In Figure, A is a long charged cylinder of linear charge density λ, length l and radius a. A hollow co-axial conducting cylinder B of of length l and radius b surrounds A.

The charge q = λl spreads uniformly on the outer surface of A. It induces – q charge on the cylinder B, which spreads on the inner surface of B. An electric field

→

E

is produced in the space between the two cylinders, which is directed radially outwards. Let us consider a co-axial cylindrical Gaussian surface of radius r. The electric flux through the cylindrical Gaussian surface is
ϕE = ∫

→

E

.

→

ds

= ∫ E ds cos 0° = E ∫ ds = E (2πrl)
The electric flux through the end faces of the cylindrical Gaussian surface is zero, as

→

E

is parallel to them. According to Gauss’s theorem,
ϕE = E (2πrl) =

q

ε0

=

λl

ε0

E =

λ

2πε0r