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 `200V//m`. At equilibrium, the angle that the pendulum makes with the vertical is:
(take `g=10m//s^(2)`)

Options

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

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the forces acting on the bob
The forces acting on the bob of the pendulum are:
1. The gravitational force (weight) acting downwards: \( F_g = mg \)
2. The electric force acting horizontally due to the electric field: \( F_e = QE \)
3. The tension in the string acting along the string, which has components in both the vertical and horizontal directions.

### Step 2: Write down the expressions for the forces
Given:
- Mass of the bob, \( m = 2 \, \text{g} = 2 \times 10^{-3} \, \text{kg} \)
- Charge of the bob, \( Q = 5.0 \, \mu C = 5.0 \times 10^{-6} \, C \)
- Electric field intensity, \( E = 200 \, \text{V/m} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

The gravitational force is:
\[ F_g = mg = (2 \times 10^{-3} \, \text{kg})(10 \, \text{m/s}^2) = 2 \times 10^{-2} \, \text{N} \]

The electric force is:
\[ F_e = QE = (5.0 \times 10^{-6} \, C)(200 \, \text{V/m}) = 1.0 \times 10^{-3} \, \text{N} \]

### Step 3: Set up the equations for equilibrium
At equilibrium, the components of the tension \( T \) can be expressed as:
1. Vertical component: \( T \cos \theta = mg \)
2. Horizontal component: \( T \sin \theta = QE \)

### Step 4: Divide the equations to find \( \tan \theta \)
Dividing the second equation by the first gives:
\[ \frac{T \sin \theta}{T \cos \theta} = \frac{QE}{mg} \]
This simplifies to:
\[ \tan \theta = \frac{QE}{mg} \]

### Step 5: Substitute the known values
Substituting the values we calculated:
\[ \tan \theta = \frac{(5.0 \times 10^{-6} \, C)(200 \, \text{V/m})}{(2 \times 10^{-3} \, \text{kg})(10 \, \text{m/s}^2)} \]
\[ \tan \theta = \frac{1.0 \times 10^{-3} \, N}{2 \times 10^{-2} \, N} = \frac{1.0}{20} = 0.05 \]

### Step 6: Calculate the angle \( \theta \)
Now, we can find \( \theta \):
\[ \theta = \tan^{-1}(0.05) \]

### Final Answer
Using a calculator, we find:
\[ \theta \approx 2.86^\circ \]