Dual Nature of Radiation and Matter - Class 12

A perfectly reflecting light sail of rest mass $m_0$ and cross-sectional area $A$ is propelled in a vacuum by a stationary laser beam of constant intensity $I$. The laser beam is directed normally to the surface of the sail. Assuming the sail moves purely translationally along the direction of the laser beam, what is the instantaneous acceleration of the sail at the moment its velocity reaches $v = \frac{3}{5}c$ (where $c$ is the speed of light in vacuum) relative to the laser?

In a photoelectric experiment, a planar metal emitter plate with work function $\Phi$ is located at $x = 0$ and a parallel collector plate is located at $x = d$. The emitter is illuminated by monochromatic light of wavelength $\lambda$. The region between the plates contains a non-uniform static electric field given by $\vec{E} = E_0 \left(1 - \frac{3x}{2d}\right) \hat{i}$, where $E_0$ is a positive constant. Assuming the photons are neutral and unaffected by the electric field, what is the minimum value of $E_0$ required to completely stop the photocurrent? (Here $e$ is the elementary charge, $h$ is Planck's constant, and $c$ is the speed of light. Assume $\frac{hc}{\lambda} > \Phi$).

An infinitely long thin straight wire with a uniform positive linear charge density $\lambda$ is placed parallel to a large, grounded, flat metal plate at a distance $d$. The entire space above the plate is filled with a transparent liquid dielectric of relative permittivity $K$. The unperturbed work function of the metal is $\Phi_0$. Due to the applied electric field, the work function is locally reduced (Schottky effect). Assuming the escaping photoelectrons are subjected to the macroscopic electric field and the modified image charge force in the dielectric, what is the maximum vacuum threshold wavelength $\lambda_{th}$ of incident light that can cause photoelectric emission from the region of the plate directly below the wire?

(Here $e$ is the elementary charge, $h$ is Planck's constant, $c$ is the speed of light in vacuum, and $\epsilon_0$ is the vacuum permittivity)

A uniform plane wave of monochromatic light of wavelength $\lambda$ and intensity $I_0$ is incident normally on an opaque screen containing a circular aperture of diameter $D$. A planar photocathode is placed parallel to the screen at a distance $L$ (where $L \gg D^2/\lambda$). The photocathode is a circular disk of radius $R$ centered on the axis of the aperture. The radius $R$ is chosen to exactly match the radius of the first dark ring of the Fraunhofer diffraction pattern produced by the aperture. The quantum efficiency of the photocathode is $\eta$ (number of photoelectrons emitted per incident photon), and the photon energy exceeds the work function. Given that the central maximum of the Airy pattern contains exactly $84\%$ of the total transmitted power, which of the following expressions gives the saturation photoelectric current $I_{sat}$? (Here $h$ is Planck's constant, $c$ is the speed of light, and $e$ is the elementary charge).

A spherical metallic cathode of radius $R$ is kept in a vacuum with surroundings at $0 \text{ K}$. It has an initial work function $\Phi_0$ and Fermi energy $E_{F0}$ at $0 \text{ K}$. The work function $\Phi$ is related to the Fermi energy $E_F$ by $\Phi = W_0 - E_F$, where $W_0$ is a temperature-independent constant representing the depth of the potential well. The Fermi energy depends on the free electron density $n$ as $E_F \propto n^{2/3}$. The cathode is illuminated by a light source of wavelength $\lambda$, from which it continuously absorbs a constant power $P$. It loses heat only by thermal radiation (assume emissivity $\epsilon = 1$). If the coefficient of linear thermal expansion of the metal is $\alpha$ (where $\alpha \ll 1$), which of the following expressions gives the stopping potential $V_s$ for the emitted photoelectrons when the cathode reaches a steady-state temperature? (Assume $\frac{hc}{\lambda} > \Phi_0$, $e$ is the elementary charge, and $\sigma$ is the Stefan-Boltzmann constant.)

A specialized photocathode of thickness $d = 1.5 \text{ \mu m}$ is illuminated by monochromatic light of photon energy $E = 6.0 \text{ eV}$. The illuminated surface is at $x = 0$. Due to a graded alloy composition, the local binding energy (minimum photon energy required to free an electron and leave it at rest at depth $x$) is given by $\Phi(x) = \Phi_0 - \alpha x$, where $\Phi_0 = 4.0 \text{ eV}$ and $\alpha = 2.0 \text{ eV}\cdot\text{\mu m}^{-1}$. Additionally, a non-uniform space-charge distribution creates a local electrostatic potential $V(x) = \frac{\beta}{e} x^2$ inside the material relative to the surface ($V(0) = 0$), where $\beta = 1.0 \text{ eV}\cdot\text{\mu m}^{-2}$ and $e$ is the elementary charge. Assuming electrons travel to the surface without any energy loss other than that due to the macroscopic electrostatic potential, what is the magnitude of the stopping potential required to completely halt the photoelectric current?

