Electric Charges and Fields Chapter-Wise Test 6

Correct answer Carries: 4.

Wrong Answer Carries: -1.

Three charges \( +3 \, \mu\text{C}, +3 \, \mu\text{C}, -6 \, \mu\text{C} \) are at the vertices of an equilateral triangle of side 2 m. What is the force on \( -6 \, \mu\text{C} \)?

\( F_1 = F_2 = 9 \times 10^9 \times \frac{3 \times 6 \times 10^{-12}}{(2)^2} = 0.0405 \, \text{N} \) (attractive).

Angle 60°. Net \( F = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos 60^\circ} = \sqrt{0.0405^2 + 0.0405^2 + 0.0016425} = 0.0702 \, \text{N} \).

0.05 N
0.06 N
0.07 N
0.08 N
3

A uniform field \( E = 4 \times 10^3 \, \text{N/C} \) is along the z-axis. What is the flux through a rectangle of 30 cm × 20 cm in the xy-plane?

Area: \( A = 0.3 \times 0.2 = 0.06 \, \text{m}^2 \).

Flux: \( \phi = E A \cos 0^\circ = 4 \times 10^3 \times 0.06 = 240 \, \text{Nm}^2/\text{C} \).

220 \( \text{Nm}^2/\text{C} \)
230 \( \text{Nm}^2/\text{C} \)
240 \( \text{Nm}^2/\text{C} \)
250 \( \text{Nm}^2/\text{C} \)
3

A uniform electric field \( E = 2 \times 10^3 \, \text{N/C} \) is along the z-axis. What is the flux through a rectangle of 30 cm × 50 cm in the xy-plane?

Area vector \( \Delta \mathbf{S} = 0.3 \times 0.5 = 0.15 \, \text{m}^2 \) along z-axis.

Flux: \( \phi = \mathbf{E} \cdot \Delta \mathbf{S} = 2 \times 10^3 \times 0.15 = 300 \, \text{Nm}^2/\text{C} \).

\( 300 \, \text{Nm}^2/\text{C} \)
\( 320 \, \text{Nm}^2/\text{C} \)
\( 350 \, \text{Nm}^2/\text{C} \)
\( 400 \, \text{Nm}^2/\text{C} \)
1

A dipole with charges \( +5 \, \mu\text{C} \) and \( -5 \, \mu\text{C} \) separated by 6 mm is in a field \( 3 \times 10^4 \, \text{N/C} \) at 30°. What is the torque?

Dipole moment: \( p = q \times 2a = 5 \times 10^{-6} \times 6 \times 10^{-3} = 3 \times 10^{-8} \, \text{C m} \).

Torque: \( \tau = p E \sin \theta = 3 \times 10^{-8} \times 3 \times 10^4 \times \sin 30^\circ = 9 \times 10^{-4} \times 0.5 = 4.5 \times 10^{-4} \, \text{N m} \).

\( 4.0 \times 10^{-4} \, \text{N m} \)
\( 4.5 \times 10^{-4} \, \text{N m} \)
\( 4.8 \times 10^{-4} \, \text{N m} \)
\( 5.0 \times 10^{-4} \, \text{N m} \)
2

A plane sheet has \( \sigma = 2.66 \times 10^{-11} \, \text{C/m}^2 \). What is the electric field near it?

\( E = \frac{\sigma}{2 \varepsilon_0} \).

\( E = \frac{2.66 \times 10^{-11}}{2 \times 8.854 \times 10^{-12}} = 1.5 \, \text{N/C} \).

\( 1.2 \, \text{N/C} \)
\( 1.5 \, \text{N/C} \)
\( 1.8 \, \text{N/C} \)
\( 2.0 \, \text{N/C} \)
2

Why does the electric field inside a uniformly charged thin spherical shell vanish, regardless of the position within it?

Gauss’s law explains this: for a spherical shell with uniform charge, a Gaussian surface inside encloses no charge (since all charge resides on the surface). Thus, the electric flux through the Gaussian surface is zero, implying the electric field inside is zero due to symmetry.

Gauss’s law
Coulomb’s law
Superposition principle
Charge quantization
1

A body gains a charge of \( -1.6 \times 10^{-7} \, \text{C} \) when rubbed. How many electrons were transferred to it?

Negative charge means electrons gained.

\( q = n e \), \( e = -1.6 \times 10^{-19} \, \text{C} \).

\( n = \frac{q}{|e|} = \frac{1.6 \times 10^{-7}}{1.6 \times 10^{-19}} = 1 \times 10^{12} \).

\( 1 \times 10^{12} \)
\( 1.2 \times 10^{12} \)
\( 1.5 \times 10^{12} \)
\( 1.8 \times 10^{12} \)
1

A conducting sphere of radius 20 cm has a surface charge density of \( 30 \, \mu\text{C/m}^2 \). What is the total charge?

Surface area: \( A = 4 \pi r^2 = 4 \pi (0.2)^2 = 0.16 \pi \, \text{m}^2 \).

Charge: \( q = \sigma A = 30 \times 10^{-6} \times 0.16 \pi = 1.507 \times 10^{-5} \, \text{C} \).

