Atoms - Class 12

A particle of mass $m$ and positive charge $q$ is fired from infinity with an initial velocity $v_0$ towards a fixed, non-conducting solid sphere of radius $R$ having a uniform total positive charge $Q$. The impact parameter of the particle is $b$, where $b < R$. Assume that $v_0$ is sufficiently large such that the deflection angle $\theta$ is very small, allowing the trajectory to be approximated as a straight line. Which of the following expressions correctly represents the small deflection angle $\theta$? (Here $\epsilon_0$ is the permittivity of free space).

At $t=0$, a particle of mass $m$ and charge $+q$ enters a region filled with a viscous medium. At this instant, it is at a very large distance from a fixed heavy nucleus of charge $+Q$, moving with an initial speed $v_0$ and an impact parameter $b$ relative to the nucleus. The medium exerts a drag force $\vec{F}_d = -\gamma \vec{v}$ on the particle, where $\gamma$ is a positive constant and $\vec{v}$ is its instantaneous velocity. The nucleus exerts a Coulombic repulsive force $\vec{F}_c = \frac{kQq}{r^2}\hat{r}$. What is the total area swept by the position vector of the particle (measured from the nucleus) from $t=0$ to $t \to \infty$?

A uniform thin spherical shell of mass $M$ and radius $R$ is free to rotate about a fixed vertical axis passing through its center $O$. A particle of mass $m = \frac{16}{9}M$ is projected from infinity in the horizontal equatorial plane of the shell, with an initial speed $v_0 = \sqrt{\frac{GM}{R}}$ and an impact parameter $b = \frac{3R}{4}$. The particle enters the shell through a small hole. Inside the shell, there is a rigid diametric spoke of negligible mass lying in the equatorial plane. The spoke is oriented such that the particle strikes it exactly at the particle's point of closest approach to the center $O$, and gets embedded in it. Assuming the shell and spoke are initially at rest, and considering only the gravitational interaction between the particle and the shell, find the final angular velocity of the system.

An alpha particle of mass $m$ and charge $+2e$ is fired with an initial speed $v_0$ directly towards a free, initially stationary Oxygen nucleus ($^{16}_{8}\text{O}$) of mass $4m$ and charge $+8e$. Assume the system is isolated and no other forces act on the particles. What is the distance of closest approach between the alpha particle and the Oxygen nucleus? (Here $\epsilon_0$ is the vacuum permittivity).

An alpha particle of mass $m$ and charge $+2e$ is projected from infinity with an initial speed $v_0$ towards a massive, fixed nucleus of charge $+Ze$. The impact parameter of the alpha particle is $b = \frac{2kZe^2}{mv_0^2}$, where $k = \frac{1}{4\pi\epsilon_0}$. What is the distance of closest approach of the alpha particle to the nucleus?

Two fixed concentric thin spherical shells of radii $R$ and $3R$ carry uniformly distributed positive charges $Q$ and $2Q$ respectively. A particle of mass $m$ and positive charge $q$ is projected from infinity directly towards the common center of the shells with an initial kinetic energy $K_0 = \frac{qQ}{3\pi\epsilon_0 R}$. What is the distance of closest approach of the particle to the center of the shells?

A projectile of mass $m$ and charge $+2e$ is fired head-on from infinity with initial kinetic energy $K_0$ towards a free, initially stationary target ion of mass $m$ and charge $+e$. To trigger an internal excitation of the target ion, the projectile must reach a distance of closest approach $d$. The electrostatic potential energy of two elementary charges $+e$ separated by $d$ is given as $U_0 = 20 \text{ eV}$. The target ion has internal energy levels described by $E_n = n E_0$ (where $n = 1, 2, 3, \dots$ and $E_0 = 15 \text{ eV}$). Due to quantum selection rules, the ion can only undergo transitions where the change in the quantum number $n$ (i.e., $\Delta n$) is an even integer. Assuming the target ion is initially in its ground state ($n=1$), what is the minimum initial kinetic energy $K_0$ required for the projectile to excite the target ion to its lowest allowed excited state?

