Vardaan Institute - Physics Solutions

Topic 1: Electrostatic Potential & Potential Difference

Part I: Fill in the Blanks

The SI unit of electrostatic potential is ________.

Solution: Volt (V)

Electrostatic potential is a ________ quantity.

Solution: Scalar

The work done in bringing a unit positive charge from infinity to a point is called the electric ________ at that point.

Solution: Potential

$1 \text{ Volt} = 1 \text{ Joule} / 1$ ________.

Solution: Coulomb (C)

If potential is the same everywhere, the work done in moving a charge is ________.

Solution: Zero

Part II: Multiple Choice Questions (MCQs)

Which of the following is the dimensional formula of electrostatic potential?

Solution: (b) $[ML^2T^{-3}A^{-1}]$

Work done in moving a charge of $2\text{ C}$ across a potential difference of $5\text{ V}$ is:

Solution: (b) $10\text{ J}$ (Since $W = qV = 2 \times 5 = 10$)

A positive charge naturally moves from:

Solution: (a) Higher to lower potential

1 electron volt ($1\text{ eV}$) is equal to:

Solution: (a) $1.6 \times 10^{-19}\text{ J}$

Part III: Match the Following

10.

Match the quantities in Column A with their appropriate unit/formula in Column B:

Solution:
(P) Electrostatic Potential $\rightarrow$ (3) Volt
(Q) Work Done $\rightarrow$ (4) Joules
(R) Charge $\rightarrow$ (1) Coulomb
(S) Potential Difference $\rightarrow$ (2) $W/q$

Topic 2: Potential due to Point Charges & Systems

Part I: Fill in the Blanks

11.

The formula for potential $V$ at a distance $r$ from a point charge $q$ is ________.

Solution: $\frac{1}{4\pi\epsilon_0}\frac{q}{r}$

12.

The electric potential on the equatorial line of an electric dipole is ________.

Solution: Zero

13.

For an electric dipole, the potential at an axial point is inversely proportional to $r^x$. The value of $x$ is ________.

Solution: $2$ (Since $V \propto 1/r^2$)

14.

The constant $\frac{1}{4\pi\epsilon_0}$ has a numerical value of ________ $\text{Nm}^2/\text{C}^2$.

Solution: $9 \times 10^9$

15.

Potential inside a hollow charged spherical conductor is ________ to the potential on its surface.

Solution: Equal (or Constant)

Part II: Multiple Choice Questions (MCQs)

16.

If the distance from a point charge is doubled, the potential becomes:

Solution: (b) Half (Since $V \propto 1/r$)

17.

The net potential at the midpoint of two equal and opposite charges is:

Solution: (b) Zero

18.

The superposition principle states that total potential is the ________ sum of individual potentials.

Solution: (b) Algebraic

Part III: True or False

19.

Electric potential due to a point charge varies inversely with the square of the distance.

Solution: False (It varies inversely with distance $r$, not $r^2$)

20.

The potential at the center of a dipole is always zero.

Solution: True

Part IV: Match the Following

21.

Match the source with its correct potential formula:

Solution:
(P) Point Charge $\rightarrow$ (3) $\frac{kq}{r}$
(Q) Dipole (Axial) $\rightarrow$ (4) $\frac{kp}{r^2}$
(R) Dipole (Equatorial) $\rightarrow$ (1) $0$
(S) Inside Hollow Sphere $\rightarrow$ (2) Constant

Topic 3: Equipotential Surfaces & E-Field Relation

Part I: Fill in the Blanks

22.

An ________ surface has the same potential at all points.

Solution: Equipotential

23.

Work done in moving a test charge on an equipotential surface is ________.

Solution: Zero

24.

Electric field lines are always ________ to equipotential surfaces.

Solution: Perpendicular ($90^\circ$)

25.

The mathematical relation between electric field and potential gradient is $E =$ ________.

Solution: $-dV/dr$

26.

The negative sign in $E = -dV/dr$ shows that electric field points in the direction of ________ potential.

Solution: Decreasing

Part II: Multiple Choice Questions (MCQs)

27.

Equipotential surfaces for an isolated point charge are:

Solution: (c) Concentric spheres

28.

Equipotential surfaces for a uniform electric field are:

Solution: (a) Parallel planes

29.

Can two equipotential surfaces intersect?

