Question

A sphere of radius $R$ has a uniform volume charge density $\rho$. The magnitude of electric field at a distance $r$ from the center of the sphere, where $r > R$, is

• A. $\dfrac{\rho}{4\pi \varepsilon_0 r^2}$
• B. $\dfrac{\rho R^2}{\varepsilon_0 r^2}$
• C. $\dfrac{\rho R^3}{\varepsilon_0 r^2}$
• D. $\dfrac{\rho R^3}{3\varepsilon_0 r^2}$
• E. $\dfrac{\rho R^2}{4\varepsilon_0 r^2}$

Hint:

Outside a uniformly charged sphere, treat it as a point charge at the center.

Answer 1

The Correct Option is D

Concept:
For $r > R$, sphere behaves like a point charge: \[ E = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r^2} \]

Step 1: Find total charge.
\[ Q = \rho \times \frac{4}{3}\pi R^3 \]

Step 2: Substitute into formula.
\[ E = \frac{1}{4\pi \varepsilon_0} \cdot \frac{\frac{4}{3}\pi R^3 \rho}{r^2} \]

Step 3: Simplify.
\[ E = \frac{\rho R^3}{3\varepsilon_0 r^2} \]