A projectile nucleus of mass $m$ and charge $+e$ is fired with an initial kinetic energy $K_0$ towards a free target nucleus of mass $M = 2m$ and charge $+3e$, which is initially at rest in the laboratory frame. During the head-on collision, the target nucleus is excited to a higher internal energy state, absorbing an excitation energy $\Delta E = \frac{K_0}{6}$. Assuming the distance of closest approach $d_{min}$ is strictly greater than the sum of their nuclear radii, what is the value of $d_{min}$? (Here $\epsilon_0$ is the permittivity of free space)

A smooth horizontal table has a small hole at the origin $O$. A smooth vertical rod passes through $O$ along the $z$-axis. A nucleus of mass $M$ and charge $+Q$ is constrained to slide on the vertical rod above the table. An electron of mass $m$ and charge $-q$ is constrained to move on the table. The system is in a gravity-free space.

A light inextensible string is attached to the nucleus, passes upwards along the $z$-axis, and wraps around a uniform solid disk of radius $R$ which is free to rotate about a fixed horizontal axis located high above the table.

Initially, the nucleus is at a height $z_0$ above the table, and the electron is in a steady circular orbit of radius $r_0$ around $O$. An external torque is applied to the disk, quasi-statically unwinding the string to lower the nucleus until it reaches the hole $O$. During this process, the electron continuously adjusts its orbit to remain in steady circular motion.

Find the total work done by the external torque. (Let $\alpha = \frac{1}{4\pi\epsilon_0} Qq$)

Consider a hypothetical hydrogen-like atom where the nucleus is not a point charge, but a non-conducting solid sphere of radius $R$. The volume charge density of the nucleus is given by $\rho(r) = \rho_0 \left(\frac{r}{R}\right)$ for $r \le R$, where $\rho_0$ is a positive constant and $r$ is the radial distance from the center. The total charge of the nucleus is $+e$. An electron of mass $m$ and charge $-e$ revolves around the center of the nucleus in a circular Bohr orbit. Assuming the $n^{\text{th}}$ Bohr orbit lies entirely inside the nucleus ($r_n < R$), which of the following expressions represents the radius $r_n$ of this orbit? (Here $\epsilon_0$ is the permittivity of free space and $\hbar = \frac{h}{2\pi}$ is the reduced Planck constant).

A small spherical particle of mass $m$ and charge $+q$ is immersed in a highly viscous liquid with a drag coefficient $b$ (Stokes' drag $\vec{F}_d = -b\vec{v}$). A fixed point charge $-Q$ is located at the origin. A uniform external electric field $\vec{E} = E_0 \hat{i}$ is applied. The particle has an equilibrium position at $x = x_0$ ($x_0 > 0$).

The particle is slightly displaced to a new position $x = x_0 + \delta x$ (where $\delta x > 0$ and $\delta x \ll x_0$). In the overdamped regime, the inertial term ($m\ddot{x}$) is negligible, and the particle's velocity $\vec{v}$ is determined entirely by the instantaneous balance of the conservative forces and the viscous drag.

Find the velocity vector $\vec{v}$ of the particle at this displaced position. (Given: $k = \frac{1}{4\pi\epsilon_0}$)

An electron of mass $m$ and charge $-e$ is constrained to move in a fixed, non-conducting circular ring of radius $R$ in the $xy$-plane. Initially, no magnetic field is present, and the electron is in a stationary state with an initial kinetic angular momentum $\vec{L}_0 = -n\hbar \hat{k}$ (where $n > 0$ is an integer and $\hat{k}$ is the unit vector along the $z$-axis). A uniform, time-varying magnetic field $\vec{B}(t) = B(t) \hat{k}$ is then applied, increasing slowly from $B(0) = 0$ to a final constant value $B_0$. Assuming the electron remains in the ring, which of the following expressions represents the exact work done by the induced electric field on the electron during this process? (Ignore self-inductance and electromagnetic radiation)

A hypothetical atom has a ground state rest mass $M$. It is initially at rest in a highly excited state. The difference in rest mass energy between this excited state and the ground state is $\Delta E = \frac{3}{2}Mc^2$. The atom emits a single photon and transitions directly to the ground state. Taking into account exact relativistic energy and momentum conservation, what is the energy $E_\gamma$ of the emitted photon? ($c$ is the speed of light in vacuum)

Consider a heavy nucleus of mass number $A \gg 1$. A student investigates the scaling laws of the nuclear energy levels using two different models for a nucleon of mass $m$.

