Question

The diagram shows a sphere of radius $R$ that carries a charge uniformly distributed throughout its volume. The volume charge density is $\rho$. 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. $Q_{enc}=4\pi \rho R^2$
• B. $Q_{enc}=\frac{4}{3}\pi \rho a^3$
• C. $Q_{enc}=4\pi \rho a^2$
• D. $Q_{enc}=\frac{4}{3}\pi \rho R^3$
• E. $Q_{enc}=\pi \rho a^2$

Answer 1

Answer: (D)

$Q_{enc}=\frac{4}{3}\pi \rho R^3$

The volume charge density is given by ,

$\rho =$charge on sphere / volume of sphere

$\rho =Q/\frac{4}{3}\pi R^3$

or $Q=\frac{4}{3}\pi \rho R^3$

we can see that the charge enclosed by gaussian surface is equal to $Q$ i.e.$Q_{enc}=Q$

therefore $Q_{enc}=\frac{4}{3}\pi \rho R^3$