1.Define electric potential at a point. Is it a scalar or a vector quantity?
2.Name the physical quantity whose SI unit is J/C.
3.Calculate the work done in moving a charge of $4 \mu\text{C}$ from infinity to a point where the potential is $10^4 \text{ V}$.
4.The work done in moving a charge of $3 \text{ C}$ between two points is $6 \text{ J}$. What is the potential difference between the two points?
5.Why is the electrostatic potential constant throughout the volume of a charged conductor and has the same value as on its surface?
6.Can electric potential at a point be zero, while the electric field is not zero? Give an example.
7.Can the electric field at a point be zero, while the electric potential is not zero? Give an example.
8.A uniform electric field of $20 \text{ N/C}$ exists in the vertically downward direction. Find the increase in electric potential as one goes up through a height of $50 \text{ cm}$.
9.Calculate the potential at a point $P$ due to a charge of $4 \times 10^{-7} \text{ C}$ located $9 \text{ cm}$ away.
10.In the above question, obtain the work done in bringing a charge of $2 \times 10^{-9} \text{ C}$ from infinity to the point $P$. Does the answer depend on the path along which the charge is brought?
11.State the superposition principle for electrostatic potential.
12.Derive an expression for the electric potential at a distance $r$ from an isolated point charge $q$.
13.Draw a graph showing the variation of electric potential $V$ with distance $r$ from a positive point charge.
14.Two charges $3 \times 10^{-8} \text{ C}$ and $-2 \times 10^{-8} \text{ C}$ are located $15 \text{ cm}$ apart. At what point on the line joining the two charges is the electric potential zero? (Take the potential at infinity to be zero).
15.Find the potential at the center of a square of side $\sqrt{2} \text{ m}$ which carries at its four corners charges $q_1 = 2 \text{ nC}$, $q_2 = 1 \text{ nC}$, $q_3 = -2 \text{ nC}$ and $q_4 = 3 \text{ nC}$.
16.What is the electrostatic potential at an equatorial point of an electric dipole? Justify your answer.
17.Write the expression for electric potential due to a short dipole at an axial point at distance $r$ from its center.
18.A short electric dipole has a dipole moment of $4 \times 10^{-9} \text{ C m}$. Determine the electric potential due to the dipole at a point $0.3 \text{ m}$ from the center of the dipole on its axis.
19.Find the potential of a metallic sphere of radius $10 \text{ cm}$ which has a charge of $5 \mu\text{C}$ uniformly distributed over it.
20.A regular hexagon of side $10 \text{ cm}$ has a charge $5 \mu\text{C}$ at each of its vertices. Calculate the potential at the center of the hexagon.
21.What is an equipotential surface?
22.Draw the equipotential surfaces for an isolated positive point charge.
23.Draw equipotential surfaces for a uniform electric field directed along the z-axis.
24.Why do two equipotential surfaces never intersect each other?
25.Show logically that the electric field is always directed perpendicular to an equipotential surface.
26.Two point charges $+q$ and $-q$ are kept at points $(-a, 0, 0)$ and $(a, 0, 0)$. Draw the equipotential surfaces of this system (an electric dipole).
27.Establish the mathematical relation between electric field $E$ and electric potential gradient $dV/dr$.
28.A uniform electric field $E$ of $300 \text{ N/C}$ is directed along the negative x-axis. Find the potential difference $V_B - V_A$ if point $A$ is at $x=0.5\text{ m}$ and point $B$ is at $x=1.2\text{ m}$.
29.Why must electrostatic field be normal to the surface at every point of a charged conductor?
30.The potential $V$ as a function of distance $x$ is given by $V = (5x^2 + 10x - 9) \text{ V}$. Find the magnitude of the electric field at $x = 1 \text{ m}$.
31.Define electrostatic potential energy of a system of point charges.
32.Does the potential energy of two positive charges increase or decrease when they are brought closer? Explain.
33.Three charges, each equal to $q$, are placed at the three corners of an equilateral triangle of side $a$. Find the total potential energy of the system.