\textbf{Model I (Incorrect Bohr-like Model):} The nucleon experiences a long-range attractive central potential $V(r) = -\frac{\alpha A}{r}$, where $\alpha$ is a positive constant. Let the ground state binding energy of the nucleon in this model scale as $A^p$.

\textbf{Model II (Correct Shell Model):} The nucleon is trapped in a 3D isotropic harmonic oscillator potential $V(r) = \frac{1}{2}m\omega^2 r^2 - V_0$, where $V_0$ is a constant well depth. The nucleus is modeled as a Fermi gas where nucleons fill all energy levels up to the Fermi level $N_F \gg 1$. The degeneracy of the $N$-th energy level is $g_N \propto N^2$. The nuclear radius $R$, known to scale as $R \propto A^{1/3}$, is defined as the classical turning point of a nucleon at the Fermi energy $E_F \approx N_F \hbar \omega$. Let the oscillator frequency $\omega$ scale as $A^q$.

Find the values of the exponents $(p, q)$.

A metal surface with a work function $W = 4.0 \text{ eV}$ is irradiated by a high-intensity laser of frequency $\omega$ and intensity $I$. The electric field of the laser is given by $E(t) = E_0 \sin(\omega t)$, where $I = \frac{1}{2} \epsilon_0 c E_0^2$. In this high-intensity regime, the electron undergoes forced oscillations before emission. The time-averaged kinetic energy of this oscillation, $U_p$, effectively increases the binding energy of the electron, such that the new effective work function becomes $W_{eff} = W + U_p$. If the energy of a single photon is $\hbar \omega = 1.5 \text{ eV}$, multi-photon absorption is required for photoemission. Determine the threshold intensity $I_{th}$ above which a minimum of $4$ photons must be simultaneously absorbed to emit a photoelectron. (Assume the initial drift velocity of the electron is zero, $m$ is the mass of the electron, and $e$ is the elementary charge).

Consider a hypothetical hydrogen-like atom where the nucleus is a uniformly charged solid sphere of radius $R_0$ and total charge $+Ze$. An electron of mass $m$ and charge $-e$ revolves in circular orbits around the center of the nucleus. Assume that $R_0$ is sufficiently large such that the first few allowed Bohr orbits lie entirely inside the nucleus ($r < R_0$). If $\lambda_n$ represents the de Broglie wavelength of the electron in the $n^{\text{th}}$ allowed orbit, what is the ratio $\frac{\lambda_2}{\lambda_1}$?

An electron of mass $m$ and charge $-e$ is moving in a circular orbit of radius $r_0$ with speed $v_0$ around a stationary nucleus. A uniform magnetic field of magnitude $B$ is slowly established perpendicular to the plane of the orbit. The direction of the magnetic field is such that the magnetic Lorentz force on the electron points towards the nucleus. Assuming the magnetic field is weak ($eB \ll m v_0/r_0$) and neglecting any radiative losses, what is the change in the kinetic energy of the electron?

A hypothetical 'gravitational atom' consists of a particle of mass $m$ orbiting a central star of mass $M$ ($M \gg m$) under purely gravitational forces. The system's energy levels are quantized according to Bohr's postulates. Let $R_0$ be the magnitude of the initial ground state energy of this system. The central star slowly and isotropically loses mass due to radiation over a very long time, until its mass becomes $M/2$. The process is slow enough that the orbital quantum number $n$ remains an adiabatic invariant. Assuming the emitted radiation does not interact with the orbiting particle, find the new ground state binding energy $E'_b$ and the work $W_g$ done by the gravitational force on the particle during this entire process.

