Question

A bob of a simple pendulum has a mass of \( 4 \, g \) and a charge of \( 20\mu C \). If it is at rest in a uniform horizontal electric field of intensity \( 1000 \, V m^{-1} \), the angle that the pendulum makes with the vertical at equilibrium is:

• A. \( \tan^{-1} \frac{1}{2} \)
• B. \( 60^\circ \)
• C. \( 30^\circ \)
• D. \( \tan^{-1} 2 \)

Hint:

In such equilibrium problems, angle is found using \( \tan\theta = \frac{\text{horizontal force}}{\text{vertical force}} \).

Answer 1

The Correct Option is A

Step 1: Identify forces acting on the bob.
The bob is under three forces:
(i) Gravitational force \( mg \) acting vertically downward
(ii) Electric force \( qE \) acting horizontally
(iii) Tension in the string

Step 2: Write expressions for forces.
Mass \( m = 4g = 4 \times 10^{-3} \, kg \)
Charge \( q = 20\mu C = 20 \times 10^{-6} \, C \)
Electric field \( E = 1000 \, V/m \)
Electric force:
\[ F_e = qE = 20 \times 10^{-6} \times 1000 = 2 \times 10^{-2} \, N \]
Gravitational force:
\[ mg = 4 \times 10^{-3} \times 10 = 4 \times 10^{-2} \, N \]

Step 3: Apply equilibrium condition.
At equilibrium, the string makes an angle \( \theta \) with vertical such that:
\[ \tan\theta = \frac{\text{horizontal force}}{\text{vertical force}} = \frac{F_e}{mg} \]

Step 4: Substitute values.
\[ \tan\theta = \frac{2 \times 10^{-2}}{4 \times 10^{-2}} = \frac{1}{2} \]

Step 5: Find angle.
\[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \]

Step 6: Conclusion.
\[ \boxed{\tan^{-1}\frac{1}{2}} \]