A simple pendulum has time period `T` The bob is given negative charge and surface below it is given positive change new time period will be

Options

A. less than T
B. greater than T
C. equal to T
D. infinite

Explanation

To solve the problem of how the time period of a simple pendulum changes when the bob is given a negative charge and the surface below it is given a positive charge, we can follow these steps:

### Step 1: Understand the Initial Time Period of the Pendulum
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

### Step 2: Analyze the Effect of Charges on the Pendulum
When the bob is given a negative charge \( -q \) and the surface below it is given a positive charge \( +Q \), an electric field \( E \) is created. The force acting on the charged bob due to this electric field is:
\[ F_e = qE \]
Since the bob has a negative charge, the direction of the force will be opposite to the direction of the electric field.

### Step 3: Calculate the Effective Gravitational Acceleration
The effective gravitational acceleration \( g_{\text{effective}} \) acting on the bob will be modified due to the electric force. The new effective gravitational force can be expressed as:
\[ mg_{\text{effective}} = mg + qE \]
Thus, we can express the effective gravitational acceleration as:
\[ g_{\text{effective}} = g + \frac{qE}{m} \]
where \( m \) is the mass of the bob.

### Step 4: Substitute into the Time Period Formula
Now, substituting \( g_{\text{effective}} \) into the time period formula, we get the new time period \( T' \):
\[ T' = 2\pi \sqrt{\frac{L}{g_{\text{effective}}}} = 2\pi \sqrt{\frac{L}{g + \frac{qE}{m}}} \]

### Step 5: Compare the New Time Period with the Original Time Period
Since \( \frac{qE}{m} \) is a positive quantity (as it represents the additional force acting on the bob), we can see that:
\[ g_{\text{effective}} > g \]
This implies that:
\[ \sqrt{\frac{L}{g_{\text{effective}}}} < \sqrt{\frac{L}{g}} \]
Therefore, the new time period \( T' \) will be:
\[ T' < T \]

### Conclusion
The new time period \( T' \) of the pendulum after charging the bob and the surface will be less than the original time period \( T \).

### Final Answer
The new time period will be **less than \( T \)**.

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