A spherical blackbody of radius $R$ is maintained at a temperature $T$ such that $k_B T \ll hc R_H$, where $R_H$ is the Rydberg constant. It is surrounded by a thin, optically thick shell of hydrogen gas in its ground state. The shell absorbs all incident photons from the blackbody that have an energy greater than or equal to the ionization energy of hydrogen. Assume that each absorbed ionizing photon causes exactly one ionization event, and the subsequent recombination cascade produces exactly one Lyman-$\alpha$ photon (transition from $n=2$ to $n=1$).

Using Wien's approximation for the spectral emissive power (power emitted per unit area per unit wavelength) $E_\lambda = \frac{2\pi hc^2}{\lambda^5} e^{-hc/\lambda k_B T}$, determine the total power $P_{L\alpha}$ emitted by the hydrogen shell in the Lyman-$\alpha$ line.

(Assume $\int_{x_0}^\infty x^n e^{-x} dx \approx x_0^n e^{-x_0}$ for $x_0 \gg 1$, where $x = \frac{hc}{\lambda k_B T}$ and $x_0$ corresponds to the ionization threshold).

A hydrogen-like ion consists of a single electron of rest mass $m_e$ and a nucleus. The sum of the rest masses of the isolated nucleons constituting the nucleus is $M_0$, and the total nuclear binding energy is $E_b$. Let $\nu_0$ be the frequency of a photon emitted during a specific electronic transition, calculated by accounting for nuclear recoil but incorrectly assuming the nuclear mass is $M_0$. Let $\nu$ be the true frequency of the photon for the same transition, correctly accounting for both nuclear recoil and the nuclear mass defect. Assuming $m_e \ll M_0$ and $E_b \ll M_0 c^2$, which of the following best approximates the fractional shift in frequency, $\frac{\nu - \nu_0}{\nu_0}$? ($c$ is the speed of light in vacuum)

A beam of atoms of mass $m$ is initially in the first excited state ($n=2$). In this state, the atoms possess an effective permanent electric dipole moment $\vec{p} = p_0 \hat{i}$ ($p_0 > 0$) whose orientation remains fixed. In the ground state ($n=1$), the dipole moment is zero. The atoms are released from rest at $t=0$ from the position $x = L$ into a region ($x > 0$) containing a non-uniform electric field $\vec{E} = -E_0 \left( \frac{x^2}{L^2} \right) \hat{i}$, where $E_0 > 0$. At time $t = \frac{\pi}{3} \sqrt{\frac{m L^2}{2 p_0 E_0}}$, an atom undergoes a transition to the ground state, emitting a photon. If the energy difference between the $n=2$ and $n=1$ states in the absence of an electric field is $\Delta E_0$, what is the frequency $\nu$ of the emitted photon? (Assume the classical velocity of the atom does not change during the emission process).

A beam of photons corresponding to the series limit of a hydrogen-like atom is used to illuminate a Young's Double Slit apparatus. The transition originates from a highly excited state with an effective lifetime $\tau$, decaying to a stable ground state. Due to the Heisenberg uncertainty principle ($\Delta E \Delta t \ge \frac{h}{4\pi}$, where $\Delta t = \tau$), the emitted light is not perfectly monochromatic but has a natural energy uncertainty $\Delta E$, corresponding to a wavelength spread $\Delta \lambda$. Assuming the fringe visibility drops to zero when the $m$-th order bright fringe of the central wavelength $\lambda$ coincides with the nearest dark fringe of wavelength $\lambda + \Delta \lambda$, what is the maximum observable order of interference $m$? (Assume $\Delta \lambda \ll \lambda$ and $c$ is the speed of light).

A stationary atomic nucleus of radius $R$ undergoes a transition to its ground state by emitting a gamma-ray photon of energy $E$. Assuming the universal nuclear mass density is $\rho_0$ and the speed of light in vacuum is $c$, which of the following expressions correctly represents the recoil speed $v$ of the nucleus? (Assume the recoil is non-relativistic and $E$ is much less than the rest mass energy of the nucleus).