A large metallic plate lying in the $x$-$y$ plane ($z=0$) has a work function $W$. It is illuminated by monochromatic light of photon energy $E_{ph}$ (where $E_{ph} > W$). Consider a photoelectron emitted from the origin with the maximum possible kinetic energy, at an angle $\theta$ with the $+z$-axis. In the region $z > 0$, a uniform electric field $\vec{E} = E_0 \hat{k}$ is present, which acts as a retarding field for the emitted electrons. Let $z_0$ be the maximum $z$-coordinate reached by this specific electron. Suppose that exactly at the plane $z = z_0$, the electric field abruptly terminates, and a non-uniform magnetic field $\vec{B} = B_0(1 + \alpha z)\hat{k}$ exists for all $z \ge z_0$. What is the radius of curvature of the electron's trajectory immediately after it enters the magnetic field region? (Assume $m$ and $-e$ are the mass and charge of the electron, respectively. Neglect gravity. Let $\Delta E = E_{ph} - W$. $E_0, B_0$, and $\alpha$ are positive constants with appropriate SI units).

An annular metal disk of inner radius $R_1 = R/e$ and outer radius $R_2 = R$ (where $e$ is the base of the natural logarithm) is placed in a vacuum chamber. The inner edge is maintained at a steady temperature $T_1 = 600 \text{ K}$ and the outer edge at $T_2 = 300 \text{ K}$. The disk is in a steady state of radial heat conduction. The work function of the metal varies with local temperature $T$ as $\Phi(T) = \Phi_0 + \alpha T$, where $\Phi_0 = 2.0 \text{ eV}$ and $\alpha = 10^{-3} \text{ eV/K}$. The entire surface of the disk is illuminated uniformly with monochromatic light of photon energy $E = 2.6 \text{ eV}$. A uniform retarding potential $V_0$ is applied between the disk and a parallel collector plate. It is observed that photoelectrons emitted from exactly half of the total surface area of the disk have sufficient kinetic energy to reach the collector. Find the value of $V_0$.

Two broad, coherent monochromatic light beams, each of wavelength $\lambda$ and uniform intensity $I_0$ (defined as power per unit area perpendicular to the direction of propagation), are incident on a flat photosensitive plate lying in the $x-y$ plane. The wave vectors of the two beams lie in the $x-z$ plane and make angles of $+\theta$ and $-\theta$ with the $z$-axis (which is normal to the plate). The beams are polarized with their electric fields parallel to the $y$-axis, and are in phase at the origin $(0,0,0)$. A rectangular photosensitive strip of width $w$ (along the $y$-axis) and length $L = \frac{\lambda}{6 \sin\theta}$ (along the $x$-axis) is marked on the plate from $x=0$ to $x=L$. The work function of the strip is $\phi$ (where $\phi < hc/\lambda$), and its quantum efficiency is $\eta$ (number of photoelectrons emitted per incident photon). Assuming all emitted photoelectrons are collected, what is the total photoelectric current $i$ generated by the strip? (Here $e$ is the elementary charge, $h$ is Planck's constant, and $c$ is the speed of light)

A hypothetical nucleus $A$ at rest undergoes a transition to a ground state nucleus $B$, emitting a single photon. The rest mass energy of nucleus $A$ is $M_A c^2 = 500 \text{ MeV}$ and the rest mass energy of nucleus $B$ is $M_B c^2 = 300 \text{ MeV}$.

The emitted photon is then incident on a free atom $C$, initially at rest, with rest mass energy $M_C c^2 = 300 \text{ MeV}$. The photon is completely absorbed, ejecting an electron and leaving behind an ion $C'$ with rest mass energy $M_{C'} c^2 = 360 \text{ MeV}$.

Assume the ejected electron is ultra-relativistic and can be treated as effectively massless such that its kinetic energy $K_e$ and momentum $p_e$ are related by $K_e \approx p_e c$. Assuming all motion occurs in one dimension, what is the maximum possible kinetic energy of the ejected electron?

A monochromatic light pulse of wavelength $\lambda$ instantaneously strikes a metal plate of work function $\Phi$ at $t=0$, emitting photoelectrons. Simultaneously, a time-varying uniform electric field $\vec{E}(t) = E_0 \left(1 - \frac{t^2}{\tau^2}\right) \hat{i}$ is applied in the region for a duration $0 \le t \le \tau$, after which the field becomes zero. The most energetic photoelectrons are emitted at $t=0$ and travel along the $+\hat{i}$ direction. What is the minimum value of $E_0$ required to ensure that no photoelectron escapes to infinity? (Let $m$ and $e$ be the mass and magnitude of charge of an electron, respectively, and $c$ be the speed of light. Assume non-relativistic mechanics.)