34.Two point charges $+4 \mu\text{C}$ and $-2 \mu\text{C}$ are separated by a distance of $1 \text{ m}$ in air. Calculate the potential energy of the system.
35.An electron and a proton are brought from infinity to a separation of $0.53 \text{ \AA}$. Calculate the potential energy of the system in electron-volts ($\text{eV}$).
36.Write the expression for the potential energy of an electric dipole of dipole moment $p$ placed in a uniform electric field $E$.
37.Under what conditions is the potential energy of a dipole in a uniform electric field (i) minimum and (ii) maximum?
38.A dipole is placed in a uniform electric field. Show that the work done in turning it from angle $\theta_1$ to $\theta_2$ is $W = pE(\cos\theta_1 - \cos\theta_2)$.
39.Find the amount of work done in rotating an electric dipole of dipole moment $3 \times 10^{-8} \text{ C m}$ from its position of stable equilibrium to the position of unstable equilibrium in a uniform electric field of intensity $10^4 \text{ N/C}$.
40.Two charges of $5 \text{ nC}$ and $-2 \text{ nC}$ are placed at $(2 \text{ cm}, 0, 0)$ and $(x \text{ cm}, 0, 0)$ in a region with no external field. If the potential energy of the system is $-0.5 \mu\text{J}$, what is the value of $x$?
41.Define electrical capacitance of a conductor. Give its SI unit.
42.A spherical drop of water has $10^{-12} \text{ C}$ charge and a surface potential of $100 \text{ V}$. Calculate its radius.
43.On what geometrical factors does the capacitance of a parallel plate capacitor depend?
44.Derive an expression for the capacitance of a parallel plate capacitor with air between its plates.
45.A parallel plate capacitor has a plate area of $25 \text{ cm}^2$ and a separation of $2 \text{ mm}$ between the plates. Calculate its capacitance in picofarads ($\text{pF}$).
46.How does the capacitance of a parallel plate capacitor change when a dielectric medium is completely introduced between its plates?
47.What is the function of a dielectric in a capacitor?
48.The capacitance of a parallel plate capacitor is $50 \text{ pF}$ and the distance between the plates is $4 \text{ mm}$. It is charged to $200 \text{ V}$ and the battery is removed. Now a dielectric slab ($K=4$) of thickness $4 \text{ mm}$ is introduced. What is the new capacitance?
49.In the previous question, what will be the new potential difference across the plates after the dielectric is inserted?
50.Two metallic spheres of radii $R$ and $2R$ are charged so that both have the same surface charge density $\sigma$. If they are connected to each other with a conducting wire, in which direction will charge flow?
51.Derive the expression for the equivalent capacitance of three capacitors $C_1$, $C_2$, and $C_3$ connected in series.
52.Derive the expression for the equivalent capacitance of three capacitors $C_1$, $C_2$, and $C_3$ connected in parallel.
53.Calculate the equivalent capacitance when two capacitors of $5 \mu\text{F}$ and $10 \mu\text{F}$ are connected in (i) series and (ii) parallel.
54.Three capacitors each of capacitance $9 \text{ pF}$ are connected in series. What is the total capacitance of the combination?
55.In the previous question, what is the potential difference across each capacitor if the combination is connected to a $120 \text{ V}$ supply?
56.Two capacitors of $20 \mu\text{F}$ and $30 \mu\text{F}$ are connected in parallel to a $50 \text{ V}$ battery. Find the charge on each capacitor.
57.Derive an expression for the energy stored in a charged capacitor in terms of $C$ and $V$.
58.A $12 \text{ pF}$ capacitor is connected to a $50 \text{ V}$ battery. How much electrostatic energy is stored in the capacitor?
59.Prove that the energy density of a parallel plate capacitor is given by $u = \frac{1}{2}\epsilon_0 E^2$.
60.A parallel plate capacitor is charged by a battery. After some time, the battery is disconnected and a dielectric slab is inserted between the plates. Will the stored energy increase, decrease, or remain the same? Justify.
