A simple pendulum consists of a small sphere of mass `m` suspended by a thread of length `l`. the sphere carries a positive charge `q`. The pendulum is placed in a uniform electric field of strength `E` directed vertically upwards. With what period will the pendulum oscillate if the electrostatic force acting on the sphere is less than the gravitational force?

Options

A. `2pi sqrt ((l)/(g-(qE)/(m))`
B. `pi sqrt ((l)/(g-(qE)/(m))`
C. `2pi sqrt ((l)/(g-(2qE)/(m))`
D. `4pi sqrt ((l)/(g-(qE)/(m))`

Explanation

To find the period of a simple pendulum consisting of a charged sphere in a uniform electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Forces Acting on the Sphere**:
- The gravitational force acting downwards is given by \( F_g = mg \), where \( m \) is the mass of the sphere and \( g \) is the acceleration due to gravity.
- The electrostatic force acting upwards due to the electric field is given by \( F_e = qE \), where \( q \) is the charge on the sphere and \( E \) is the strength of the electric field.

2. **Determine the Net Force**:
- Since the electrostatic force is less than the gravitational force, we can write the net force acting on the sphere as:
\[ F_{\text{net}} = F_g - F_e = mg - qE \]

3. **Calculate Effective Acceleration**:
- According to Newton's second law, the net force is also equal to the mass times the acceleration (\( F = ma \)). Thus, we can express the net force as:
\[ ma = mg - qE \]
- Dividing both sides by \( m \) gives us the effective acceleration:
\[ a = g - \frac{qE}{m} \]
- We denote this effective acceleration as \( g_{\text{effective}} \):
\[ g_{\text{effective}} = g - \frac{qE}{m} \]

4. **Substitute into the Formula for Period**:
- The period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{l}{g_{\text{effective}}}} \]
- Substituting our expression for \( g_{\text{effective}} \):
\[ T = 2\pi \sqrt{\frac{l}{g - \frac{qE}{m}}} \]

5. **Final Expression**:
- Therefore, the period of the pendulum oscillating in the electric field is:
\[ T = 2\pi \sqrt{\frac{l}{g - \frac{qE}{m}}} \]