Solution: (b) No (It would mean two different potentials at one point)

30.

If $V = 5x$ volts, the magnitude of the electric field $E$ along the x-axis is:

Solution: (a) $5\text{ V/m}$ ($E = |-\frac{dV}{dx}|$)

Part III: True or False

31.

The surface of a charged perfect conductor is an equipotential surface.

Solution: True

32.

Potential gradient is a vector quantity.

Solution: True (Electric field is the negative potential gradient)

Section D: Electrostatic Potential Energy

Part I: Fill in the Blanks

33.

The potential energy of a system of two charges is given by $U =$ ________.

Solution: $\frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r}$

34.

The SI unit of electrostatic potential energy is ________.

Solution: Joules (J)

35.

Potential energy of a dipole in an external electric field is $U =$ ________.

Solution: $-pE\cos\theta$ or $-\vec{p}\cdot\vec{E}$

36.

A dipole is in stable equilibrium when the angle between $\vec{p}$ and $\vec{E}$ is ________ degrees.

Solution: $0$

37.

Work done in rotating a dipole from $\theta_1$ to $\theta_2$ is $W =$ ________.

Solution: $pE(\cos\theta_1 - \cos\theta_2)$

Part II: Multiple Choice Questions (MCQs)

38.

If two like charges are brought closer, the potential energy of the system:

Solution: (a) Increases (Work is done against repulsion)

39.

If an electron and proton are brought closer, the potential energy:

Solution: (b) Decreases

40.

The potential energy of a dipole is minimum when it is aligned:

Solution: (c) Parallel to E ($\theta = 0^\circ$)

Part III: Match the Following

41.

Match the Dipole Alignment with its Potential Energy ($U$):

Solution:
(P) $\theta = 0^\circ$ (Stable) $\rightarrow$ (2) $-pE$
(Q) $\theta = 90^\circ$ $\rightarrow$ (3) $0$
(R) $\theta = 180^\circ$ (Unstable) $\rightarrow$ (1) $+pE$

Topic 4: Electrostatic Potential Energy

Part I: Fill in the Blanks

42.

Capacitance $C$ is the ratio of ________ to potential difference $V$.

Solution: Charge ($Q$)

43.

The SI unit of capacitance is ________.

Solution: Farad (F)

44.

$1\mu\text{F} =$ ________ Farads.

Solution: $10^{-6}$

45.

Capacitance of an isolated spherical conductor of radius $R$ is $C =$ ________.

Solution: $4\pi\epsilon_0 R$

46.

Capacitance of a parallel plate capacitor in vacuum is $C =$ ________.

Solution: $\frac{\epsilon_0 A}{d}$

47.

When a dielectric is inserted between the plates, capacitance ________.

Solution: Increases (by factor $K$)

48.

In a series combination of capacitors, ________ remains the same across each.

Solution: Charge ($Q$)

49.

In a parallel combination of capacitors, ________ remains the same across each.

Solution: Potential Difference ($V$)

Part II: Multiple Choice Questions (MCQs)

50.

Equivalent capacitance of $C_1$ and $C_2$ in parallel is:

Solution: (a) $C_1 + C_2$

51.

Equivalent capacitance of two $4\mu\text{F}$ capacitors in series is:

Solution: (b) $2\mu\text{F}$ ($\frac{1}{C_s} = \frac{1}{4} + \frac{1}{4}$)

52.

The energy stored in a capacitor is given by:

Solution: (d) All of these

53.

If the distance between plates of a parallel plate capacitor is halved, capacitance:

Solution: (b) Doubles ($C \propto 1/d$)

54.

The value of dielectric constant ($K$) for a perfect conductor is:

Solution: (c) Infinity

Part III: Match the Following

55.

Match the quantities with their formulas:

Solution:
(P) Energy Density ($u$) $\rightarrow$ (2) $\frac{1}{2}\epsilon_0 E^2$
(Q) Energy Stored ($U$) $\rightarrow$ (1) $\frac{1}{2}CV^2$
(R) Series Equivalent ($C_s$) $\rightarrow$ (4) $(\frac{1}{C_1} + \frac{1}{C_2})^{-1}$
(S) Parallel Equivalent ($C_p$) $\rightarrow$ (3) $C_1 + C_2$

Topic 5: Capacitance & Combinations

Part I: Fill in the Blanks

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