Question

Consider a solid sphere of radius R which has volume charge density given by the expression $\rho = \rho_0 \left( 1 - \frac{r}{R} \right)$ where $\rho_0$ is a constant and $r$ is the distance from the center. There is no charge outside the sphere. Consider potential at infinite distance from the solid sphere to be zero. What is the distance of a point from center of solid sphere where electric field is maximum?

• A. $\frac{R}{4}$
• B. $\frac{R}{3}$
• C. $\frac{2R}{3}$
• D. $\frac{R}{2}$

Answer 1

Answer: (C)

$\frac{2R}{3}$

Answer 2

To find the point where the electric field is maximum, we need to:

1. Calculate the electric field inside the sphere ($r \le R$) using Gauss's Law. This involves finding the enclosed charge $Q_{enclosed}(r)$ and then applying Gauss's Law to find $E(r)$.

2. Calculate the electric field outside the sphere ($r > R$) using Gauss's Law with the total charge $Q_{total}$.

3. Find the maximum of $E(r)$ within the sphere ($0 \le r \le R$) by taking the derivative of $E(r)$ with respect to $r$, setting the derivative to zero, and solving for $r$.

4. Compare the value of $E(r)$ at the critical point found in step 3 with the value at the boundary $r = R$ and consider the behavior for $r > R$ to determine the absolute maximum.

The electric field is maximum at $r = \frac{2R}{3}$.