Question

The diagram shows a sphere that has a charge density described by the function $=\frac{A}{r}$.
Which of the following expressions gives the electric field strength at point $P$, a point that lies inside the sphere and is at a distance $r$ from the center of the sphere?

• A. $E=\frac{A}{2{ϵ}_{0}r}$
• B. $E=\frac{A}{2{ϵ}_0}$
• C. $E=\frac{2A}{{ϵ}_0}$
• D. $E=\frac{A}{3{ϵ}_0}$
• E. $E=\frac{A}{4\pi {ϵ}_{0}r}$

Answer 1

The correct option is B $E=\frac{A}{2{ϵ}_0}$

The charge density of the sphere is given by $\rho =\frac{A}{r}$.

Hence charge enclosed in a spherical volume of infinitesimal width=$\rho 4\pi r^{2}dr$

$⟹E\left(4\pi r^2\right)=\frac{{\int }_0^r\rho 4\pi r^{2}dr}{{ϵ}_0}$

$⟹E=\frac{{\int }_0^r\left(\frac{A}{r}4\pi r^{2}dr\right)}{4\pi r^2{ϵ}_0}$

$=\frac{A}{2{ϵ}_0}$