Question

Inside a uniformly charged spherical shell, the value of the electric field at distance r from the center is:

• A. 0
• B. kQ/r
• C. kQ/r\(^2\)
• D. Constant

Hint:

This is a classic result of Gauss's Law. It also applies to the inside of any hollow conductor in electrostatic equilibrium, a principle used in Faraday cages for electrostatic shielding.

Answer 1

The Correct Option is A

Step 1: Apply Gauss's Law for a point inside the shell. Consider a spherical Gaussian surface of radius \(r\) (where \(r\) is less than the radius of the shell) concentric with the charged shell. Gauss's Law states that the net electric flux through this surface is equal to the enclosed charge divided by \(\epsilon_0\): \[ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0} \]
Step 2: Determine the enclosed charge. For a spherical shell, all the charge resides on the surface of the shell. The Gaussian surface at radius \(r\) inside the shell encloses no charge. \[ Q_{enclosed} = 0 \]
Step 3: Solve for the electric field. Since \(Q_{enclosed} = 0\), the net flux must be zero. \[ \oint \vec{E} \cdot d\vec{A} = 0 \] Due to the spherical symmetry, if the electric field \(\vec{E}\) were non-zero, it would have a constant magnitude on the Gaussian surface. The integral would be \(E \times (4\pi r^2)\). For this to be zero, the electric field \(E\) must be zero. \[ E = 0 \] Thus, the electric field is zero everywhere inside a uniformly charged spherical shell.