1.The electric potential in a region is given by $V(x, y, z) = 6x - 8xy^2 - 8y + 6yz$. Calculate the magnitude of the electric field at the point $(1, 1, 1)$ m.
2.Two concentric spherical conducting shells of radii $R$ and $2R$ carry charges $Q$ and $2Q$ respectively. If the inner shell is earthed, find the new charge on the inner shell.
3.In the previous problem, what is the charge on the inner shell if the outer shell is earthed instead of the inner one?
4.A non-conducting sphere of radius $R$ has a volume charge density $\rho = \rho_0 (1 - r/R)$, where $r$ is the distance from the center. Find the electric potential at the center of the sphere.
5.A uniform electric field $E$ exists in the x-y plane at an angle of $30^\circ$ with the x-axis. Find the potential difference $V_A - V_B$ between two points $A(2, 0)$ and $B(0, 4)$.
6.An electric dipole of moment $p$ is placed at the origin along the x-axis. Find the locus of points in the x-y plane where the electric potential is zero.
7.A point charge $q$ is placed at a distance $r$ from the center of an uncharged conducting sphere of radius $R$ ($r > R$). Find the potential of the sphere.
8.Three point charges $q, -2q, q$ are located at $(0, a, 0), (0, 0, 0), (0, -a, 0)$ respectively. Find the expression for potential at a distant point $(r, \theta)$ in the x-y plane. What type of multipole is this?
9.A thin wire of length $L$ carries a uniform linear charge density $\lambda$. Find the electric potential at a point located at a distance $d$ from one of its ends, along the line of the wire.
10.Two identical thin rings, each of radius $R$, are coaxially placed at a distance $R$. They carry charges $Q_1$ and $Q_2$. Find the work done in moving a charge $q$ from the center of one ring to the center of the other.
11.Assume that an electric field $\vec{E} = 30x^2 \hat{i}$ exists in space. Calculate the potential difference $V_A - V_O$, where $V_O$ is the potential at the origin and $V_A$ is the potential at $x = 2 \text{ m}$.
12.Four charges $q, -q, q, -q$ are placed at the corners $A, B, C, D$ of a square of side $a$. Find the electric potential at the center. Also find the electric field at the center.
13.A positive charge $Q$ is uniformly distributed over a circular ring of radius $R$. A particle of mass $m$ and negative charge $-q$ is placed at its center. If it is displaced slightly along the axis and released, show that it will undergo SHM and find its time period.
14.Two infinite plane parallel sheets of charge having charge densities $\sigma_1$ and $\sigma_2$ ($\sigma_1 > \sigma_2$) are placed at a distance $d$ apart. Find the potential difference between them.
15.Determine the electric potential at the surface of a gold nucleus. The radius is $6.6 \times 10^{-15} \text{ m}$ and atomic number $Z = 79$. Given $1 / (4\pi\epsilon_0) = 9 \times 10^9 \text{ N m}^2 / \text{C}^2$.
16.A uniformly charged thin spherical shell of radius $R$ carries a total charge $Q$. A small circular hole of radius $r$ ($r \ll R$) is cut from it. Find the electric field and potential at the center of the hole.
17.An equipotential surface is defined by $x^2 + y^2 + z^2 = c^2$. What is the corresponding shape of the electric field lines?
18.If $V = \frac{A}{r} + \frac{B}{r^2}$, find the expression for electric field intensity $\vec{E}$.
19.A solid conducting sphere having a charge $Q$ is surrounded by an uncharged concentric conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be $V$. If the shell is now given a charge of $-3Q$, what will be the new potential difference?
20.A charge $+q$ is fixed at the origin. A dipole of moment $\vec{p} = p_0 \hat{i}$ is placed at $(x, 0)$. Find the force exerted by the point charge on the dipole.
21.Three point charges $q, 2q, 8q$ are to be placed on a straight line of length $9 \text{ cm}$. Find the positions where the charges should be placed such that the potential energy of this system is minimum.
22.Calculate the self-potential energy of a uniformly charged solid sphere of radius $R$ and total charge $Q$.
23.Two point charges $Q_1$ and $Q_2$ are placed at distance $r$ and $2r$ from the origin on the x-axis. If the electric potential energy of the system including a charge $q$ at the origin is zero, find the ratio $Q_1/Q_2$.
24.An alpha particle with kinetic energy $10 \text{ MeV}$ is heading towards a stationary tin nucleus ($Z=50$). Calculate the distance of closest approach.
25.Two identical particles of mass $m$ and charge $q$ are projected towards each other from infinity with speed $v$ each. Find the minimum separation between them.
26.A particle of mass $m$ and charge $q$ is released from rest in a uniform electric field $E$. Find its kinetic energy after it has traveled a distance $x$.
27.A dipole with dipole moment $p$ is placed in a non-uniform electric field $\vec{E} = \alpha x \hat{i}$. Find the net force on the dipole if it is placed at the origin and aligned with the x-axis.
28.Two dipoles $\vec{p}_1$ and $\vec{p}_2$ are placed along the same axis at a distance $r$ apart. Find the interaction potential energy between them.
29.Find the work done by the external agent in slowly rotating an electric dipole of moment $\vec{p}$ from parallel to anti-parallel position in a uniform electric field $\vec{E}$.
30.A charge $Q$ is uniformly distributed over a spherical volume of radius $R$. Determine the electrostatic energy density at a distance $r < R$ from the center.
