There is a uniformly charged non conducting solid sphere made of material of dielectric constant one. If electric potential at infinity be zero. Then the potential at its surface is V. if we take electric potential at its surface to be zero. Then the potential at the centre will be

Options

A. `(3V)/(2)`
B. `(V)/(2)`
C. V
D. zero

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the electric potential of a uniformly charged non-conducting solid sphere
The electric potential \( V \) at a distance \( r \) from the center of a uniformly charged non-conducting solid sphere of radius \( R \) and total charge \( Q \) is given by the formula:
\[ V(r) = \begin{cases} \frac{kQ}{R} \left( \frac{3R^2 - r^2}{2R^2} \right) & \text{for } r \leq R \\ \frac{kQ}{r} & \text{for } r > R \end{cases} \]
where \( k \) is Coulomb's constant.

### Step 2: Calculate the potential at the surface of the sphere
At the surface of the sphere, \( r = R \):
\[ V(R) = \frac{kQ}{R} \left( \frac{3R^2 - R^2}{2R^2} \right) = \frac{kQ}{R} \left( \frac{2R^2}{2R^2} \right) = \frac{kQ}{R} \]
Thus, the potential at the surface is \( V = \frac{kQ}{R} \).

### Step 3: Set the potential at the surface to zero
If we take the potential at the surface to be zero, we have:
\[ V_{\text{surface}} = 0 \]
This means that the potential at the center of the sphere needs to be recalculated relative to this new reference point.

### Step 4: Calculate the potential at the center of the sphere
At the center of the sphere, \( r = 0 \):
\[ V(0) = \frac{kQ}{R} \left( \frac{3R^2 - 0^2}{2R^2} \right) = \frac{kQ}{R} \left( \frac{3R^2}{2R^2} \right) = \frac{3kQ}{2R} \]

### Step 5: Relate the potential at the center to the surface potential
Since we set the potential at the surface to zero:
\[ V_{\text{center}} = \frac{3kQ}{2R} \]
Now, since \( V = \frac{kQ}{R} \), we can express the potential at the center in terms of \( V \):
\[ V_{\text{center}} = \frac{3}{2} V \]

### Step 6: Final calculation with the new reference point
Since we have set the surface potential to zero, the potential at the center becomes:
\[ V_{\text{center}} - V_{\text{surface}} = \frac{3}{2} V - 0 = \frac{3}{2} V \]

### Step 7: Conclusion
Thus, if the potential at the surface is taken as zero, the potential at the center will be:
\[ V_{\text{center}} = \frac{3}{2} V \]