\( 1.2 \times 10^{-5} \, \text{C} \)
\( 1.4 \times 10^{-5} \, \text{C} \)
\( 1.51 \times 10^{-5} \, \text{C} \)
\( 1.6 \times 10^{-5} \, \text{C} \)
3

A conducting sphere of radius 25 cm has an electric field of \( 2 \times 10^3 \, \text{N/C} \) at 50 cm from its center. What is the charge?

\( E = \frac{k q}{r^2} \).

\( 2 \times 10^3 = 9 \times 10^9 \times \frac{q}{(0.5)^2} \).

\( q = \frac{2 \times 10^3 \times 0.25}{9 \times 10^9} = 5.56 \times 10^{-8} \, \text{C} \).

\( 4.0 \times 10^{-8} \, \text{C} \)
\( 5.0 \times 10^{-8} \, \text{C} \)
\( 5.2 \times 10^{-8} \, \text{C} \)
\( 5.56 \times 10^{-8} \, \text{C} \)
4

A thin spherical shell of radius 31 cm has \( q = 20 \, \mu\text{C} \). What is the electric field at 35 cm from the center?

Outside shell: \( E = \frac{k q}{r^2} \).

\( E = 9 \times 10^9 \times \frac{20 \times 10^{-6}}{(0.35)^2} = 9 \times 10^9 \times \frac{20 \times 10^{-6}}{0.1225} = 1.469 \times 10^6 \, \text{N/C} \).

\( 1.4 \times 10^6 \, \text{N/C} \)
\( 1.45 \times 10^6 \, \text{N/C} \)
\( 1.46 \times 10^6 \, \text{N/C} \)
\( 1.47 \times 10^6 \, \text{N/C} \)
4

A charge of \( 9 \, \mu\text{C} \) is enclosed in a cube of edge 40 cm. What is the flux through one face?

Total flux: \( \phi = \frac{q}{\varepsilon_0} = \frac{9 \times 10^{-6}}{8.854 \times 10^{-12}} = 1.016 \times 10^6 \, \text{Nm}^2/\text{C} \).

Flux per face (6 faces): \( \phi_{\text{face}} = \frac{1.016 \times 10^6}{6} = 1.693 \times 10^5 \, \text{Nm}^2/\text{C} \).

\( 1.5 \times 10^5 \, \text{Nm}^2/\text{C} \)
\( 1.69 \times 10^5 \, \text{Nm}^2/\text{C} \)
\( 1.8 \times 10^5 \, \text{Nm}^2/\text{C} \)
\( 2.0 \times 10^5 \, \text{Nm}^2/\text{C} \)
2

A closed surface has a net flux of \( 3.39 \times 10^5 \, \text{Nm}^2/\text{C} \). What is the charge enclosed?

\( \phi = \frac{q}{\varepsilon_0} \).

\( q = \phi \varepsilon_0 = 3.39 \times 10^5 \times 8.854 \times 10^{-12} = 3 \times 10^{-6} \, \text{C} = 3 \, \mu\text{C} \).

\( 2.8 \, \mu\text{C} \)
\( 2.9 \, \mu\text{C} \)
\( 3.0 \, \mu\text{C} \)
\( 3.1 \, \mu\text{C} \)
3

A dipole with charges \( +4 \, \mu\text{C} \) and \( -4 \, \mu\text{C} \) separated by 8 mm is in a field \( 5 \times 10^4 \, \text{N/C} \) at 45°. What is the torque?

Dipole moment: \( p = q \times 2a = 4 \times 10^{-6} \times 8 \times 10^{-3} = 3.2 \times 10^{-8} \, \text{C m} \).

Torque: \( \tau = p E \sin \theta = 3.2 \times 10^{-8} \times 5 \times 10^4 \times \sin 45^\circ = 1.6 \times 10^{-3} \times \frac{\sqrt{2}}{2} = 1.13 \times 10^{-3} \, \text{N m} \).

\( 1.0 \times 10^{-3} \, \text{N m} \)
\( 1.13 \times 10^{-3} \, \text{N m} \)
\( 1.2 \times 10^{-3} \, \text{N m} \)
\( 1.3 \times 10^{-3} \, \text{N m} \)
2

A charge of \( 7 \, \mu\text{C} \) is at the center of a cube of edge 35 cm. What is the total flux through the cube?

Total flux: \( \phi = \frac{q}{\varepsilon_0} \).

\( \phi = \frac{7 \times 10^{-6}}{8.854 \times 10^{-12}} = 7.91 \times 10^5 \, \text{Nm}^2/\text{C} \).

\( 7.5 \times 10^5 \, \text{Nm}^2/\text{C} \)
\( 7.91 \times 10^5 \, \text{Nm}^2/\text{C} \)
\( 8.2 \times 10^5 \, \text{Nm}^2/\text{C} \)
\( 8.5 \times 10^5 \, \text{Nm}^2/\text{C} \)
2

A glass rod loses \( 9.6 \times 10^{-8} \, \text{C} \) of charge when rubbed with silk. How many electrons were transferred from it?

Losing charge means electrons are removed, so charge is positive.

\( q = n e \), \( e = 1.6 \times 10^{-19} \, \text{C} \).

\( n = \frac{q}{|e|} = \frac{9.6 \times 10^{-8}}{1.6 \times 10^{-19}} = 6 \times 10^{11} \).

\( 6 \times 10^{11} \)
\( 6.5 \times 10^{11} \)
\( 7 \times 10^{11} \)
\( 7.5 \times 10^{11} \)
1

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