1.
A particle of specific charge $\alpha = q/m$ is projected from the origin with velocity $\vec{v} = v_0\hat{i}$ in a region containing uniform crossed fields $\vec{E} = E_0\hat{j}$ and $\vec{B} = B_0\hat{k}$. Derive the parametric equations of its trajectory $(x(t), y(t))$ and identify the condition for it to follow a cycloidal path.
2.
A uniform magnetic field $\vec{B} = B_0\hat{k}$ exists in a cylindrical region of radius $R$. A proton is projected radially inward from the edge of the cylinder with velocity $v$. Determine the minimum velocity $v_{min}$ required for the proton to pass through the exact center of the cylinder.
3.
A particle of mass $m$ and charge $q$ is released from rest in a region where parallel electric and magnetic fields $\vec{E} = E_0\hat{k}$ and $\vec{B} = B_0\hat{k}$ exist. If an initial perturbation gives it a small transverse velocity $v_0\hat{i}$, derive the expression for the pitch of its helical path as a function of time $t$.
4.
Two identical charges $+q$ and mass $m$ are projected simultaneously from the origin with the same speed $v_0$, but in mutually perpendicular directions in the xy-plane. A uniform magnetic field $\vec{B} = B_0\hat{k}$ is present. Calculate the maximum separation between them during their subsequent motion.
5.
A charged particle enters a region of width $d$ containing a uniform magnetic field $\vec{B}$ perpendicular to its initial velocity $v$. Derive the minimum condition on $v$ such that the particle successfully crosses the magnetic field region without being reflected back.
6.
A non-conducting disc of radius $R$ carries a uniform surface charge density $\sigma$. It rotates with an angular velocity $\omega$ about its central axis. Calculate the exact magnetic force experienced by it when placed in a uniform radial magnetic field $B_r = B_0$ diverging from its center.
7.
A positive charge $q$ moves along the x-axis with a relativistic speed $v$. Derive the expression for the transverse magnetic field $B_y$ observed by a stationary observer at a distance $r$ on the y-axis, taking into account relativistic corrections.
8.
An electron is trapped in a "magnetic bottle" formed by a non-uniform magnetic field $B(z)$ that increases towards the ends. Conceptually and mathematically prove the condition for magnetic mirroring, determining the critical angle $\theta_c$ of the velocity vector to the z-axis that prevents escape.
9.
A charged particle moves in a region where $\vec{B} = \alpha y \hat{k}$ (a linearly varying magnetic field). If it is projected from the origin with velocity $v_0\hat{i}$, obtain an equation relating its x-coordinate to its y-coordinate during the motion.
10.
In a mass spectrometer, ions of mass $m_1$ and $m_2$ ($m_1 < m_2$) are accelerated by a potential $V$ and enter a uniform magnetic field $B$ perpendicularly. Derive the expression for the spatial separation between the points where they strike the photographic plate after completing a semicircle.
11.
Derive the magnetic field at the center of a regular polygon of $n$ sides, inscribed in a circle of radius $R$, carrying a steady current $I$. Use this to show that as $n \to \infty$, the field approaches that of a circular loop.
12.
A thin insulating disc of radius $R$ is uniformly charged with total charge $Q$. It rotates with angular velocity $\omega$ about an axis passing through its center and perpendicular to its plane. Calculate the magnetic field at a point on its axis at a distance $z$ from the center.
13.
Calculate the magnetic field at the center of an Archimedean spiral loop defined by polar equation $r = a + b\theta$ (where $0 \le \theta \le 2\pi$), carrying a current $I$.
14.
An infinitely long wire is bent at a right angle at the origin. One semi-infinite part lies along the positive x-axis and the other along the positive y-axis. Find the magnetic field vector at a point $(a, a, 0)$.
15.
A solid sphere of radius $R$ has a uniform volume charge density $\rho$. If it rotates about its z-axis diameter with angular velocity $\omega$, evaluate the magnetic field produced exactly at its center.
