1.Define electric potential. Why is the electric potential inside a hollow charged spherical conductor constant and equal to its value on the surface?
2.Two point charges $5 \times 10^{-8} \text{ C}$ and $-3 \times 10^{-8} \text{ C}$ are located $16 \text{ cm}$ apart. Find the points on the line joining the two charges where the electric potential is zero.
3.A uniformly charged conducting sphere of $2.4 \text{ m}$ diameter has a surface charge density of $80.0 \mu\text{C/m}^2$. Find the total charge on the sphere and the potential at its surface.
4.Draw a graph showing the variation of electric potential $V$ and electric field $E$ with distance $r$ from the center of a uniformly charged hollow sphere.
5.Calculate the work done in bringing a point charge $q$ from infinity to the center of a uniformly charged ring of radius $R$ and total charge $Q$.
6.The electric potential at a point $(x, y, z)$ is given by $V = -x^2y - xz^3 + 4$. The electric field $\vec{E}$ at that point is given by?
7.Can the electric potential be zero at a point where the electric field is not zero? Give a practical example to support your answer.
8.Calculate the electric potential at the center of a square of side $\sqrt{2} \text{ m}$, having charges $100\mu\text{C}$, $-50\mu\text{C}$, $20\mu\text{C}$, and $-60\mu\text{C}$ at its four corners.
9.A charge of $24 \mu\text{C}$ is given to a hollow metallic sphere of radius $0.2 \text{ m}$. Find the potential at a point $0.1 \text{ m}$ from the center of the sphere.
10.Show that the work done in moving a charge along any closed path in an electrostatic field is zero. What property of the electrostatic field does this signify?
11.Derive an expression for the electric potential at a point on the axial line of an electric dipole.
12.Show mathematically that the potential at any point on the equatorial line of an electric dipole is zero.
13.An electric dipole of length $4 \text{ cm}$, when placed with its axis making an angle of $60^\circ$ with a uniform electric field, experiences a torque of $4\sqrt{3} \text{ Nm}$. Calculate the potential energy of the dipole, if it has a charge of $\pm 8 \text{ nC}$.
14.Derive the expression for the electric potential at any general point $(r, \theta)$ due to a short electric dipole.
15.A regular hexagon of side $a$ has a charge $Q$ at each of its vertices. Determine the electric potential and the electric field at the center of the hexagon.
16.Twenty-seven spherical drops of radius $3 \text{ mm}$ and carrying $10^{-12} \text{ C}$ of charge each are combined to form a single drop. Find the capacitance and potential of the bigger drop.
17.Two concentric spherical shells of radii $R$ and $2R$ are given charges $Q_1$ and $Q_2$ respectively. Find the potential of the inner shell.
18.A point charge $q$ is placed at the origin. Let $\vec{E}_A$, $\vec{E}_B$, and $\vec{E}_C$ be the electric fields at three points $A (1, 2, 3)$, $B (1, 1, -1)$, and $C (2, 2, 2)$ due to charge $q$. Then what is the relation between the potentials at these points?
19.Draw the equipotential surfaces due to an electric dipole. Locate the points where the potential due to the dipole is zero.
20.Two point charges $q$ and $-2q$ are kept at distance $d$ apart. Find the locus of points where the electric potential is zero.
21.State two important properties of equipotential surfaces. Why do equipotential surfaces due to a point charge get closer as we move towards the charge?
22.Draw the equipotential surfaces for (i) a uniform electric field, and (ii) two identical positive charges placed at a small distance apart.
23.Prove that the electric field is in the direction in which the potential decreases steepest.
24.A uniform electric field $E$ exists between two parallel plates. A point charge $q$ is moved from one plate to another in two different paths: a straight line and a semicircle. Compare the work done in both cases.
25.The electric potential $V(x)$ in a region around the origin is given by $V(x) = 4x^2$ volts. The electric charge enclosed in a cube of $1 \text{ m}$ side centered at the origin is (in terms of $\epsilon_0$)?
26.Two identical conducting spheres $A$ and $B$, separated by a large distance, carry charges $+Q$ and $-2Q$ respectively. They are connected by a conducting wire. What is the final charge on sphere $A$?
27.Calculate the work done in taking a charge of $2 \times 10^{-9} \text{ C}$ from $A$ to $B$ via $C$ in an electric field $\vec{E} = 20\hat{i} \text{ V/m}$, if coordinates of $A$, $B$, and $C$ are $(0, 0)$, $(2, 0)$, and $(2, 2)$ respectively (all in meters).
28.Show that the potential gradient is a vector quantity. What does its negative sign indicate?
29.Equipotential surfaces are shown in a region. Can the electric field be zero at any point in this region? Explain.
30.A hollow metal sphere of radius $5 \text{ cm}$ is charged such that the potential on its surface is $10 \text{ V}$. What is the potential at the center of the sphere?
31.Derive an expression for the potential energy of a system of two point charges $q_1$ and $q_2$ located at $\vec{r}_1$ and $\vec{r}_2$ in an external electric field $\vec{E}$.
32.Three point charges $+q$, $+2q$, and $-3q$ are placed at the vertices of an equilateral triangle of side $a$. Calculate the work done to dissociate the system.