31.Four identical charges $q$ are brought from infinity to the corners of a tetrahedron of side length $a$. Calculate the total work done.
32.A proton is accelerated through a potential difference of $1 \text{ MV}$. Calculate its final momentum and de Broglie wavelength.
33.A drop of water carrying a charge of $3 \times 10^{-6} \text{ C}$ is kept in equilibrium by a vertical electric field of $10^5 \text{ V/m}$. Find the mass of the drop. ($g = 10 \text{ m/s}^2$)
34.Two small spheres, each of mass $m$ and charge $q$, are suspended from a common point by insulating threads of length $l$. If the angle between the threads is $2\theta$, express $q$ in terms of $m, l, \theta, g$.
35.Find the interaction energy of a point charge $q$ placed at the center of a uniformly charged cubic surface of side $a$ and total charge $Q$.
36.A positive point charge $q$ is fixed at origin. A dipole $\vec{p}$ is placed at $\vec{r}$ with its moment directed radially outward. Is the equilibrium stable or unstable? Find the interaction energy.
37.A spherical shell of radius $R_1$ with uniform charge $q$ has a point charge $q_0$ at its center. Find the work done to expand the shell to a radius $R_2$.
38.Determine the work done in bringing a charge $q$ from infinity to the center of the base of a charged solid cone of base radius $R$, height $h$, and uniform volume charge density $\rho$.
39.Two concentric shells of radii $r_1$ and $r_2$ ($r_2 > r_1$) have charges $q_1$ and $q_2$. What is the electrostatic potential energy of this system?
40.A charged particle is projected with velocity $v_0$ directly towards a fixed point charge of same sign. If $v_0$ is doubled, how does the distance of closest approach change?
41.A parallel plate capacitor is filled with a dielectric whose dielectric constant varies as $K(x) = K_0 + \alpha x$, where $x$ is the distance from one plate. If the plate separation is $d$ and area is $A$, find the capacitance.
42.Find the equivalent capacitance of an infinite ladder network of capacitors consisting of alternating series capacitors $C_1$ and parallel capacitors $C_2$.
43.A capacitor is made of two square plates of side $a$ separated by distance $d$. A dielectric slab of constant $K$ and thickness $d$ is inserted a distance $x$ into the capacitor. Find the capacitance as a function of $x$.
44.In the previous problem, if the capacitor is connected to a constant voltage source $V$, find the force exerted by the electric field on the dielectric slab.
45.In Problem 43, if the capacitor was charged to $Q$ and isolated before the slab was inserted, find the force on the dielectric slab as a function of $x$.
46.Twelve identical capacitors of capacitance $C$ are connected along the edges of a cube. Find the equivalent capacitance across a body diagonal.
47.For the same cube of capacitors, find the equivalent capacitance across a face diagonal.
48.A parallel plate capacitor is completely filled with two dielectrics of constants $K_1$ and $K_2$. In case (a) they split the thickness $d/2$ each. In case (b) they split the area $A/2$ each. Find the ratio of equivalent capacitances $C_a / C_b$.
49.A spherical capacitor consists of inner radius $a$ and outer radius $b$. The space between them is filled with a dielectric of constant $K = K_0 / r$, where $r$ is the distance from the center. Find its capacitance.
50.Three conducting plates A, B, and C of area $A$ are kept parallel to each other at distances $d$ and $2d$ respectively. A and C are connected by a wire. Find the equivalent capacitance between B and the A-C combination.
51.Two capacitors $C_1 = 2 \mu\text{F}$ and $C_2 = 4 \mu\text{F}$ are charged to $100 \text{ V}$ and $50 \text{ V}$ respectively. They are connected in parallel such that plates of opposite polarity are joined. Find the heat dissipated.
52.A parallel plate capacitor has plates of area $A$ and separation $d$. A metal slab of thickness $t$ ($t < d$) and area $A$ is inserted midway between the plates. What is the new capacitance?
53.If the breakdown voltage of a material is $V_b$ and its dielectric constant is $K$, what is the maximum energy that can be stored in a parallel plate capacitor of volume $V_{ol}$ filled with this material?
54.A capacitor $C$ is charged to $V_0$. It is then connected across an uncharged inductor $L$. What is the maximum current in the circuit? (LC Oscillation context applied to electrostatics energy conservation).
55.Find the equivalent capacitance between points A and B of an infinite grid of identical capacitors $C$ formed by a square lattice.
56.A spherical drop of capacitance $1 \mu\text{F}$ is broken into 8 identical smaller drops. Find the capacitance of each smaller drop.
57.A variable capacitor has $N$ parallel semi-circular plates of radius $R$, separated by distance $d$. Alternate plates are connected together to form a rotor and stator. If the rotor is turned through angle $\theta$, find the capacitance.
58.A parallel plate capacitor is charged to $V$. A dielectric slab of constant $K$ is introduced. The battery is kept connected. Find the ratio of energy stored before and after.
59.An air capacitor of capacitance $C_0$ is connected to a battery of voltage $V$. A dielectric of constant $K$ is inserted while the battery remains connected. Find the work done by the battery during the process.
60.A capacitor with air between its plates has a capacitance of $8 \mu\text{F}$. The separation is now divided equally into two halves. The lower half is filled with dielectric $K=4$ and the upper half with $K=2$. Find the new capacitance.
