1.Two identical pith balls, each of mass $m$ and charge $q$, are suspended from a common point by two strings of length $l$. If the angle between the strings is $2\theta$, derive an expression for the charge $q$ in terms of $m, l, \theta,$ and $g$.
2.In the above setup, if the entire system is placed in a satellite in a gravity-free space, what will be the angle between the strings and the tension in each string?
3.A point charge $q$ of mass $m$ is constrained to move along the x-axis. Two identical positive charges $+Q$ are fixed at $(0, a)$ and $(0, -a)$. If $q$ is displaced slightly along the x-axis ($x \ll a$), show that it executes Simple Harmonic Motion (SHM) if $q$ is negative, and find its time period.
4.A particle of mass $m$ and charge $-q$ enters a uniform electric field $\vec{E}$ with velocity $v_0$ perpendicular to the field. Find the transverse deflection $y$ after it has traveled a horizontal distance $x$.
5.Two free positive charges $4q$ and $q$ are separated by a distance $l$. What charge $Q$ must be placed between them so that the entire system is in equilibrium? State the magnitude, sign, and exact position of $Q$.
6.Is the equilibrium of the charge $Q$ in Q5 stable, unstable, or neutral when displaced strictly along the line joining the charges?
7.A simple pendulum of length $L$ has a bob of mass $m$ carrying a charge $+q$. A uniform electric field $E$ acts vertically downwards. Find the new time period of small oscillations.
8.Solve Q7 if the uniform electric field $E$ is applied horizontally instead.
9.A bead of mass $m$ and charge $q$ can slide without friction on a horizontal wire. A charge $+Q$ is fixed at a distance $d$ below the wire. Find the velocity of the bead as a function of its horizontal position $x$ if it is released from rest at $x = x_0$. (Assume only electrostatic forces).
10.An electron is projected with an initial speed $v_0$ directly towards the center of a uniformly charged stationary proton. Find the distance of closest approach using Coulomb forces.
11.Derive the expression for the electric field at a distance $x$ from the center, along the axis of a uniformly charged circular ring of radius $R$ and total charge $Q$.
12.Using the result from Q11, find the value of $x$ where the electric field is maximum, and find the value of $E_{max}$.
13.A uniformly charged semi-circular arc of radius $R$ carries a linear charge density $\lambda$. Calculate the electric field at the center of the semicircle.
14.A wire of length $L$ carries a total charge $Q$ uniformly distributed. Find the electric field at a point $P$ located at a distance $a$ along the axis of the wire, from one of its ends.
15.A uniformly charged finite line segment has linear charge density $\lambda$ and length $2L$. Find the electric field at a distance $y$ from its center on its perpendicular bisector.
16.Show that as $L \to \infty$ in Q15, the electric field reduces to the standard formula for an infinite line charge.
17.A circular disc of radius $R$ has a uniform surface charge density $\sigma$. Using integration of rings, derive the electric field at an axial distance $x$ from its center.
18.Show that for points very close to the disc ($x \ll R$), the disc behaves like an infinite plane sheet of charge.
19.A non-conducting solid hemisphere of radius $R$ has a uniform volume charge density $\rho$. Find the electric field at the center of its flat base.
20.A thread carries a linear charge density $\lambda = \lambda_0 \cos\theta$ over a semicircular arc of radius $R$. Find the net electric field at the center of the arc.
21.A point charge $q$ is placed at distance $a/2$ directly above the center of a square of side $a$. What is the electric flux through the square?
22.A point charge $q$ is placed at the center of one of the faces of a cube of edge $a$. Find the flux emerging from the opposite face of the cube.
23.A charge $Q$ is located at an edge (midpoint of a side) of a cube. What is the total flux passing through the cube?
24.Using the concept of solid angles, write the formula for the flux through a circular disc of radius $R$ due to a point charge $q$ placed at an axial distance $x$ from the disc.
25.A solid insulating sphere of radius $R$ has a non-uniform volume charge density given by $\rho = \rho_0 (r/R)$, where $r$ is the radial distance. Use Gauss's Law to find the electric field at $r < R$ and $r > R$.
26.A long solid dielectric cylinder of radius $R$ has a uniform volume charge density $\rho$. Find the electric field at a distance $r$ ($r < R$) from its axis.
27.An infinite sheet of charge has a circular hole of radius $R$. A particle of mass $m$ and charge $-q$ is placed on the axis of the hole at a distance $x$ ($x \ll R$). Prove it will undergo SHM and find its frequency.
28.An uncharged spherical thick conducting shell has inner radius $R_1$ and outer radius $R_2$. A point charge $+q$ is placed at the center. Find the surface charge densities on the inner and outer surfaces.
29.In Q28, plot the variation of electric field $E$ as a function of distance $r$ from the center for $0 < r < \infty$.
30.A spherical cavity of radius $a$ is made inside a solid non-conducting uniformly charged sphere of radius $R$ and volume charge density $\rho$. The center of the cavity is at a distance vector $\vec{d}$ from the center of the sphere. Prove that the electric field inside the cavity is uniform.
31.Find the electrostatic force of interaction between two short collinear dipoles of moments $p_1$ and $p_2$ separated by a distance $r$.
32.In Q31, how does the force depend on the distance $r$? ($F \propto 1/r^n$. Find $n$).
33.Find the torque exerted by a short dipole $p_1$ on another short dipole $p_2$ placed at an equatorial distance $r$, such that $\vec{p_1}$ is parallel to $\vec{p_2}$.