A stationary point radioactive source contains $N_0$ nuclei with a decay constant $\lambda$. Each nuclear decay instantaneously and isotropically emits a single gamma photon of energy $E_0$. A detector of area $A$ is placed at a distance $d$ from the source ($d \gg \sqrt{A}$), oriented such that its surface is perpendicular to the incident photons. Assume the detector has $100\%$ efficiency for incident photons and the medium between the source and detector is a perfect vacuum. Consider a time interval $\Delta t = \frac{4\pi d^2}{N_0 \lambda A}$, and assume $\lambda \Delta t \ll 1$ so that the source activity remains constant. What is the probability that the detector registers exactly one gamma photon during the interval $\Delta t$?

A particle of mass $m$ is released from rest at the origin $x = 0$. It moves along the positive $x$-axis under the influence of a position-dependent force $F(x) = \alpha x^n$, where $\alpha$ and $n$ are positive constants. If $\lambda(x)$ represents the de Broglie wavelength of the particle at a coordinate $x > 0$, which of the following expressions correctly gives the spatial derivative of the de Broglie wavelength, $\frac{d\lambda}{dx}$?

A non-linear polyatomic ideal gas, consisting of molecules of mass $m$, is in thermal equilibrium at absolute temperature $T$. Each molecule has exactly $6$ active degrees of freedom ($3$ translational and $3$ rotational, with all vibrational modes frozen). If $E$ represents the average total energy per molecule, which of the following expressions gives the effective thermal de Broglie wavelength ($\lambda$) of the gas molecules based on their root-mean-square momentum? (Here $h$ is Planck's constant)

A particle of mass $m$ moves in a closed elliptical orbit around a fixed massive body of mass $M$ ($M \gg m$). The semi-major axis of the orbit is $a$ and its eccentricity is $e$. Let $\lambda$ be the instantaneous de Broglie wavelength of the particle. Which of the following expressions represents the integral $\oint \frac{1}{\lambda} ds$ evaluated over one complete orbit? (Here $ds$ is the infinitesimal arc length along the orbit, $G$ is the universal gravitational constant, and $h$ is Planck's constant).

An isolated parallel-plate capacitor in vacuum carries a constant free charge, creating a potential difference $V$ between its plates. An electron of mass $m$ and charge $-e$ is released from rest from the negative plate. It accelerates across the gap and reaches the positive plate with a de Broglie wavelength $\lambda_0$. The space between the plates is then completely filled with a linear, homogeneous dielectric liquid of dielectric constant $K$. The free charge on the plates remains unchanged. If an identical electron is now released from rest from the negative plate, what is its final de Broglie wavelength just before reaching the positive plate? (Assume the dielectric offers no viscous drag and does not alter the electron's rest mass).

In the classical Drude model of a metallic conductor, the free electrons are treated as an ideal gas at absolute temperature $T$. The relaxation time $\tau$ of the electrons due to lattice scattering is given to be inversely proportional to the temperature ($\tau \propto T^{-1}$). A uniform electric field $E$ is applied across the conductor, establishing a steady drift velocity $v_d$. Let $\lambda_{dB}$ be the de Broglie wavelength of the electrons, and $\lambda_{mfp}$ be their mean free path between successive collisions. Assuming the thermal velocity $v_{th}$ is much greater than $v_d$, which of the following correctly represents the temperature dependence of $\lambda_{dB}$ and the ratio $R = \frac{\lambda_{dB}}{\lambda_{mfp}}$?

A beam of non-relativistic electrons is accelerated from rest through a time-varying potential difference $V(t) = V_0 e^{-\alpha t}$, where $V_0$ and $\alpha$ are positive constants. The electron beam is incident normally on a single slit of width $a$ and produces a Fraunhofer diffraction pattern on a distant screen. What is the magnitude of the angular velocity of the first diffraction minimum at the instant when its angular position is $\theta = 30^\circ$?

In a semi-classical model, the electron (mass $m_e$) in the ground state of a hydrogen atom is treated as a continuous mass cloud rather than a point particle. The mass density $\rho_m(\vec{r})$ of this cloud is given by $\rho_m(\vec{r}) = m_e |\Psi(\vec{r})|^2$, where the normalized wave function is $\Psi(\vec{r}) = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}$ and $a_0$ is the Bohr radius. An external impulsive torque $\vec{\tau}(t) = J_0 \delta(t) \hat{k}$ is applied to this electron cloud, where $J_0$ is a constant angular impulse and $\delta(t)$ is the Dirac delta function. Assuming the cloud rotates as a rigid body about the z-axis immediately after the impulse, what is the acquired angular velocity $\vec{\omega}$?

A parallel beam of light propagates in the $-z$ direction and strikes the curved surface of a perfectly reflecting hemisphere of radius $R$. The flat base of the hemisphere lies in the $xy$-plane with its center at the origin. The intensity of the incident beam is non-uniform and varies with the radial distance $ho = \sqrt{x^2 + y^2}$ from the $z$-axis as $I(\rho) = I_0 \left(\frac{\rho}{R}\right)^2$, where $I_0$ is a positive constant. If $c$ is the speed of light in vacuum, what is the magnitude of the net force exerted by the radiation on the hemisphere?

