A pendulum bob of mass m charge q is at rest with its string making an angle `theta` with the vertical in a uniform horizontal electric field E. The tension in the string in

Options

A. `(mg)/(sintheta)`
B. `mg`
C. `(qE)/(sintheta)`
D. `(qE)/(costheta)`

Explanation

To find the tension in the string of a pendulum bob with mass \( m \) and charge \( q \) in a uniform horizontal electric field \( E \), we can follow these steps:

### Step 1: Analyze the Forces Acting on the Pendulum Bob
The forces acting on the pendulum bob are:
1. The gravitational force \( \vec{F_g} = mg \) acting vertically downward.
2. The tension \( \vec{T} \) in the string, which acts along the string at an angle \( \theta \) with the vertical.
3. The electric force \( \vec{F_e} = q\vec{E} \) acting horizontally due to the electric field.

### Step 2: Resolve the Tension into Components
The tension \( \vec{T} \) can be resolved into two components:
- The vertical component: \( T \cos \theta \)
- The horizontal component: \( T \sin \theta \)

### Step 3: Set Up the Equations
In the vertical direction, the forces must balance:
\[ T \cos \theta = mg \quad \text{(1)} \]

In the horizontal direction, the electric force must balance the horizontal component of the tension:
\[ T \sin \theta = qE \quad \text{(2)} \]

### Step 4: Solve for Tension
From equation (1), we can express \( T \):
\[ T = \frac{mg}{\cos \theta} \quad \text{(3)} \]

From equation (2), we can also express \( T \):
\[ T = \frac{qE}{\sin \theta} \quad \text{(4)} \]

### Step 5: Equate the Two Expressions for Tension
Setting equations (3) and (4) equal to each other gives:
\[ \frac{mg}{\cos \theta} = \frac{qE}{\sin \theta} \]

### Step 6: Rearranging to Find Tension
Cross-multiplying gives:
\[ mg \sin \theta = qE \cos \theta \]

Now, we can solve for \( T \):
\[ T = \frac{qE}{\sin \theta} \]

### Final Answer
Thus, the tension in the string is:
\[ T = \frac{qE}{\sin \theta} \]