In the above problem, find the electric field and electric potential at the centre of the sphere due to induced charges on the sphere?

Options

A. `(1)/(4pi in_(0)) (Q)/(R^(2)) and 0`
B. `0 and (1)/(4pi in_(0)) (Q)/(R )`
C. `(1)/(4pi in_(0)) (Q)/(R^(2)) and (1)/(4pi in_(0)) (Q)/(R )`
D. 0 and 0

Answer

To find the electric field and electric potential at the center of a sphere due to induced charges on the sphere, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a conducting sphere with a charge \( Q \) placed at a distance \( R \) from its center. The presence of this external charge will induce charges on the surface of the conductor.

2. **Induced Charges**:
- The external charge \( Q \) will induce negative charges on the side of the sphere closest to the charge and positive charges on the side farthest from it. This distribution of charges will be such that the total induced charge on the sphere remains zero since the sphere is a conductor.

3. **Electric Field Inside the Conductor**:
- According to electrostatic principles, the electric field inside a conductor in electrostatic equilibrium is always zero. Therefore, the electric field at the center of the sphere due to the induced charges is:
\[ E = 0 \, \text{N/C} \]

4. **Electric Potential Inside the Conductor**:
- The electric potential inside a conductor is constant and equal to the potential on its surface. Since the net charge on the conductor is zero (as the induced charges balance each other), the potential at the center of the sphere due to the induced charges is also zero:
\[ V = 0 \, \text{V} \]

5. **Final Answers**:
- Therefore, the electric field at the center of the sphere due to the induced charges is \( 0 \, \text{N/C} \) and the electric potential at the center is \( 0 \, \text{V} \).

### Summary:
- Electric Field \( E = 0 \, \text{N/C} \)
- Electric Potential \( V = 0 \, \text{V} \)