A photocell with a sensitive area $A$ is illuminated by a monochromatic light source of wavelength $\lambda$. The intensity of the incident light varies with time $t$ as $I(t) = I_0 \sin^2(\omega t)$, where $I_0$ is the peak intensity and $\omega$ is a positive constant. The photocell is connected in series with an initially uncharged capacitor of capacitance $C$ and an ideal DC voltage source $V_0$. The voltage $V_0$ is sufficiently large to ensure that all emitted photoelectrons are instantaneously swept to the anode, maintaining a saturation current at all times. If the quantum efficiency of the photocell is $\eta$ (defined as the number of photoelectrons emitted per incident photon) and the work function of the cathode material is $\Phi < \frac{hc}{\lambda}$, what is the total charge accumulated on the capacitor from $t = 0$ to $t = \frac{2\pi}{\omega}$? (Here $h$ is Planck's constant, $c$ is the speed of light, and $e$ is the elementary charge).

A solid, highly thermally conductive sphere of radius $R$ is placed in deep space at an ambient temperature of $0\text{ K}$. A broad, unidirectional monochromatic laser beam of wavelength $\lambda_0$ and uniform photon flux $\Phi$ (number of photons per unit area per unit time) illuminates the sphere. The surface of the sphere has a wavelength-dependent spectral hemispherical absorptivity $\alpha(\lambda)$ given by $\alpha(\lambda) = \alpha_1$ for $\lambda < \lambda_c$ and $\alpha(\lambda) = \alpha_2$ for $\lambda \ge \lambda_c$. Given that $\lambda_0 < \lambda_c$ and the sphere's steady-state temperature $T$ is such that virtually all of its thermal emission occurs at wavelengths $\lambda > \lambda_c$, which of the following expressions gives the steady-state temperature $T$ of the sphere? (Assume $h$ is Planck's constant, $c$ is the speed of light, and $\sigma$ is the Stefan-Boltzmann constant).

In a Young's Double Slit Experiment operating in a low photon-flux regime, a detector records photon counts over a fixed time interval $\tau$. The expected number of coherent signal photons detected at a classical interference maximum is $N_s$, while at a classical minimum it is zero. Additionally, a uniform incoherent background light contributes an expected $N_b = 10$ photons to the detector at any position during $\tau$. The photon arrivals follow independent Poisson statistics, where the variance of a count equals its expected value.

An experimenter measures the total photon count $C_{max}$ at a maximum and $C_{min}$ at an adjacent minimum in separate, independent intervals of length $\tau$. The fringes are defined to be 'just resolvable' if the expected value of the difference $\langle C_{max} - C_{min} \rangle$ is exactly equal to the standard deviation of this difference. Determine the required value of $N_s$ to satisfy this resolvability condition.

A perfectly black spherical target of radius $R$ ($\text{m}$), density $\rho$ ($\text{kg m}^{-3}$), and specific heat capacity $s$ ($\text{J kg}^{-1}\text{K}^{-1}$) is initially at temperature $T_0$ ($\text{K}$). It is placed in a vacuum chamber with surroundings at $0\text{ K}$ and cools purely by thermal radiation. The instantaneous radiated power $P(t)$ ($\text{W}$) is continuously measured. In a parallel Davisson-Germer diffraction setup, electrons of mass $m_e$ ($\text{kg}$) are accelerated from rest such that their final kinetic energy $K$ ($\text{J}$) at any time $t$ ($\text{s}$) is exactly equal to $P(t) \cdot t_0$, where $t_0$ ($\text{s}$) is a constant time factor. Which of the following expressions correctly represents the de Broglie wavelength $\lambda(t)$ ($\text{m}$) of these electrons as a function of time $t$? (Assume $\sigma$ is the Stefan-Boltzmann constant in $\text{W m}^{-2}\text{K}^{-4}$ and $h$ is Planck's constant in $\text{J s}$).

A beam of electrons is accelerated from rest and directed normally onto the surface of a simple cubic crystal. The surface atoms form a 2D square lattice with a lattice constant $D = \sqrt{2} \text{ \AA}$. The lowest-order non-zero diffraction maximum in the azimuthal direction $\phi = 45^\circ$ (measured from a principal lattice axis) is observed at a polar angle $\theta = 60^\circ$ (measured from the surface normal).

The kinetic energy of these incident electrons exactly matches the energy of a photon emitted during the $n=3 \to n=1$ transition of a hydrogen-like ion embedded in a dense plasma. Due to plasma screening, the effective Rydberg energy of this ion is modified to $E_R^*$. Assuming the electron diffraction follows the 2D surface Laue condition and taking $\frac{h^2}{2m_e} = 150 \text{ eV \AA}^2$, what is the value of $E_R^*$?

