Question

The electric potential inside a charged conducting sphere is constant. The charge distribution inside the sphere will be:

• A. Uniform
• B. Non uniform
• C. Zero charge
• D. Radially increasing

Hint:

For a conductor in electrostatic equilibrium: 1. The electric field inside is zero. 2. The electric potential inside is constant and equal to the potential on the surface. 3. Any net charge resides entirely on the surface.

Answer 1

The Correct Option is C

Step 1: Relate electric potential and electric field. The electric field \(\vec{E}\) is related to the electric potential \(V\) by \(\vec{E} = -\vec{\nabla}V\). In one dimension, this is \(E = -dV/dr\).
Step 2: Apply the given condition. We are given that the potential \(V\) is constant inside the sphere. If \(V\) is constant, its derivative with respect to position must be zero. \[ \frac{dV}{dr} = 0 \implies E = 0 \] So, the electric field inside the conductor is zero.
Step 3: Relate electric field to charge distribution using Gauss's Law. Gauss's Law in differential form (Poisson's equation) is \(\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0\), where \(\rho\) is the volume charge density. Since \(\vec{E} = 0\) everywhere inside the sphere, its divergence \(\vec{\nabla} \cdot \vec{E}\) must also be zero. \[ \vec{\nabla} \cdot (0) = 0 \implies \frac{\rho}{\epsilon_0} = 0 \implies \rho = 0 \] This means there is no net charge at any point inside the volume of the conductor. Any net charge on a conductor must reside on its surface.