Question

A conducting sphere of radius $R$ is given a charge $Q$. The electric potential and the electric field at the centre of the sphere respectively are

• A. Zero and $ \frac{Q}{4\pi\varepsilon _0R^2} $
• B. $ \frac{Q}{4\pi \varepsilon _0R} $ and Zero
• C. $ \frac{Q}{4\pi \varepsilon _0R}$ and $\frac{Q}{4\pi \varepsilon _0R^2} $
• D. Both are Zero

Answer 1

The Correct Option is B

The question involves finding the electric potential and the electric field at the center of a conducting sphere with charge $Q$ and radius $R$.

1. Concept of a Conducting Sphere: In a conducting sphere, the charge is uniformly distributed on its surface when it is charged. According to electrostatic principles, the entire charge of a conductor is on its outer surface, and the interior of a conductor is an equipotential region.

2. Electric Potential at the Centre:

   The electric potential $V$ at any point inside a conducting sphere is constant and equal to the potential on its surface because the sphere is equipotential. The formula for potential on the surface of the sphere is:

   V = \frac{Q}{4\pi \varepsilon_0 R}

   Therefore, the electric potential at the center of the sphere is also \frac{Q}{4\pi \varepsilon_0 R} .

3. Electric Field at the Centre:

   Inside a conductor in electrostatic equilibrium, the electric field is zero. This is because any internal electric field would cause the electrons to move, which contradicts the condition of electrostatic equilibrium.

   Therefore, the electric field at the center of the conducting sphere is zero.

4. Conclusion: From the above explanations, the electric potential at the center of the sphere is \frac{Q}{4\pi \varepsilon_0 R} , and the electric field is zero. Therefore, the correct answer is:

   • $ \frac{Q}{4\pi \varepsilon _0R} $ and Zero