A long charged cylinder of linear charged density λ is surrounded by a hollow co-axial conducting cylinder.

Question:

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

Solution:

Charge density of the long charged cylinder of length L and radius r is λ.

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,

$\phi=E(2 \pi d) L$

Where, = 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

$\therefore \phi=E(2 \pi d L)=\frac{q}{\epsilon_{0}}$

Where,

q = Charge on the inner sphere of the outer cylinder

$\epsilon_{0}=$ Permittivity of free space

$E(2 \pi d L)=\frac{\lambda L}{\epsilon_{0}}$

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

Therefore, the electric field in the space between the two cylinders is $\frac{\lambda}{2 \pi \epsilon_{0} d}$.