Electrostatic Potential and Capacitance Chapter-Wise Test 1

Correct answer Carries: 4.

Wrong Answer Carries: -1.

In a series combination of capacitors, why does each capacitor have the same charge but different potential differences?

In a series combination, capacitors are connected end-to-end, forming a single path for charge flow. When a voltage is applied, the same charge \( Q \) accumulates on each capacitor (as charge conservation ensures the same \( Q \) passes through each during charging). However, the potential difference across each capacitor is \( V = Q/C \), so \( V \) varies inversely with capacitance \( C \), leading to different potential differences unless the capacitances are equal.

Due to different plate separations
Due to varying dielectric constants
Due to different plate areas
Due to charge conservation and capacitance differences
4

A dielectric slab is inserted into a parallel plate capacitor while maintaining a constant charge. Why does the potential difference between the plates increase if the slab is removed partially?

With a constant charge \( Q \), capacitance \( C = \frac{\varepsilon_0 A}{d} \) without dielectric, and \( C' = \frac{K \varepsilon_0 A}{d} \) with dielectric (\( K > 1 \)). When the dielectric is fully inserted, \( C \) increases, reducing \( V = \frac{Q}{C} \). If the slab is partially removed, the effective capacitance decreases (as less dielectric area contributes \( K \)), approaching the air value. Since \( Q \) is constant, a decrease in \( C \) leads to an increase in \( V \), as \( V = \frac{Q}{C} \).

The electric field becomes non-uniform
The effective capacitance decreases
The charge on the plates increases
The dielectric induces more charges
1

Two charges \( 20 \, \mu\text{C} \) and \( -8 \, \mu\text{C} \) are placed 20 cm apart. What is the potential energy of the system? (Take \( \frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \, \text{Nm}^2 \text{C}^{-2} \)).

\( U = \frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r} = 9 \times 10^9 \times \frac{20 \times 10^{-6} \times (-8 \times 10^{-6})}{0.2} \).

\( U = 9 \times 10^9 \times \frac{-160 \times 10^{-12}}{0.2} = -7.2 \, \text{J} \).

-7 J
-7.5 J
-7.2 J
-8 J
3

Why is the electric field inside a hollow conducting shell zero when no charges are present inside, regardless of the external field?

In electrostatic equilibrium, the electric field inside a conductor is zero. For a hollow conducting shell, charges reside on the surfaces. If no charges are inside, applying Gauss’s law to a surface inside the shell shows no enclosed charge (\( \oint E \cdot dA = \frac{q_{\text{enc}}}{\varepsilon_0} \), \( q_{\text{enc}} = 0 \)), so \( E = 0 \) inside. External fields induce charges on the outer surface, but the inner cavity remains shielded, ensuring zero field inside, a property known as electrostatic shielding.

The shell absorbs the external field
No charges exist to create a field inside
The shell's inner surface cancels the external field
The shell's thickness prevents field penetration
2

A spherical conductor of radius 2 cm has a charge of \( 2 \times 10^{-8} \, \text{C} \). What is the potential at its surface? (Take \( \frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \, \text{Nm}^2 \text{C}^{-2} \)).

Potential at the surface: \( V = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{R} \).

\( V = 9 \times 10^9 \times \frac{2 \times 10^{-8}}{0.02} = 9 \times 10^9 \times 10^{-6} = 9000 \, \text{V} \).

9000 V
8000 V
10000 V
12000 V
1

A parallel plate capacitor has plates of area \( 0.08 \, \text{m}^2 \) and separation 0.4 mm in air. What is its capacitance? (Take \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2 \text{N}^{-1} \text{m}^{-2} \)).

\( C = \frac{\varepsilon_0 A}{d} = \frac{8.85 \times 10^{-12} \times 0.08}{0.4 \times 10^{-3}} = 1.77 \times 10^{-9} \, \text{F} = 1770 \, \text{pF} \).

1770 pF
1500 pF
2000 pF
1000 pF
1

A \( 4 \, \mu\text{F} \) capacitor is charged to \( 250 \, \text{V} \). What is the energy stored in it?

\( U = \frac{1}{2} C V^2 = \frac{1}{2} \times 4 \times 10^{-6} \times (250)^2 \).

\( U = \frac{1}{2} \times 4 \times 10^{-6} \times 62500 = 0.125 \, \text{J} \).

0.1 J
0.15 J
0.125 J
0.2 J
3

Two charges \( 28 \, \mu\text{C} \) and \( -14 \, \mu\text{C} \) are placed 28 cm apart. What is the potential energy of the system? (Take \( \frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \, \text{Nm}^2 \text{C}^{-2} \)).

\( U = \frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r} = 9 \times 10^9 \times \frac{28 \times 10^{-6} \times (-14 \times 10^{-6})}{0.28} \).

\( U = 9 \times 10^9 \times \frac{-392 \times 10^{-12}}{0.28} = -12.6 \, \text{J} \).

-12 J
-13 J
-12.6 J
-14 J
3

A spherical conductor of radius 12 cm has a charge of \( 3 \times 10^{-8} \, \text{C} \). What is the electric field at 30 cm from the center? (Take \( \frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \, \text{Nm}^2 \text{C}^{-2} \)).

For \( r = 0.3 \, \text{m} > R = 0.12 \, \text{m} \), \( E = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r^2} \).

\( E = 9 \times 10^9 \times \frac{3 \times 10^{-8}}{(0.3)^2} = 9 \times 10^9 \times \frac{3 \times 10^{-8}}{0.09} = 3 \times 10^3 \, \text{N/C} \).

2.5 × 10³ N/C
3.5 × 10³ N/C
3 × 10³ N/C
4 × 10³ N/C
3

Five capacitors of \( 10 \, \mu\text{F} \) each are in series. What is the equivalent capacitance?

\( \frac{1}{C} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{5}{10} \).

\( C = \frac{10}{5} = 2 \, \mu\text{F} \).

1 µF
1.5 µF
2.5 µF
2 µF
4

Performance Summary

Score:

Category Details
Total Attempts:
Total Skipped:
Total Wrong Answers:
Total Correct Answers:
Time Taken:
Average Time Taken per Question:
Accuracy:
0