Question

Which one of the following graphs represents the variation of electric potential with distance \( r \) from the center of a non-conducting charged sphere of radius \( R \)?

• A.
• B.
• C.
• D.

Hint:

For a uniformly charged non-conducting sphere: - Inside (\( r<R \)): \( V \) follows a quadratic relation. - Outside (\( r>R \)): \( V \) follows \( V \propto \frac{1}{r} \). - The transition at \( r = R \) is smooth.

Answer 1

The Correct Option is D

Step 1: Understanding the Electric Potential of a Non-Conducting Sphere For a uniformly charged non-conducting sphere: - Inside the sphere (\( r<R \)): The electric potential \( V \) at a distance \( r \) from the center is given by: \[ V = \frac{kQ}{2R} \left( 3 - \frac{r^2}{R^2} \right) \] This shows a parabolic decrease from the center to the surface. - Outside the sphere (\( r \geq R \)): The sphere behaves like a point charge, and the potential follows: \[ V = \frac{kQ}{r} \] which represents an inverse relationship with distance. 
Step 2: Identifying the Correct Graph - Inside the sphere (\( r<R \)), \( V \) follows a quadratic relation. - Outside the sphere (\( r>R \)), \( V \) follows an inverse relation \( V \propto \frac{1}{r} \), which shows a smooth decrease. Among the given graphs, Graph D correctly represents this behavior: - A parabolic decrease inside the sphere. - A smooth inverse decrease outside the sphere.