The electirc field strength at a distance `r` from the centre of a charged sphere of radius `r` is `E`. If `r gt R`, how much work will be done in bringing a test charge `q_(0)` from infinity to that point, is

Options

A. `q_(0)RE`
B. `(1)/(2)q_(0)RE`
C. `q_(0)rE`
D. `(1)/(2)q_(0)rE`

Explanation

To find the work done in bringing a test charge \( q_0 \) from infinity to a point at a distance \( r \) from the center of a charged sphere (where \( r > R \)), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Electric Field**:
The electric field \( E \) at a distance \( r \) from the center of a charged sphere (where \( r > R \)) is given. For a charged sphere, the electric field outside the sphere behaves as if all the charge were concentrated at the center.

2. **Determine the Electric Potential**:
The electric potential \( V \) at a distance \( r \) from the center of the sphere can be calculated using the relationship between electric field and potential:
\[ V = E \cdot r \]
Here, \( E \) is the electric field strength at distance \( r \).

3. **Calculate the Work Done**:
The work done \( W \) in bringing a test charge \( q_0 \) from infinity to a point at distance \( r \) is equal to the change in potential energy. The potential energy \( U \) at distance \( r \) is given by:
\[ U = q_0 \cdot V \]
Substituting the expression for potential \( V \):
\[ U = q_0 \cdot (E \cdot r) \]

4. **Final Expression for Work Done**:
Since the potential energy at infinity is zero (as the potential is defined to be zero at infinity), the work done \( W \) is simply:
\[ W = U - U_{\infty} = q_0 \cdot (E \cdot r) - 0 = q_0 \cdot E \cdot r \]

### Conclusion:
The work done in bringing a test charge \( q_0 \) from infinity to a distance \( r \) from the center of a charged sphere is:
\[ W = q_0 \cdot E \cdot r \]