33.Determine the electrostatic potential energy of a system consisting of two charges $7 \mu\text{C}$ and $-2 \mu\text{C}$ (and with no external field) placed at $(-9 \text{ cm}, 0, 0)$ and $(9 \text{ cm}, 0, 0)$ respectively.
34.How much work is required to separate the two charges in the previous question infinitely away from each other?
35.An electric dipole of moment $p$ is placed in a uniform electric field $E$. Derive an expression for its potential energy $U = -\vec{p} \cdot \vec{E}$.
36.A molecule of a substance has a permanent electric dipole moment of magnitude $10^{-29} \text{ C m}$. A mole of this substance is polarised by applying a strong electrostatic field of $10^6 \text{ V/m}$. Find the heat released by the substance if the field is suddenly changed in direction by $60^\circ$.
37.Four charges $-q$, $-q$, $+q$, and $+q$ are placed at the corners $A, B, C, D$ of a square of side $a$. Deduce the expression for the work done in assembling these charges.
38.A point charge $q$ is surrounded by eight identical charges $q$ placed at the corners of a cube of side $L$. Evaluate the potential energy of the central charge.
39.In which orientation is a dipole placed in a uniform electric field in (i) stable, and (ii) unstable equilibrium? Represent the potential energy in both cases.
40.If the potential energy of two equal point charges placed at a distance $r$ apart is $U$, what will be the potential energy if the distance between them is doubled and the charges are halved?
41.Explain the principle of a capacitor. Deduce the expression for the capacitance of a parallel plate capacitor when a dielectric slab is inserted between its plates.
42.A parallel plate capacitor with air between the plates has a capacitance of $8 \text{ pF}$. What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant $6$?
43.Derive the expression for the capacitance of a parallel plate capacitor having a conducting slab of thickness $t < d$ placed between the plates.
44.What is dielectric polarization? Explain the effect of a dielectric on the capacitance of a capacitor.
45.A $12 \text{ pF}$ capacitor is connected to a $50 \text{ V}$ battery. How much electrostatic energy is stored in the capacitor?
46.Two metallic spheres of radii $a$ and $b$ are separated by a large distance. They are connected by a wire. Find the ratio of their final surface charge densities.
47.A parallel plate capacitor is charged by a battery. The battery is disconnected and a dielectric slab is inserted to completely fill the space between the plates. How will (i) its capacitance, (ii) electric field between plates, and (iii) energy stored change?
48.A parallel plate capacitor is charged by a battery. While the battery remains connected, a dielectric slab is inserted. How will (i) the charge on the plates, and (ii) the energy stored change?
49.Derive an expression for the capacitance of a spherical capacitor having concentric spherical shells of radii $a$ and $b$ ($b > a$).
50.Calculate the capacitance of the earth, considering it to be a spherical conductor of radius $6400 \text{ km}$.
51.Derive the expression for the energy loss when two charged capacitors $C_1$ and $C_2$, charged to potentials $V_1$ and $V_2$ respectively, are connected in parallel. Where does this energy go?
52.A $600 \text{ pF}$ capacitor is charged by a $200 \text{ V}$ supply. It is then disconnected from the supply and is connected to another uncharged $600 \text{ pF}$ capacitor. How much electrostatic energy is lost in the process?
53.Three capacitors of capacitances $2 \mu\text{F}$, $3 \mu\text{F}$, and $4 \mu\text{F}$ are connected in parallel. What is the equivalent capacitance? Determine the charge on each capacitor if the combination is connected to a $100 \text{ V}$ supply.
54.Five capacitors of $10 \mu\text{F}$ each are connected in a Wheatstone bridge network. What is the equivalent capacitance across the opposite corners?
55.Two identical parallel plate capacitors $A$ and $B$ are connected to a battery of $V$ volts with the switch closed. The switch is now opened and the free space between the plates of the capacitors is filled with a dielectric of dielectric constant $K$. Find the ratio of the total electrostatic energy stored in both capacitors before and after the introduction of the dielectric.
56.A network of four $10 \mu\text{F}$ capacitors is connected to a $500 \text{ V}$ supply. Three are connected in series ($C_1, C_2, C_3$) and the fourth ($C_4$) is connected in parallel across the series combination. Determine (a) the equivalent capacitance of the network and (b) the charge on each capacitor.
57.What is the common potential when two capacitors $C_1$ and $C_2$ charged to potentials $V_1$ and $V_2$ are connected in parallel?
58.A $4 \mu\text{F}$ capacitor is charged by a $200 \text{ V}$ supply. It is then disconnected from the supply, and is connected to another uncharged $2 \mu\text{F}$ capacitor. How much electrostatic energy of the first capacitor is dissipated in the form of heat and electromagnetic radiation?
59.Calculate the energy density of a parallel plate capacitor. Show that the total energy stored in the capacitor is the volume integral of the energy density.
60.Two capacitors of unknown capacitances $C_1$ and $C_2$ are connected first in series and then in parallel across a battery of $100 \text{ V}$. If the energy stored in the two combinations is $0.045 \text{ J}$ and $0.25 \text{ J}$ respectively, determine the value of $C_1$ and $C_2$.