16.
Two identical parallel circular coils of radius $R$ and $N$ turns are separated by a distance $R$ (Helmholtz coils). They carry current $I$ in the same direction. Prove that the first and second derivatives of the magnetic field with respect to axial distance $x$ vanish at the midpoint between them.
17.
Find the magnetic field at the focus of a parabolic wire $y^2 = 4ax$ carrying a current $I$, stretching from $y = -\infty$ to $y = +\infty$.
18.
A current $I$ flows along the perimeter of an elliptical loop with semi-major axis $a$ and semi-minor axis $b$. Determine the exact magnetic field at the geometric center of the ellipse.
19.
A tightly wound flat spiral coil of inner radius $a$ and outer radius $b$ has $N$ total turns. It carries a current $I$. Integrate to find the magnetic field at the center of the spiral.
20.
Three infinitely long straight wires lying along the x, y, and z coordinate axes carry identical currents $I$. Determine the magnitude and direction of the magnetic field at point $(a, a, a)$.
21.
An infinitely long solid cylindrical conductor of radius $R$ carries a current whose volume current density varies with radial distance as $\vec{J} = J_0(r/R)\hat{k}$. Determine the magnetic field $\vec{B}$ for regions $r < R$ and $r > R$.
22.
A long conducting cylinder of radius $R$ has a cylindrical off-center cavity of radius $a$. The axis of the cavity is displaced from the main axis by vector $\vec{d}$. If the cylinder carries a uniform current density $\vec{J}$, prove using superposition that the magnetic field inside the cavity is perfectly uniform. Find its magnitude.
23.
Find the magnetic field in the space between two infinite parallel sheets of current located at $z = -a$ and $z = +a$. The upper sheet carries surface current density $\vec{K} = K_0\hat{i}$ and the lower sheet carries $\vec{K} = -K_0\hat{i}$.
24.
A toroid has a rectangular cross-section with inner radius $a$, outer radius $b$, and height $h$. It consists of $N$ tightly wound turns carrying current $I$. Calculate the total magnetic flux $\Phi_B$ enclosed by the cross-section of the toroid.
25.
An infinite conducting slab of thickness $2d$ (lying between $y = -d$ and $y = +d$) carries a uniform volume current density $\vec{J} = J_0\hat{i}$. Derive expressions for the magnetic field $\vec{B}$ everywhere (inside and outside the slab) as a function of $y$.
26.
Consider a coaxial cable where the inner solid conductor of radius $a$ carries current $I_0$ uniformly, and the outer cylindrical shell of inner radius $b$ and outer radius $c$ carries return current $I_0$ with a non-uniform density $\vec{J} = \alpha/r$. Find $\alpha$ and compute the field in the outer shell ($b < r < c$).
27.
A hypothetical solenoid extends from $x = 0$ to $x = \infty$ (a semi-infinite solenoid). It has $n$ turns per unit length and carries current $I$. Evaluate the magnetic field perfectly at the face $x = 0$ using Ampere's law logic combined with symmetry arguments.
28.
A long straight wire carries a steady current. Is it mathematically valid to apply Ampere's Law taking a square Amperian loop around it? Does it yield a solvable form for $\vec{B}$? Rigorously explain the limitations of Ampere's Law geometry.
29.
Calculate the work done by the magnetic field in moving a unit magnetic monopole (hypothetical) completely around a wire carrying a current of 1 A. (Relate to Dirac's quantization condition conceptually).
30.
A non-conducting infinite cylinder of radius $R$ has a uniform surface charge density $\sigma$. It accelerates from rest with angular acceleration $\alpha$. Formulate the time-varying magnetic field generated inside the cylinder.
31.
Calculate the force per unit length exerted on a long straight wire carrying current $I_1$, placed parallel to and at a distance $d$ from an infinitely wide, flat conducting sheet carrying a uniform surface current density $K$.
