Question

The magnitude of the electric field $E$ in the annular region of a charged cylindrical capacitor:

• A. is same throughout
• B. is higher near the outer cylinder than near the inner cylinder
• C. varies as $\frac{1}{r}$, where $r$ is the distance from the axis
• D. varies as $\frac{1}{r^2}$, where $r$ is the distance from the axis

Answer 1

Answer: (C)

varies as $\frac{1}{r}$, where $r$ is the distance from the axis

Consider a hypothetical cylinder of radius $r$ and length $l$ in the annular region of a charged cylindrical capacitor.
By Gauss's law, the filed in the annular region , $E.2\pi rl=\frac{q_{en}}{{ϵ}_0}=\frac{\lambda l}{{ϵ}_0}$, where $\lambda =$ linear charge density.
or, $E=\frac{\lambda }{2\pi {ϵ}_{0}r}$
$\Rightarrow$ $E\propto \frac{1}{r}$