wavelets have uniform amplitude across the slit.

A parallel plate capacitor consists of two circular plates of radius $R$ separated by a distance $d$. The space between the plates is filled with a non-magnetic dielectric material whose relative permittivity varies with time as $\epsilon_r(t) = 1 + \alpha \sin(\omega t)$, where $\alpha$ is a positive dimensionless constant ($\alpha \ll 1$). An alternating voltage $V(t) = V_0 \cos(\omega t)$ is applied across the plates. Neglecting edge effects, what is the peak magnitude of the induced magnetic field at the edge of the capacitor plates (at a radial distance $R$ from the central axis) specifically for the component oscillating at angular frequency $2\omega$? (Let $c$ be the speed of light in vacuum)

A circuit consists of a voltage source $V(t) = V_0 + V_m \cos(\omega t)$, an inductor of inductance $L = 1 \text{ mH}$, a capacitor of capacitance $C = 10 \text{ \mu F}$, and a non-linear resistor. The non-linear resistor has a voltage-current relationship given by $V_R = k I_R^2$ (for $I_R > 0$), where $k = 2 \text{ V/A}^2$. The inductor is connected in series with the voltage source. The capacitor and the non-linear resistor are connected in parallel with each other, and this parallel combination is connected in series with the inductor. Given $V_0 = 50 \text{ V}$ and assuming $V_m \ll V_0$ (small-signal regime), find the angular frequency $\omega$ at which the steady-state AC component of the current drawn from the source is exactly in phase with the AC component of the source voltage.

An insulated calorimeter of constant heat capacity $C$ contains a coil of constant resistance $R$ and a temperature-dependent inductance $L(T) = \frac{\alpha}{T}$, where $T$ is the absolute temperature of the system and $\alpha$ is a positive constant. The coil is connected to an external AC voltage source of constant RMS value $V_{rms}$ and angular frequency $\omega$. At the initial temperature $T_0$, the inductive reactance of the coil is exactly equal to its resistance $R$. Assuming all electrical energy dissipated by the coil is entirely converted into heat within the calorimeter, find the time $t$ required for the system's temperature to reach $2T_0$.

A very long ideal solenoid of radius $R$ and $n$ turns per unit length carries a high-frequency alternating current $I(t) = I_0 \cos(\omega t)$. The interior of the solenoid is completely filled with a non-magnetic, non-conducting dielectric material of relative permittivity $\epsilon_r$. The speed of light in vacuum is $c$. Assuming $\omega R \ll c$, which of the following expressions best represents the effective inductive reactance per unit length of the solenoid when accounting for the first-order correction due to Maxwell's displacement current?

An air-filled parallel plate capacitor is constructed using two identical flat metal plates, each of initial area $A_0$ and coefficient of linear thermal expansion $\alpha$. The plates are held parallel to each other at an initial separation $d_0$ by a composite insulating spacer that keeps the plates perfectly parallel. The spacer consists of Material 1 (Young's modulus $2Y$, total cross-sectional area $S$, coefficient of linear expansion $\alpha$) and Material 2 (Young's modulus $Y$, total cross-sectional area $S$, coefficient of linear expansion $3\alpha$). Both materials are rigidly fastened to the plates at their ends, forcing them to have the same length at all temperatures. The capacitor is connected to an AC voltage source of constant angular frequency $\omega$. If the temperature of the entire assembly is increased by $\Delta T$, what is the fractional change in the capacitive reactance ($\frac{\Delta X_C}{X_{C0}}$) of the capacitor? (Assume $\Delta T$ is small enough that higher-order terms are negligible).

A series $L-C$ circuit consists of an inductor with inductance $L = 10 \text{ mH}$ and a parallel-plate capacitor. The vacuum capacitance of the capacitor (without any dielectric) is $C_0 = 1.0 \text{ \mu F}$. The space between the capacitor plates is completely filled with a dispersive liquid dielectric. The relative permittivity (dielectric constant) of this material varies with the angular frequency $\omega$ of the applied AC source according to the relation $k(\omega) = 4 + \frac{\beta}{\omega^2}$, where $\beta = 36 \times 10^6 \text{ rad}^2\text{s}^{-2}$. What is the resonant angular frequency $\omega_r$ of this circuit? (Assume resonance occurs when the total reactive impedance of the circuit is zero).

An ideal voltage source outputs a periodic non-sinusoidal signal given by $V(t) = 120 \left[ \sin(100 t) + \frac{1}{2}\sin(200 t) + \frac{1}{3}\sin(300 t) \right] \text{ V}$ where $t$ is in seconds. The source is connected across a pure capacitor of capacitance $C = 50 \text{ \mu F}$. What is the root-mean-square (RMS) value of the total current $I_{rms}$ flowing through the circuit?

An AC voltage source $V(t) = V_0 \sin(\omega t)$ with peak voltage $V_0 = 20 \ \text{V}$ and angular frequency $\omega = 1000 \ \text{rad/s}$ has an internal impedance consisting of a resistor $R_1 = 5 \ \Omega$ and an inductor $L_1 = 10 \ \text{mH}$ in series. The source is connected to a load consisting of a variable resistor $R_2$, a fixed inductor $L_2 = 40 \ \text{mH}$, and a variable capacitor $C$ all connected in series. The values of $R_2$ and $C$ are adjusted such that the average power transferred to the load is maximized. Under this condition, determine the required capacitance $C$ and the peak voltage $V_{C0}$ across the capacitor.

A series LCR circuit with resistance $R$, inductance $L$, and capacitance $C$ is driven by an AC voltage source $V(t) = V_0 \sin(\omega t)$. The circuit is tuned to resonance. An electron (rest mass $m_0$, charge $e$) is accelerated from rest through a static potential difference equal to the peak voltage across the capacitor. After acceleration, the electron enters a region of uniform magnetic field $B$ perpendicular to its velocity, moving in a circular path. Which of the following represents the magnetic dipole moment of the revolving electron? (Let $c$ be the speed of light in vacuum).