32.
A rigid semi-circular wire of radius $R$ carrying current $I_2$ lies in the xy-plane. It is situated with its diameter parallel to a long straight wire carrying current $I_1$ along the x-axis at a distance $d$ ($d > R$). Determine the net magnetic force on the semi-circular loop.
33.
Three infinitely long parallel wires pass through the vertices of an equilateral triangle of side $a$ in the xy-plane. They carry identical currents $I$ in the same direction (+z). Find the magnitude and direction of the net magnetic force per unit length on one of the wires.
34.
A flexible, infinitely long conducting ribbon of width $w$ carries a uniform current $I$ distributed over its width. Formulate the force per unit length it exerts on an adjacent parallel thin wire carrying current $I_0$ at a distance $a$ from the near edge of the ribbon.
35.
A flexible conducting circular ring of radius $R$ and mass per unit length $\lambda$ carries a steady current $I$. It is placed in a uniform external magnetic field $B$ perpendicular to its plane. Formulate the tension $T$ generated within the ring. Under what condition will the ring physically snap (assuming breaking tension $T_{max}$)?
36.
Two very long straight wires, one along the x-axis carrying $I_1$ and another at $z=d$ parallel to the y-axis carrying $I_2$, are mutually skew. Calculate the total torque exerted by the first wire on a finite length $2L$ of the second wire centered around the z-axis.
37.
An infinitely long wire carries a current $I_1$. A finite wire of length $L$ carrying current $I_2$ is placed coplanar to it, but aligned at an angle $\theta$ relative to the infinite wire. The closest end is at distance $a$. Set up the integral and solve for the total magnetic force on the finite wire.
38.
A square loop of side $a$ carrying current $I_2$ is moving with constant velocity $v$ away from a very long straight wire carrying current $I_1$. Determine the instantaneous magnetic force opposing its motion when the near side is at distance $r$.
39.
A conducting liquid column of radius $R$ carries a heavy axial current $I$. Explain the "Pinch Effect" (magnetic self-compression) using the Lorentz force, and calculate the radial pressure difference required to balance this magnetic pinch.
40.
Find the exact expression for the mutual interaction force between two identical, small magnetic dipoles (moment $\vec{m}$) separated by distance $r$, assuming their moments are strictly collinear and parallel.
41.
A uniformly charged solid sphere (charge $Q$, mass $M$, radius $R$) rotates with angular velocity $\omega$. Prove that its magnetic dipole moment $\vec{\mu}$ and angular momentum $\vec{L}$ satisfy the relation $\vec{\mu} = \frac{Q}{2M}\vec{L}$. Does this relation hold for a rotating disc?
42.
A planar current loop is formed by connecting points $(a,0,0)$, $(0,a,0)$, and $(0,0,a)$ with straight wires carrying current $I$. The loop is placed in a uniform magnetic field $\vec{B} = B_0(\hat{i} + \hat{j} + \hat{k})$. Calculate the net torque on this 3D triangular loop.
43.
A small magnetic dipole of moment $\vec{m} = m_0\hat{i}$ is located at the origin. Another identical dipole is placed at $(r, 0, 0)$. Determine the minimum external work required to slowly rotate the second dipole by 90° so its moment points in the $+\hat{j}$ direction.
44.
A rigid wire loop of arbitrary shape area $A$ carries a current $I$ in a non-uniform magnetic field $\vec{B} = \alpha x \hat{i} + \beta y \hat{j} + \gamma z \hat{k}$. If the loop lies strictly in the xy-plane, determine the net translational force acting on it. (Hint: Use Maxwell's equation $\nabla \cdot \vec{B} = 0$).
45.
A magnetic needle (dipole moment $\mu$, moment of inertia $I_{cm}$) is placed in a uniform magnetic field $B$. If displaced by a small angle $\theta$ and released, it undergoes Simple Harmonic Motion. Derive the exact expression for its time period $T$.
