The bob of a simple pendulum has mass `2g` and a charge of `5.0muC`. It is at rest in a uniform horizontal electric field of intensity `2000V//m`. At equilibrium, the angle that the pendulum makes with the vertical is: (take `g=10m//s^(2)`)

Options

A. `tan^(-1)(2.0)`
B. `tan^(-1)(0.2)`
C. `tan^(-1)(5.0)`
D. `tan^(-1)(0.5)`

Answer

To solve the problem, we need to find the angle θ that the pendulum bob makes with the vertical when it is at equilibrium in a uniform horizontal electric field.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Bob:**
- The bob has a mass \( m = 2 \, \text{g} = 0.002 \, \text{kg} \).
- The charge on the bob is \( Q = 5.0 \, \mu\text{C} = 5.0 \times 10^{-6} \, \text{C} \).
- The electric field intensity is \( E = 2000 \, \text{V/m} \).
- The gravitational force acting on the bob is given by \( F_g = mg = 0.002 \times 10 = 0.02 \, \text{N} \).

2. **Calculate the Electric Force:**
- The electric force \( F_e \) acting on the bob due to the electric field is given by:
\[ F_e = Q \cdot E = (5.0 \times 10^{-6}) \times 2000 = 0.01 \, \text{N} \]

3. **Set Up the Equilibrium Condition:**
- At equilibrium, the forces acting on the bob can be resolved into two components:
- The gravitational force \( F_g \) acts downward.
- The electric force \( F_e \) acts horizontally (to the right).
- The pendulum will make an angle \( \theta \) with the vertical.

4. **Use Trigonometry to Relate Forces:**
- In the equilibrium position, we can use the tangent of the angle \( \theta \):
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{F_e}{F_g} \]
- Substituting the values:
\[ \tan \theta = \frac{0.01}{0.02} = \frac{1}{2} \]

5. **Calculate the Angle \( \theta \):**
- Now, we can find \( \theta \) using the inverse tangent function:
\[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \]

6. **Final Answer:**
- Using a calculator to find \( \tan^{-1}(0.5) \):
\[ \theta \approx 26.57^\circ \]

### Conclusion:
The angle that the pendulum makes with the vertical at equilibrium is approximately \( 26.57^\circ \).