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 ?

Explanation

From fig. A is a long charged cylinder of linear charge density `lambda`, length l and radius a. a hollow co-axial conducting cylinder B of length L and radius b surrounds.

The charge `q = lambda` spreads uniforunly on the outer surface of `lambda`. It induces - q charge on the cylinder B. Which spreads on the inner surface of B. An electric field `vec(E)` is produced in the space between the two cylinders which is directed radically out weards. Let us consider a co-axial cylindrical gasussian surface of radius r. The electric flux through the cylindrical Gaussian surface is
`phi_(E ) = int vec(E ) vec(ds) = int E ds cos `theta^(@)` = E int ds = E (2 pi r L)` .The Electric flux through the end faces of the cylindrical gaussion surface is zero as `vec(E )` is parallel to them. According to Gauss's theorem
`phi_(E ) = E (2 pi r L) = (q)/(epsilon_(0)) = (lambda L)/(epsilon_(0)) E = (lambda)/(2 pi epsilon_(0)r)`