Question

Potential at the surface of a uniformly charged non-conducting sphere is V. Then the potential at its centre is

• A. 0
• B. \(\frac{V}{2}\)
• C. \(2V\)
• D. \(\frac{3V}{2}\)

Answer 1

The Correct Option is D

For a uniformly charged non-conducting sphere, the potential \(V(r)\) inside the sphere (at a distance \(r\) from the center) is given by:

\[ V(r) = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{R} \left( \frac{3}{2} - \frac{r^2}{2R^2} \right) \]

Here:

• \(Q\) is the total charge on the sphere.
• \(R\) is the radius of the sphere.
• \(r\) is the distance from the center (for the center, \(r = 0\)).

At the center (\(r = 0\)):

\[ V_{\text{center}} = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{R} \cdot \frac{3}{2} \]

At the surface (\(r = R\)), the potential is:

\[ V_{\text{surface}} = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{R} \]

From this, we see:

\[ V_{\text{center}} = \frac{3}{2} V_{\text{surface}} \]

Therefore, the potential at the center is:

\[ V_{\text{center}} = \frac{3V}{2} \]

Answer

The potential at the center of the sphere is:

\(\frac{3V}{2}\)