A cubical container of volume $V$ holds $N$ atoms of an ideal monatomic gas, each of mass $m$. The gas is initially in thermal equilibrium at temperature $T$ with pressure $P_0$. At time $t=0$, every atom simultaneously emits a single photon of energy $E$ in the $+x$ direction. Assume the emission is instantaneous and the atoms undergo elastic collisions with the container walls without colliding with each other. Let $\Delta P_x$ be the change in the average pressure exerted by the gas on the two walls perpendicular to the x-axis, and $\Delta P_y$ be the change in pressure on the walls perpendicular to the y-axis. Which of the following is correct? (Here $c$ is the speed of light and $n = N/V$)

A particle of mass $M$ is moving in a horizontal plane along a circular path of radius $R$ with constant angular speed $\omega$, constrained by a massless radial string. At a certain instant, the string snaps and simultaneously the particle ejects a fragment of mass $m$ tangentially forward. In the reference frame moving with the instantaneous velocity of the particle just before ejection, the ejected fragment has a speed $u$ in the forward direction. What is the kinetic energy of the remaining fragment in the laboratory frame immediately after the ejection?

A free, initially stationary atom of mass $M$ (in its ground state) undergoes an electronic transition from an excited state to the ground state. The energy difference between the excited state and the ground state is $\Delta E$. Assuming $\Delta E \ll Mc^2$, where $c$ is the speed of light in vacuum, which of the following expressions best represents the exact energy $E_\gamma$ of the emitted photon, accounting for the recoil of the atom?

Two isolated particles of masses $m$ and $2m$ are initially at a distance $R$ from each other in free space. In the laboratory frame, they are simultaneously projected directly away from each other along the line joining them, both with a speed $v_0 = \sqrt{\frac{3Gm}{4R}}$, where $G$ is the universal gravitational constant. What is the maximum separation $r_{\text{max}}$ between the two masses during their subsequent motion?

In a theoretical model of a positronium atom (a bound system of an electron and a positron, each of mass $m_e$ and charge magnitude $e$), the system's interaction with a weak uniform external magnetic field $\vec{B}$ is governed by the effective one-body relative Hamiltonian $H_{rel} = \frac{(\vec{p} - e\vec{A})^2}{2\mu} + V(r)$, where $\mu$ is the reduced mass of the system, $\vec{p}$ is the relative momentum, and $\vec{A} = \frac{1}{2}(\vec{B} \times \vec{r})$ is the vector potential. Under this model, what is the magnitude of the energy splitting $\Delta E$ between adjacent orbital magnetic sublevels (states differing by magnetic quantum number $\Delta m_l = 1$)?

A uniform straight rod of mass $M=3m$ and length $2R$ is in free space. A small ring of mass $m$ can slide without friction along the rod. Initially, the ring is held at a distance of $R/2$ from the center of the rod, and the entire system is rotating with an angular velocity $\omega_0$ about an axis passing through the system's center of mass and perpendicular to the rod. The ring is then released. What is the areal velocity of the ring relative to the center of the rod at the instant the ring reaches the end of the rod?

A diatomic molecule is modeled as two point masses $m_1 = m \text{ (kg)}$ and $m_2 = 2m \text{ (kg)}$ connected by a massless spring of force constant $k \text{ (N/m)}$ and natural length $r_0 \text{ (m)}$. The molecule rotates freely in space with a constant angular momentum $J \text{ (kg}\cdot\text{m}^2\text{/s)}$ about an axis passing through its center of mass and perpendicular to the internuclear axis. Due to the rotation, the equilibrium separation between the masses extends to $r_e \text{ (m)}$. What is the angular frequency $\omega \text{ (rad/s)}$ of small radial oscillations of the masses about this new equilibrium separation $r_e$?

Two small particles, $P_1$ and $P_2$, having masses $m$ and $2m$, and charges $+q$ and $+2q$ respectively, are connected by an ideal massless spring of spring constant $k$ and natural length zero. The system is placed in a uniform magnetic field $\vec{B} = B \hat{k}$. The particles are projected such that they undergo steady circular motion in the $xy$-plane about their common center of mass, which remains at rest. If the particles rotate in the counter-clockwise direction, what is the steady angular frequency $\omega$ of their relative motion?