In a precision photoelectric experiment, a photon of energy $E$ interacts with a stationary free atom of mass $M$ and ionization energy $\Phi$. The interaction occurs over a finite time interval $\tau$, during which internal electromagnetic forces $\vec{F}(t)$ mediate the momentum transfer between the ejecting electron and the parent ion. Assuming the momentum of the incident photon is negligible, what is the exact magnitude of the stopping potential $V_0$ required to halt the emitted electron of mass $m$? (Let $e$ be the elementary charge).

A photoelectric cell consists of two large parallel planar electrodes in vacuum. The emitter plate is illuminated by monochromatic light of photon energy $E = 6.0 \text{ eV}$. The work function of the emitter material is $\Phi = 2.0 \text{ eV}$. A variable voltage source is connected across the electrodes and adjusted to apply a retarding potential difference exactly equal to the stopping potential $V_0$, such that the photocurrent just drops to zero. The voltage source is then disconnected, leaving the electrodes isolated. Subsequently, the entire gap between the electrodes is filled with a transparent, non-conducting liquid of dielectric constant $\epsilon_r = 4$. Assuming the work function of the emitter and the incident photon energy remain unchanged, what is the maximum kinetic energy of the first few photoelectrons upon reaching the collector plate?

In a quantum-classical hybrid experiment, an electron (charge $e$) is accelerated from rest through the instantaneous potential difference across an inductor (inductance $L$) in a series $RL$ circuit. The circuit is connected to a DC battery (EMF $V$) and a resistor (resistance $R$). The switch is closed at $t=0$. The instantaneous kinetic energy $E(t)$ acquired by the electron is determined by the induced EMF across the inductor at that instant. Due to the rapid transient decay of the magnetic field, the energy of the electron has an uncertainty $\Delta E$ coupled to the uncertainty in its emission time $\Delta t$. Assuming the energy uncertainty is given by the first-order variation $\Delta E = \left| \frac{dE}{dt} \right|_{t=0} \Delta t$, find the minimum uncertainty in the emission time $\Delta t$ according to the Heisenberg uncertainty principle $\Delta E \Delta t \ge \frac{\hbar}{2}$. (All quantities are in SI units, and $\hbar$ is the reduced Planck constant).

A particle of mass $m$ is trapped in a one-dimensional non-linear potential well given by $U(x) = \frac{1}{4} k x^4$, where $k$ is a positive constant. Classically, the lowest energy state is at $x=0$, where the restoring force is zero. However, quantum mechanically, the particle's position spread $a$ and momentum spread $p$ are constrained by the simplified Heisenberg uncertainty relation $a p = \hbar$. By approximating the particle's total energy as $E \approx \frac{p^2}{2m} + \frac{1}{4} k a^4$, determine the effective magnitude of the restoring force $F_{\text{eff}} = \left| \frac{dU}{da} \right|$ when the particle is in its state of minimum total energy.

A photon of initial energy $E_{\gamma}$ is emitted from a stationary source at the origin $x=0$ at time $t=0$, traveling along the $+x$-axis in a vacuum. An observer of mass $m$ and charge $+q$ is initially at rest at $x=L$. At $t=0$, a uniform electric field $\vec{E} = E \hat{i}$ is switched on. The photon, being electrically neutral, is unaffected by the electric field. Assuming the observer's velocity remains strictly non-relativistic ($v \ll c$), what is the energy of the photon as measured by the observer at the exact instant the photon intercepts them? (Express the answer to the first order in $v/c$, where $c$ is the speed of light).

A continuous laser beam propagates along the $z$-axis through a dilute, neutral gas of constant electric susceptibility $\chi$. The time-averaged intensity of the beam varies radially as $I(r) = I_0 e^{-r^2/w^2}$, where $r$ is the radial distance from the beam axis, $I_0$ is the peak intensity, and $w$ is the beam waist parameter. Although photons are electrically neutral, their oscillating electric fields induce a time-varying polarization in the gas molecules, resulting in a net time-averaged macroscopic volume force density $\vec{f}$ exerted on the gas. Assuming the gas is sufficiently dilute such that the speed of light in the medium is approximately $c$ (the speed of light in vacuum), what is the magnitude of the maximum radial force density exerted on the gas?

A reverse-biased silicon p-n junction is uniformly illuminated by monochromatic light of wavelength $\lambda = 620 \text{ nm}$ and intensity $I = 3.2 \text{ W/m}^2$. The incident photon flux $\Phi_0$ generates electron-hole pairs at a uniform volume rate $G = \alpha \Phi_0$ throughout the device, where $\alpha = 4 \times 10^3 \text{ m}^{-1}$ is the absorption coefficient. The depletion region of the junction has a width $W = 2.0 \text{ \mu m}$. The minority carrier diffusion lengths in the p-type and n-type neutral regions are $L_n = 8.0 \text{ \mu m}$ and $L_p = 15.0 \text{ \mu m}$, respectively. Assuming steady-state conditions and that all minority carriers generated within one diffusion length of the depletion region reach the junction to contribute to the current, what is the magnitude of the total steady-state photocurrent density $J_{ph}$?

