Find the electric field outside the cylinder, a distance r from the axis using Gauss's law, if a long & straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of λ, and the cylinder has a net charge per unit length of 2λ?

Concept:

For outside points, a hollow metal cylinder behaves as if an equal magnitude linear charge density is placed on its axis.

Formula:

Electric field due to an infinite line charge density λ = λ/2πϵ₀r (Using Gauss's law)

Calculations:

Given:

The linear charge density of wire = λ 

The linear charge density of a hollow metal cylinder = 2λ 

The electric field at distance r from axis due to hollow metal cylinder of linear charge density 2λ:

E1= \(\frac{{2\lambda }}{{2\pi {\varepsilon _0}r}}\) (radially outwards)

The electric field at distance r from wire having linear charge density λ :

E2​ = \(\frac{{\lambda }}{{2\pi {\varepsilon _0}r}}\) (radially outwards)

As, the electric field is a vector quantity so the total Electric Field,

E = E₁ + E₂

⇒ E =  \(\frac{{2\lambda }}{{2\pi {\varepsilon _0}r}}+\frac{{\lambda }}{{2\pi {\varepsilon _0}r}}\)

⇒ E = \(\frac{{3\lambda }}{{2\pi {\varepsilon _0}r}}\) (radially outwards)