An atom of mass $m$, initially in its ground state, falls from rest at a very large distance towards a spherically symmetric planet of mass $M$. At a distance $R$ from the planet's center, the atom absorbs a photon traveling radially outward. The local frequency of the photon at the point of absorption is $\nu$. The absorption excites the atom to an internal state with energy $\Delta E$ above the ground state and brings the atom instantaneously to rest. Assuming non-relativistic mechanics for the atom's motion, which of the following expressions gives the required photon frequency $\nu$? (Here $h$ is Planck's constant, $c$ is the speed of light, and $G$ is the universal gravitational constant).

A perfectly absorbing spherical dust particle of radius $R$ and initial rest mass $m_0$ is initially at absolute zero ($T = 0\text{ K}$) in deep space. It is illuminated by a unidirectional plane electromagnetic wave (laser) of constant intensity $I$. As the particle absorbs the radiation, it heats up and emits blackbody radiation isotropically in its instantaneous rest frame. Eventually, the particle reaches a thermal steady state at temperature $T_{eq}$. Assume the particle's internal thermal energy is given by $U(T) = \beta T^4$, where $\beta$ is a positive constant, and its velocity remains non-relativistic ($v \ll c$) throughout. Which of the following gives the particle's steady-state rest mass $m_{eq}$ and its instantaneous acceleration $a$ at this steady state? (Here, $c$ is the speed of light and $\sigma$ is the Stefan-Boltzmann constant).

A gas discharge tube contains a mixture of Hydrogen ($^1\text{H}$) and Deuterium ($^2\text{H}$) atoms. The atoms are excited and emit light corresponding to the first line of the Balmer series ($n=3 \to n=2$). This emitted light is passed through a Michelson interferometer. As the movable mirror is displaced, the interference fringes periodically wash out (reach minimum visibility) due to the superposition of the slightly different wavelengths from the two isotopes. Assuming the mass of the proton is $M$, the mass of the electron is $m$ (with $M \gg m$), and the mass of the Deuterium nucleus is $2M$, what is the minimum non-zero path difference $\Delta x$ introduced by the interferometer at which the fringes completely wash out? (Let $R_\infty$ be the Rydberg constant for an infinitely massive nucleus).

A monatomic ideal gas is maintained in a container of initial volume $V_0$ at an initial absolute temperature $T_0$. The atoms of the gas have a ground state energy $E_0 = 0$ and a first excited state energy $E_1 = \Delta E$, where $\Delta E \gg k_B T_0$ ($k_B$ is the Boltzmann constant). The gas undergoes a reversible adiabatic expansion to a final volume $V_f = 8 V_0$. Assuming the total emission intensity $I$ (total power radiated by the entire gas) of the spectral line corresponding to the transition $E_1 \rightarrow E_0$ is directly proportional to the total number of atoms in the excited state, which of the following expressions represents the ratio of the final emission intensity $I_f$ to the initial emission intensity $I_0$?

For a hypothetical hydrogen-like ion, the transition corresponding to the first line of the Balmer series emits a photon of wavelength $\lambda_{B1} = \frac{36h}{5 m_e c}$, where $h$ is Planck's constant, $m_e$ is the electron rest mass, and $c$ is the speed of light in vacuum. A linearly polarized photon emitted from the Lyman series limit of this same ion is incident on a free electron initially at rest. The photon undergoes Compton backscattering (scattering angle $\theta = 180^\circ$). Calculate the de Broglie wavelength $\lambda_e$ of the recoiling electron.

A light beam consists of a linearly polarized component of intensity $I_0$ and an unpolarized component of intensity $I_0$. The beam passes through an ideal analyzer rotating at a constant angular velocity $\omega$. The emergent light illuminates a special photovoltaic cell. The cell generates an open-circuit EMF $E(t) = \alpha I(t)$ and has an internal resistance $r(t) = \beta I(t)$, where $I(t)$ is the instantaneous intensity of the incident light, and $\alpha, \beta$ are positive constants. The cell is connected across a fixed coil of radius $a$, having $N$ turns, self-inductance $L$, and zero ohmic resistance. A small metallic ring of radius $b$ ($b \ll a$) and resistance $R_0$ is placed concentrically and coplanar with the coil. Assuming the system has been operating for a long time, what is the time-averaged power dissipated due to eddy currents in the small metallic ring?

