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: Establish the relationship between electric potential and electric field. The electric field \(\vec{E}\) is the negative gradient of the electric potential \(V\), expressed as \(\vec{E} = -\vec{abla}V\). In one dimension, this simplifies to \(E = -dV/dr\).
Step 2: Apply the given condition of constant potential. Given that the potential \(V\) is constant within the sphere, its derivative with respect to position is zero. Thus, \(dV/dr = 0\), which implies \(E = 0\). Consequently, the electric field inside the conductor is zero.
Step 3: Relate the electric field to charge distribution using Gauss's Law. Gauss's Law in differential form, also known as Poisson's equation, is \(\vec{abla} \cdot \vec{E} = \rho/\epsilon_0\), where \(\rho\) is the volume charge density. Since \(\vec{E} = 0\) throughout the interior of the sphere, its divergence, \(\vec{abla} \cdot \vec{E}\), must also be zero. This leads to \(\vec{abla} \cdot (0) = 0\), which means \(\rho/\epsilon_0 = 0\), and therefore \(\rho = 0\). This indicates that there is no net charge within the volume of the conductor. Any net charge present on a conductor must reside on its surface.