Question

Consider a thick cylinder of inner radius $a$ and outer radius $b$ has conductivity $\sigma $ . A battery of emf $V$ is connected between the inner and outer surface of the cylinder. Find the current flowing, per unit length of the cylinder.

• A. $\frac{4 \pi \sigma}{\ell \text{n} \left(\text{b} / \text{a}\right)} \text{V}$
• B. $\frac{4 \pi \sigma}{\left(\text{b} + \text{a}\right)} \text{V}$
• C. $\frac{2 \pi \sigma}{\left(\text{b} + \text{a}\right)} \text{V}$
• D. $\frac{2 \pi \sigma }{\ell \text{n} \left(\text{b} / \text{a}\right)} \text{V}$

Answer 1

Answer: (D)

$V=\int\limits _{a}^{b}E.dl=\lambda 2\pi \left(\epsilon \right)_{0}ln\left(\right. \frac{b}{a} \left.\right)$
where $\lambda $ is the linear charge density on the inner cylinder.

$Now; J=\sigma E$
required $\frac{i}{l}=\sigma .\frac{\lambda .2 \pi \epsilon _{0}}{r}\times 2\pi r$
$=\sigma .\frac{V}{ln \frac{b}{a}}\times 2\pi $
$=\frac{2 \pi \sigma }{ln \frac{b}{a}}.V$