Two distant, mutually incoherent point sources $S_1$ and $S_2$ emit linearly polarized light of wavelength $\lambda = 500\text{ nm}$. The electric field vector of the emitted light from both sources makes an angle of $45^\circ$ with the vertical. A telescope is used to observe these sources. The telescope has a rectangular aperture of width $a = 2.0 \times 10^{-3}\text{ m}$ (horizontal) and height $b = 5.0 \times 10^{-3}\text{ m}$ (vertical). A special mask is placed over the aperture: the left half ($-a/2 \le x \le 0$) is covered by an ideal linear polarizer with its transmission axis vertical, and the right half ($0 \le x \le a/2$) is covered by an ideal linear polarizer with its transmission axis horizontal. What is the minimum angular separation $\theta_{min}$ between the two sources along the horizontal direction such that they are just resolved according to the Rayleigh criterion?

An alpha particle of mass $m$ and charge $q$ is projected from infinity with an initial kinetic energy $K_0 = \frac{qQ}{2\pi\epsilon_0 R}$ towards a fixed, uniformly charged non-conducting solid sphere of radius $R$ and total charge $Q$ ($q, Q > 0$). The impact parameter of the alpha particle is $b = \frac{R}{2}$. Assuming the sphere remains fixed and unperturbed, what is the distance of closest approach, $r_{\text{min}}$, of the alpha particle to the center of the sphere?

A particle of rest mass $m_0$ and charge $q$ is projected from the origin with an initial velocity $\vec{v}_0 = \frac{2\sqrt{6}}{5}c \hat{i}$, where $c$ is the speed of light in vacuum. The region $x > 0$ contains a uniform electric field $\vec{E} = E_0 \hat{j}$. After a certain time, the particle's velocity vector makes an angle of $45^\circ$ with the x-axis. Neglecting gravity and radiation losses, what is the transverse displacement $y$ of the particle at this instant?

A fixed, uniformly charged oblate spheroid with total charge $+Q$ is centered at the origin of a Cartesian coordinate system. Its axis of rotational symmetry lies along the $z$-axis. A point charge $+q$ of mass $m$ is projected from a large distance ($x \to -\infty$) with an initial velocity vector $\vec{v} = v_0 \hat{i}$. The initial position of the particle is given by the coordinates $(-\infty, b, c)$, where $b$ and $c$ are positive distances. Let $\vec{L}$ be the angular momentum vector of the particle about the origin. Which of the following correctly describes the conservation of angular momentum for this system during the particle's motion?

A particle of mass $m$ (in $\text{kg}$) moves in a horizontal $x$-$y$ plane under the influence of a conservative central attractive force $\vec{F}_c = -f(r) \hat{r}$ (in $\text{N}$) directed towards the origin $O$. The region is filled with a viscous medium that exerts a drag force on the particle given by $\vec{F}_d = -\frac{\beta}{r^2} \vec{v}$, where $\beta$ is a positive constant (in $\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-1}$), $r$ is the distance from the origin (in $\text{m}$), and $\vec{v}$ is the velocity vector of the particle (in $\text{m/s}$). At time $t=0$, the particle is at a distance $r_0$ (in $\text{m}$) from the origin and is projected with a velocity $v_0$ (in $\text{m/s}$) perpendicular to its position vector. The particle eventually loses all its transverse velocity. Assuming the particle does not reach the origin before its transverse velocity vanishes, what is the total angle $\Delta \theta$ (in $\text{rad}$) swept by the position vector of the particle from $t=0$ until its motion becomes purely radial?

A thin spherical shell of radius $R$ is fixed in space with its center at the origin. It carries a non-uniform surface charge density $\sigma(\theta) = \sigma_0(1 + \cos\theta)$, where $\theta$ is the polar angle measured from the positive $z$-axis, and $\sigma_0$ is a positive constant. A particle of mass $m$ and positive charge $q$ is fired from $z = +\infty$ along the $z$-axis towards the origin. The initial angular momentum of the particle about the origin is zero, and its initial kinetic energy is $K = \frac{15 q \sigma_0 R}{16 \epsilon_0}$. Assuming no energy is lost to radiation, find the $z$-coordinate of the particle at its distance of closest approach to the origin.

