Question

The diagram shows a sphere of radius $R$ that carries a charge $Q$ uniformly distributed throughout its volume. A Gaussian sphere of radius $a$ is imagined that is concentric to the charged sphere.
If $a<R$, what is the charge enclosed in the Gaussian surface?

• A. $Q_{enc}=Q\frac{a^4}{R^4}$
• B. $Q_{enc}=Q\frac{a}{R}$
• C. $Q_{enc}=Q\frac{a^2}{R^2}$
• D. $Q_{enc}=Q\frac{a^3}{R^3}$
• E. $Q_{enc}=Q\frac{a^5}{R^5}$

Answer 1

Answer: (D)

$Q_{enc}=Q\frac{a^3}{R^3}$

Total charge on the whole sphere $=Q$

Thus charge per unit volume (or charge density) $\rho =\frac{Q}{\left(4\pi R^3\right)/3}$

$\therefore$ Charge inside the sphere of radius $a$, $Q_{enc}=\rho \times V_a=\frac{Q}{\left(4\pi R^3\right)/3}\times \frac{4\pi a^3}{3}=Q\frac{a^3}{R^3}$