Question

Question 2.17: 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?

Class 12 - Physics - Electrostatic Potential And Capacitance Electrostatic Potential And Capacitance Page-88

Answer 1

Let A is a long charged cylinder of linear charge density $\lambda$ , length l and radius a is given. also we assume that a hollow co-axial conducting cylinder B of length l and radius b surrounded A.

now, charge = linear charge density × length of wire. so, $\blue{q=\lambda l}$ distributes uniformly in outer surface of A. Let consider the $\blue{\bf{Gaussian\:surface}}$of radius r where a < r < b
so, the electric flux through the cylinderical $\blue{\bf{Gaussian\: surface}}$ is $\red{\phi=\int E.ds}=\int Edscos0\\=E\int ds=E(2πrl)$
[ because surface area of cylinder = 2πrl]

now, A/C to Gaussian theorem,
$\bf{\phi=\frac{q}{\epsilon_0}}$
E(2πrl) = $\frac{q}{\epsilon_0}$
$E(2\pi rl)=\frac{\lambda l}{\epsilon_0}\\\\E=\frac{\lambda}{2\pi\epsilon_0r}$

hence, $\green{E=\frac{\lambda}{2\pi\epsilon_0r}}$

Answer 2

Answer:

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,

ϕ=E(2πd)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

∴ϕ=E(2πdL) = q/ε₀

Where,

q= Charge on the inner sphere of the outer cylinder

ε₀= Permittivity of free space

E(2πdL) = λL/ε₀

E = λ/2πε₀d

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

λ/2πε₀d