Question

The magnitude of electric field $\overrightarrow{E}$ 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 $r$, 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

We know $\int \overrightarrow{E}\cdot d\overrightarrow{A}=\frac{Q_{enclosed}}{{\epsilon }_0.}$
Taking a Gaussian cylinder, as shown in the figure:
$E\cdot 2\pi rt=\frac{\sigma 2\pi at}{{\epsilon }_0}$, where $\sigma$ is the charge density in inner cylinder
$\Rightarrow$ $E=\frac{\sigma a}{{\epsilon }_{0}r}$
$\therefore E\propto \frac{1}{r}$