46.
A non-conducting thin ring of radius $R$ and mass $m$ carries a uniform linear charge density $\lambda$. It is given an initial angular velocity $\omega_0$ and placed in a rough horizontal plane in the presence of a vertical magnetic field $B$. Formulate its deceleration profile and time taken to stop.
47.
Explain mathematically the phenomenon of Larmor Precession. Derive the precession angular frequency $\omega_L$ of a magnetic dipole with gyromagnetic ratio $\gamma$ placed in a strong uniform magnetic field $B_0$.
48.
A spherical shell of radius $R$ carries a surface charge density $\sigma = \sigma_0 \cos\theta$ (where $\theta$ is the polar angle). If it rotates with $\omega$ about the z-axis, determine its effective magnetic dipole moment.
49.
An electron moves in a circular orbit under a central force $F = -kr$. Determine the ratio of its orbital magnetic moment to its orbital angular momentum. Does it differ from the standard Bohr orbit?
50.
A small rectangular coil of $N$ turns, area $A$, carrying current $I$ is suspended from an arm of a delicate balance. A long straight wire carrying $I_0$ passes symmetrically directly underneath it. Determine the change in balance weight reading due to the localized dipole-wire interaction.
51.
A Ballistic Galvanometer is designed to measure transient charge $Q$ rather than steady current. Formulate the differential equation of its motion and prove that the maximum angular deflection (throw) $\theta_0$ is strictly proportional to $Q$, assuming $Q$ passes in a time $t \ll$ time period of oscillation.
52.
A moving coil galvanometer is heavily shunted to act as an ammeter. Explain the physical mechanism of "Electromagnetic Damping" induced by the shunt and formulate the damping torque $T_D$ using Faraday's/Lenz's laws assuming angular velocity $\omega$.
53.
The suspension wire of a galvanometer has a torsional constant $C = \frac{\pi \eta r^4}{2l}$. If the wire's length is decreased by 10% and radius is decreased by 5% due to thermal contraction, calculate the percentage error introduced in its current sensitivity readings.
54.
A multi-range voltmeter is constructed using a single galvanometer (Resistance $R_g$, full scale current $I_g$) by connecting three resistors $R_1, R_2, R_3$ in a specialized tapped series configuration. Derive the specific values for $R_1, R_2, R_3$ to yield ranges $V_1, V_2,$ and $V_3$.
55.
An ammeter is constructed from a galvanometer ($R_g$, $I_g$). If an external temperature increase causes the copper coil's resistance to rise by $\alpha \Delta T$ but the manganin shunt remains constant, establish an expression for the fractional error in the ammeter reading at the higher temperature.
56.
In the absence of a perfectly radial magnetic field (e.g., a uniform unidirectional field $B_0$), derive the non-linear relationship between the steady current $I$ and the angular deflection $\theta$ of the coil.
57.
Thomson's Half-Deflection Method: A galvanometer of resistance $G$ is in series with resistance $R$ and a battery, giving deflection $\theta$. A shunt $S$ is applied across $G$, and $R$ is adjusted to $R'$ to make the deflection $\theta/2$. Derive the formula connecting $G, S, R,$ and $R'$.
58.
To critically damp a moving coil galvanometer, the total circuit resistance must drop below a critical threshold. Formulate the characteristic equation of the damped rotational motion $I\ddot{\theta} + b\dot{\theta} + k\theta = 0$ and find the condition on $b$ for critical damping.
59.
If two identical moving coil galvanometers are connected exactly in series in a circuit, how does their combined effective voltage sensitivity compare to the voltage sensitivity of a single isolated unit? Provide mathematical proof.
60.
A galvanometer has $N$ turns, area $A$, and suspension constant $k$. A small mirror attached to the coil reflects a laser beam onto a scale at distance $D$. If the current is $I$, derive the exact displacement $y$ of the laser spot on the flat scale, and identify the small-angle approximation limit.