(Take the energy of a photon as $E = \frac{hc}{\lambda}$, $hc = 1240 \text{ eV}\cdot\text{nm}$, and elementary charge $e = 1.6 \times 10^{-19} \text{ C}$)

A stationary nucleus $X$ (nuclear rest mass $M_X$) undergoes $\beta^-$ decay to an excited state of nucleus $Y$. The emitted $\beta^-$ particles are observed to have a continuous kinetic energy spectrum with a maximum value $K_{max}$. The excited nucleus subsequently decays to its ground state $Y$ (nuclear rest mass $M_Y$) by emitting a single $\gamma$-photon of energy $E_\gamma$. Assuming the antineutrino is massless, neglecting the recoil kinetic energy of the nucleus during the $\beta^-$ decay, and using the non-relativistic approximation for the recoil during $\gamma$ emission ($E_\gamma \ll M_Y c^2$), which of the following is the correct expression for the rest mass energy of the initial nucleus $X$ ($M_X c^2$)? Let $m_e$ be the rest mass of the electron and $c$ be the speed of light in vacuum.

A photocell is connected in series with a load resistor $R_L = 200 \ \text{k}\Omega$ and an ideal DC voltage source of EMF $E = 15 \ \text{V}$. The potential difference across the photocell, $V_p$, is measured by finding the null point on a potentiometer wire of length $L = 100 \ \text{cm}$. The potentiometer wire is connected across a separate ideal standard cell of $10 \ \text{V}$. In the operating region, the photocell is near saturation, and its non-ideal current-voltage characteristic is given by $I = I_0 + \frac{V_p}{r_d}$, where $I_0$ is the ideal saturation current proportional to the incident light intensity, and $r_d = 200 \ \text{k}\Omega$ is the dynamic resistance of the photocell. Initially, the photocell is illuminated such that $I_0 = 25 \ \mu\text{A}$, and the potentiometer null point is obtained at a distance $x_1$ from the high-potential end. The light intensity is then increased such that $I_0$ becomes $45 \ \mu\text{A}$, and the new null point is at $x_2$. Calculate the magnitude of the shift in the null point, $|\Delta x| = |x_2 - x_1|$.

A metallic photocathode with a vacuum work function $\phi_0$ (in $\text{J}$) is placed at the origin $x=0$. A semiconductor depletion region of absolute permittivity $\epsilon$ (in $\text{F}\cdot\text{m}^{-1}$) extends from $x=0$ to $x=w$ (in $\text{m}$). The depletion region contains a uniform positive space charge density $\rho = e N_D$, where $e$ is the elementary charge (in $\text{C}$) and $N_D$ is the donor number density (in $\text{m}^{-3}$). The electric field vanishes at $x=w$. Due to the Schottky effect, the internal electric field magnitude $E_0$ at the interface lowers the potential barrier for electron emission by $\Delta V = \sqrt{\frac{e E_0}{4\pi\epsilon}}$ (in $\text{V}$). What is the minimum incident photon frequency $\nu_{th}$ (in $\text{Hz}$) required to observe a non-zero photoelectric saturation current? (Here $h$ is Planck's constant in $\text{J}\cdot\text{s}$)

A thermally insulated cylinder with a massless, frictionless piston contains an ideal diatomic gas ($f=5$). Inside the cylinder, a photoelectric cathode of work function $\phi = 2.0 \text{ eV}$ is illuminated by monochromatic light of wavelength $\lambda = 310 \text{ nm}$. A small collection voltage maintains a saturation photoelectric current $I_s = 14.0 \text{ mA}$ but does negligible work on the electrons. Assume all emitted photoelectrons possess the maximum possible kinetic energy and transfer this energy entirely to the gas molecules via inelastic collisions before reaching the anode. The gas expands isobarically against a constant external pressure $P_0 = 100 \text{ kPa}$. Calculate the rate of change of the gas volume, $\frac{dV}{dt}$. (Take $hc = 1240 \text{ eV}\cdot\text{nm}$)

A solid non-conducting sphere of radius $R$ has a positive volume charge density $\rho(r) = \rho_0 \left(\frac{r}{R}\right)^2$, where $r$ is the radial distance from the center and $\rho_0$ is a positive constant. The sphere is kept in a vacuum. The material of the sphere has a photoelectric work function $\Phi$. A monochromatic beam of light of frequency $\nu$ is incident on the surface of the sphere, causing the emission of photoelectrons. Assuming the sphere's macroscopic charge distribution remains unperturbed, what is the maximum kinetic energy of the emitted photoelectrons when they reach a distance of infinity from the sphere? (Here $e$ is the elementary charge, $h$ is Planck's constant, and $\epsilon_0$ is the permittivity of free space. Assume $h\nu$ is sufficiently large to allow electrons to reach infinity).