Two identical point charges, each of mass $m$ and charge $q$, are initially separated by a very large distance in free space. One charge is initially at rest, and the other is projected towards it with an initial speed $v_0$ and an impact parameter $b$. Given the relation $b = \frac{2 q^2}{4 \pi \epsilon_0 m v_0^2}$, what is the minimum separation distance between the two charges during their subsequent motion? (Assume all physical quantities are expressed in standard SI units).

An alpha particle (mass $m$, charge $+2e$) is projected from a distance $d$ directly towards a fixed, infinite non-conducting sheet with a uniform positive surface charge density $\sigma$. What is the minimum initial speed $v_0$ required for the alpha particle to just reach the surface of the sheet? (Assume the space is vacuum with permittivity $\varepsilon_0$ and neglect gravity).

An alpha particle of mass $m$ and charge $+2e$ is projected from infinity with an initial speed $v_0$ and an impact parameter $b$ towards a fixed heavy nucleus of charge $+Ze$ located at the origin. The entire region contains a static, non-uniform magnetic field given by $\vec{B} = B_0 \left( \frac{x}{a} \hat{i} - \frac{y}{a} \hat{j} \right)$, where $B_0$ and $a$ are positive constants. Due to the combined electric and magnetic forces, the alpha particle follows a complex 3D trajectory. If $r_{min}$ is the distance of closest approach to the nucleus, what is the speed $v$ of the alpha particle at this point of closest approach? (Assume the nucleus remains stationary, $\epsilon_0$ is the vacuum permittivity, and ignore electromagnetic radiation).

A solid, isolated conducting sphere of radius $R$ is fixed in space and carries a net positive charge $Q = \frac{7}{18}q$. A point particle of mass $m$ and positive charge $q$ is projected from infinity directly towards the center of the sphere. What is the minimum initial kinetic energy $K_0$ required for the particle to eventually collide with the surface of the sphere? \n\n(Assume the system is in vacuum with permittivity $\epsilon_0$ and ignore gravity. The potential energy of the particle must account for the polarization of the conducting sphere).

Consider a hypothetical atom consisting of a stationary nucleus of mass $M$ and charge $+Ze$, and an electron of mass $m$ and charge $-e$. Assume that both the electrostatic and gravitational forces between the nucleus and the electron are significant and act simultaneously. Using the Bohr model of quantization of angular momentum, determine the magnitude of the total energy of the electron in its $n$-th allowed circular orbit. Let $h$ be Planck's constant, $\hbar = \frac{h}{2\pi}$, $G$ be the universal gravitational constant, and $\epsilon_0$ be the permittivity of free space. All quantities are in SI units and the final energy is expressed in Joules ($\text{J}$).

A hypothetical atom consists of a massive stationary nucleus of radius $R$ and total charge $+Q$. The charge within the nucleus is distributed with a spherically symmetric volume charge density $\rho(r) = c r$ for $r \le R$, where $c$ is a positive constant. A particle of mass $m$ and charge $-q$ orbits the nucleus in a circular path of radius $r_n$ (where $r_n < R$). Assuming the Bohr quantization condition for angular momentum is valid for this system, the time period $T_n$ of the particle in the $n^{\text{th}}$ allowed orbit is proportional to $n^x$. What is the value of $x$?

An electron (mass $m$, charge $-e$) is constrained to move along a frictionless, non-conducting circular ring of radius $r$. Initially, there is no magnetic field, and the electron is in a quantized stationary state with a mechanical angular momentum $\vec{L}_i = n\hbar \hat{k}$ (where $n$ is a positive integer and $\hbar = h/2\pi$). A uniform time-varying magnetic field $\vec{B}(t) = B(t) \hat{k}$ is then applied across the region, slowly increasing from $B=0$ to a constant final value $B=B_0$. Which of the following expressions represents the final mechanical angular momentum $\vec{L}_f$ of the electron?