Question

What is the total charge inside the sphere?

• A. $\frac{4}{3}\pi {\rho }_0{R}^3$
• B. $\frac{3}{3}\pi {\rho }_0{R}^3$
• C. $\frac{2}{3}\pi {\rho }_0{R}^3$
• D. $\frac{1}{3}\pi {\rho }_0{R}^3$

Answer 1

The correct option is B $\frac{3}{3}\pi {\rho }_0{R}^3$

Take a spherical element of radius $r$ and width $dr$.

The charge density of this elemental volume is $\rho ={\rho }_o\frac{r}{R}$ and the volume is $V=area\times thickness=4\pi r^{2}dr$

Hence, charge on this spherical shell is

$dq=\rho .V$

$dq={\rho }_o\frac{r}{R}4\pi r^{2}dr$

$⟹dq=\frac{4\pi {\rho }_o}{R}{r}^{3}dr$

$⟹q=\int dq={\int }_0^R\frac{4\pi {\rho }_o}{R}{r}^{3}dr$

$⟹q=\frac{4\pi {\rho }_o}{R}\frac{R^4}{4}$

$⟹q=\pi {\rho }_o{R}^3$