A cylindrical region of radius $R$ contains a uniform magnetic field $\vec{B}$ directed into the plane of the paper. The magnitude of the magnetic field varies with time $t$ as $B = \alpha t$, where $\alpha$ is a positive constant. A smooth, evacuated, semi-circular insulating tube of radius $R$ is placed along the lower half of the boundary of this region. The tube connects points $P$ and $Q$, which lie on the right and left ends of the horizontal diameter, respectively. A metal plate located at $P$ has a work function $\phi$ and is illuminated by monochromatic light of frequency $\nu$. Photoelectrons emitted from the plate are constrained to move through the tube to reach a collector at $Q$. What is the minimum stopping potential $V_0 = V_P - V_Q$ required to prevent even the most energetic photoelectrons from reaching $Q$? (Assume $h$ is Planck's constant and $e$ is the elementary charge).

A monochromatic light beam of wavelength $\lambda$ passes through a long single slit of width $a$ and length $L$ ($L \gg a$). The resulting diffraction pattern falls on a flat photoelectric plate placed at a distance $D$ ($D \gg a$) from the slit. The work function of the plate is $\phi$ (where $\phi < hc/\lambda$), and its quantum efficiency is $\eta$ (defined as the number of photoelectrons emitted per incident photon). The peak intensity of the central diffraction maximum on the plate is $I_0$. Assuming the small-angle approximation ($\sin \theta \approx \theta$), what is the total number of photoelectrons emitted per second from the region of the plate corresponding strictly to the central maximum?

(Given: $\int_{-\pi}^{\pi} \left(\frac{\sin x}{x}\right)^2 dx = \alpha$, where $h$ is Planck's constant and $c$ is the speed of light.)

In a photoelectric experiment, an emitter plate at $x=0$ is illuminated by light of frequency $f$. The work function of the emitter is $\Phi$ (where $hf > \Phi$). A collector plate is located at $x=d$. Consider an electron emitted at $t=0$ with the maximum possible kinetic energy, traveling directly along the $x$-axis towards the collector. The potential difference $V(t) = V_{\text{collector}} - V_{\text{emitter}}$ is initially zero. At a time $t_0 = \frac{d}{3} \sqrt{\frac{m}{2(hf - \Phi)}}$ (where $m$ is the mass of the electron), the potential difference is instantaneously switched to a constant positive value $V_0$, and remains $V_0$ thereafter. Assuming the electric field between the plates is always spatially uniform, what is the kinetic energy of this electron at the instant it reaches the collector?

A solid non-conducting sphere of uniform charge $Q$ (Coulombs), initial radius $R$ (meters), and bare mechanical mass $m$ (kg) is moving with a constant non-relativistic velocity $v_0$ (m/s) along the x-axis in a field-free vacuum. An internal mechanism causes the sphere to slowly expand radially to a new radius $2R$ while maintaining a uniform charge distribution. Assume no external forces act on the system and radiation losses are negligible. Let $c$ (m/s) be the speed of light in vacuum and $\epsilon_0$ (F/m) be the vacuum permittivity. If the total momentum of the system is given by $\vec{P} = m\vec{v} + \vec{p}_{em}$, where $\vec{p}_{em} = \epsilon_0 \int (\vec{E} \times \vec{B}) dV$ is the electromagnetic field momentum (with $\vec{E}$ and $\vec{B}$ being the fields generated by the sphere), what is the final velocity $v_f$ of the sphere?

A uniform magnetic field $\vec{B} = B_0(1 + \beta t) \hat{k}$ exists in a cylindrical region of radius $R$ whose axis coincides with the $z$-axis. A particle of mass $m$ and positive charge $q$ is located at a point $(r_0, 0, 0)$ at time $t = 0$, where $r_0 < R$. At this instant, the particle is projected with an initial velocity $\vec{v} = v_0 \hat{i}$. Assuming $v_0 > \frac{r_0 \beta}{2}$, what is the instantaneous radius of curvature of the particle's trajectory at $t = 0$?

In a modified Young's Double Slit Experiment, a primary source placed symmetrically with respect to two narrow slits $S_1$ and $S_2$ emits a single, ultra-short light pulse of duration $\tau$. The pulse has a rectangular time envelope, meaning its intensity is constant for a duration $\tau$ and zero otherwise. The slits are separated by a distance $d$, and the screen is placed at a distance $D$ ($D \gg d$). The intensity of the pulse arriving from each slit individually at the screen is $I_0$.

A detector at a point $P$ on the screen, located at a distance $y$ from the central maximum, measures the total time-integrated energy per unit area (fluence, $F$). If $y$ is such that the path difference $\Delta x = \frac{yd}{D}$ satisfies $0 < \Delta x < c\tau$ (where $c$ is the speed of light in vacuum), which of the following expressions correctly represents the fluence $F$ at point $P$?

(Assume the phase difference between the waves during their overlap is $\Delta \phi = \frac{2\pi yd}{\lambda D}$, where $\lambda$ is the wavelength of the light).

