Question

A long charged cylinder of linear charged density $\lambda$ is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders?

Answer 1

Charge density of the long charged cylinder of length L and radius r is $\lambda$.

Another cylinder of same length surrounds the pervious cylinder. The radius of this cylinder is R.

Let E be the electric field produced in the space between the two cylinders.

Electric flux through the Gaussian surface is given by Gauss’s theorem as,

$\varphi =E\left(2\pi d\right)L$

Where, $d=$ Distance of a point from the common axis of the cylinders

Let q be the total charge on the cylinder.

It can be written as

$\therefore \varphi =E\left(2\pi dL\right)=\frac{q}{{\in }_0}$

Where,

$q=$ Charge on the inner sphere of the outer cylinder

${\in }_0=$ Permittivity of free space

$E\left(2\pi dL\right)=\frac{\lambda L}{{\in }_0}$

$E=\frac{\lambda }{2\pi {\in }_{0}d}$

Therefore, the electric field in the space between the two cylinders is

$\frac{\lambda }{2\pi {\in }_{0}d}$.