Question

The diagram shows the cross-section of a long cylinder having radius $R$. The charge carried by the cylinder is NOT uniform; the charge density varies with radius according to the equation $=A\sqrt{r}$.
Which of the following expressions gives the electric field strength at point $P$, a point that lies outside the cylinder and is a distance $r$ from the center of the cylinder?

• A. $\frac{A\sqrt{R}}{\pi {ϵ}_{0}r}$
• B. $\frac{A{R}^{\frac{3}{2}}}{2{ϵ}_{0}r}$
• C. $\frac{A{R}^2}{2{ϵ}_0{r}^{\frac{3}{2}}}$
• D. $\frac{3}{2}\frac{A{R}^{\frac{3}{2}}}{{ϵ}_{0}r}$
• E. $\frac{Ar}{2{ϵ}_0\sqrt{R}}$

Answer 1

The correct option is D $\frac{3}{2}\frac{A{R}^{\frac{3}{2}}}{{ϵ}_{0}r}$

In this diagram, radius $=R$

equation $=A\sqrt{r}$ at point $P$, that lies outside the cylinder and is the distance $=r$ from the center of the cylinder.

The electric field of the cylinder $=\frac{3}{2}\frac{A\sqrt{R}}{{ϵ}_{0}r}\times R$

$=\frac{3}{2}\frac{A{R}^{3/2}}{{ϵ}_{0}r}$