A large reservoir maintained at a constant pressure $P_0 \text{ (N/m}^2\text{)}$ drives an ideal, incompressible fluid of density $\rho \text{ (kg/m}^3\text{)}$ through a horizontal pipe of uniform cross-sectional area $A \text{ (m}^2\text{)}$ and length $L \text{ (m)}$. The fluid discharges into the atmosphere at pressure $P_{\text{atm}} \text{ (N/m}^2\text{)}$. Initially, the exit aperture has an area $A$, and the fluid flows steadily with a velocity $v_0 \text{ (m/s)}$. At $t=0 \text{ s}$, a dynamic mechanism instantaneously constricts the exit aperture to an area $A/2$. The length of the constricted section is negligible. What is the magnitude of the instantaneous acceleration $\text{(m/s}^2\text{)}$ of the fluid column in the pipe immediately after the constriction at $t=0^+$?

A spherical blackbody of radius $R$ is maintained at a constant temperature $T$. It is surrounded by a concentric, perfectly reflecting spherical shell of radius $r$ ($r > R$). The shell is expanding radially outward at a constant velocity $v$. Assuming the space between the blackbody and the shell is vacuum, determine the total force $F$ exerted by the radiation on the expanding shell. (Given: Stefan-Boltzmann constant $\sigma$, speed of light $c$)

A thin, perfectly absorbing spherical shell of radius $R$ is uniformly charged such that its electrostatic self-energy is $U$. An isotropic point light source emitting a constant electromagnetic power $P$ is located at the center of the shell. The shell is divided into two identical hemispheres by a hypothetical plane. Assuming the system is in gravity-free space and ignoring any thermal re-emission from the shell, what is the minimum total external force required to hold the two hemispheres together?

A p-n junction photodiode has a built-in potential $V_{bi} = 0.8 \text{ V}$. It is illuminated by a constant photon flux, resulting in a uniform generation rate $G$ of electron-hole pairs per unit volume throughout the depletion region. Assume the photocurrent $I$ is entirely due to the drift of these generated carriers across the depletion region. At a reverse bias voltage $V_{R1} = 2.8 \text{ V}$, the steady-state photocurrent is $I_1 = 2.0 \text{ \mu A}$ and the average drift velocity of electrons is $v_1 = 1.0 \times 10^4 \text{ m/s}$. If the reverse bias voltage is increased to $V_{R2} = 13.6 \text{ V}$ while keeping the photon flux constant, determine the new photocurrent $I_2$ and the new average drift velocity $v_2$. (Assume the electron mobility $\mu_e$ is constant, and the depletion width $W$ follows the relation $W \propto \sqrt{V_{bi} + V_R}$).

A uniform solid cylinder of initial mass $M_0$ and initial radius $R_0$ is initially at rest in free space, free to rotate about its fixed central longitudinal axis. A laser beam of constant power $P$ is continuously adjusted so that it always strikes the curved surface of the cylinder tangentially. The cylinder perfectly absorbs the laser photons, and the resulting radiation pressure exerts a torque.

The absorbed laser energy causes the material at the edge to ablate (vaporize) continuously. The latent heat of ablation of the material is $L$ (energy required to ablate unit mass). The ablated gas is ejected radially outwards in the rotating frame of the cylinder. As mass is lost, the cylinder remains a uniform solid cylinder of constant length, but its radius $R(t)$ decreases.

What is the angular velocity $\omega$ of the cylinder at the instant its mass reduces to $\frac{M_0}{2}$?

(Given: $c$ is the speed of light. Ignore any relativistic mass corrections for the cylinder.)

A thin opaque cylindrical disk of initial density $\rho_0$ and initial face area $A_0$ at temperature $T_0$ is suspended in a transparent coolant gas maintained at $T_0$. A laser beam of constant total power $P$ is completely absorbed by one flat face of the disk. The disk loses heat to the gas exclusively from this illuminated face via convection, with a constant convective heat transfer coefficient $h$. The coefficient of linear expansion of the disk material is $\alpha$. Assume the face area expands as $A = A_0(1 + 2\alpha\Delta T)$ and the density varies as $\rho = \rho_0(1 - 3\alpha\Delta T)$, where $\Delta T = T - T_0$. Which of the following expressions gives the steady-state density of the disk?

A rigid vacuum phototube of total mass $M$ is located in deep space (a gravity-free region). Inside the tube, a planar cathode with work function $\Phi$ and a parallel planar anode are separated by a distance $d$. A highly intense laser beam of frequency $\nu$ (where $h\nu > \Phi$), uniform intensity $I$, and cross-sectional area $A$ is incident normally on the perfectly reflecting outer surface of the anode end of the tube. The radiation pressure from the laser accelerates the entire phototube along the axis perpendicular to the electrodes. If $m$ and $-e$ are the mass and charge of an electron respectively, and $c$ is the speed of light, what is the exact stopping potential $V_s$ required to just prevent the most energetic photoelectrons from reaching the anode? (Assume the mass of the emitted electrons is negligible compared to $M$).