34.An electric dipole of moment $\vec{p}$ is placed at the origin oriented along the x-axis. Find the angle made by the electric field vector with the x-axis at a point $(r, \theta)$ in polar coordinates.
35.A dipole with moment $\vec{p} = p \hat{i}$ is placed in a non-uniform electric field $\vec{E} = \alpha x \hat{i}$. Find the net force on the dipole.
36.A dipole with moment $p$ is suspended by a string of moment of inertia $I$ in a uniform electric field $E$. Derive its time period for small angular oscillations.
37.Two point charges $+q$ and $-q$ are moving in a circular path of radius $R$ with angular velocity $\omega$ at diametrically opposite points. Is this system an electric dipole? What is its average dipole moment over one full cycle?
38.Calculate the work done to rotate a dipole from $\theta = 60^\circ$ to $\theta = 180^\circ$ in a uniform field $E$.
39.A small electric dipole is placed at the center of a uniformly charged ring of radius $R$ and charge $Q$. The dipole is constrained to rotate in the plane of the ring. What is the net torque on the dipole?
40.What is the electric flux through a closed Gaussian sphere of radius $R$ centered at the origin, if a short dipole $\vec{p}$ is kept at $(R/2, 0, 0)$?
41.Two charges $2 \mu\text{C}$ and $8 \mu\text{C}$ are placed $15\text{ cm}$ apart. The electric field is zero at a point from the $2 \mu\text{C}$ charge at a distance of $x\text{ cm}$. Find $x$.
42.A charge $Q = 10 \mu\text{C}$ is distributed uniformly over a thin spherical shell of radius $R = 10\text{ cm}$. The electric field at a distance of $5\text{ cm}$ from the center is $Y \times 10^5\text{ N/C}$. Find $Y$.
43.A solid metallic sphere of radius $a$ is surrounded by a concentric metallic shell of inner radius $2a$ and outer radius $3a$. A charge $Q$ is given to the inner sphere and $-2Q$ to the outer shell. The ratio of electric field at $r = 1.5a$ to $r = 4a$ is $x : 1$. Find $|x|$.
44.An infinite line charge has $\lambda = 4 \times 10^{-6}\text{ C/m}$. An electron is revolving in a circular orbit around it. The kinetic energy of the electron is $K \times 10^{-15}\text{ J}$. Find $K$.
45.A charge $q = 8.85 \mu\text{C}$ is placed at the center of a cube. The flux passing through one face is $Z \times 10^5\text{ N m}^2\text{/C}$. Find $Z$.
46.Two perfectly identical conducting spheres carrying charges $+5\text{ mC}$ and $-1\text{ mC}$ are placed at a distance $r$. They attract with force $F$. They are touched and placed back at distance $r$. They now repel with force $F'$. The ratio $F/F'$ is $N$. Find $N$.
47.A water drop of mass $10^{-6}\text{ kg}$ and charge $10^{-6}\text{ C}$ is suspended in a uniform electric field. The magnitude of the electric field is $E\text{ V/m}$. Find $E$. (Take $g = 10\text{ m/s}^2$).
48.A short dipole has $p = 4 \times 10^{-9}\text{ C m}$. The electric field at an axial point at $r = 0.2\text{ m}$ is $E \times 10^3\text{ N/C}$. Find $E$.
49.An electric field is given by $\vec{E} = (10x\hat{i} + 20y\hat{j})\text{ N/C}$. The flux passing through a square of side $1\text{ m}$ lying in the y-z plane at $x=2\text{ m}$ is $\Phi$. Find $\Phi$.
50.If the dielectric constant of a medium between two point charges increases by $100\%$, the electrostatic force between them decreases by $P\%$. Find $P$.
51.Two infinite plane sheets carrying uniform charge densities $+\sigma$ and $-\sigma$ are parallel. The ratio of the electric field between them to the field outside is $X$. Find $X$ (use $\infty$ if not defined, or write the exact integer).
52.A square wire frame of side $L$ carries total charge $Q$. The electric field at the center is $E$. Find $E$.
53.The force of repulsion between two point charges is $100\text{ N}$. If the distance between them is increased by $25\%$, the new force is $F\text{ N}$. Find $F$.
54.A simple pendulum with bob mass $m$ and charge $q$ is in a horizontal uniform field $E$. At equilibrium, the string makes angle $\theta = 45^\circ$ with the vertical. Find the ratio $qE/mg$.
55.A charge $q$ is placed at distance $R$ from the center of an uncharged conducting solid sphere of radius $R/2$. The electric field at the center of the sphere due to the *induced charges* has magnitude $k q / (R^x)$. Find $x$.
56.Two charges $4q$ and $-q$ are at $(0,0)$ and $(a,0)$. A third charge is in equilibrium at point $x$. Find $x$ in terms of $a$.
57.A solid insulating sphere has $\rho(r) = \rho_0 (1 - r/R)$. At what value of $r$ ($r < R$) is the electric field maximum? Express as a fraction of $R$.
58.An electric dipole experiences a maximum torque of $10\text{ Nm}$ in a field. What is the work done to rotate it from stable to unstable equilibrium?
59.A charge $Q$ is distributed over two concentric hollow spheres of radii $r$ and $R$ ($R>r$) such that surface charge densities are equal. Find the charge on the inner sphere.
60.A very long straight thread carrying linear charge density $\lambda$ penetrates a cube of side $L$ perpendicularly through the centers of two opposite faces. Find the electric flux through the cube.
