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)

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

$Now;J=\sigma E$
required $\frac{i}{l}=\sigma .\frac{\lambda .2\pi {ϵ}_0}{r}\times 2\pi r$
$=\sigma .\frac{V}{ln\frac{b}{a}}\times 2\pi$
$=\frac{2\pi \sigma }{ln\frac{b}{a}}.V$