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 `lambda`, and the cylinder has a net charge per unit length of `2lambda`. From this information, use Gauss's law to find (i) the charge per unit length on the inner and outer surfaces of the cylinder and (ii) the electric field outside the cylinder, a distance r from the axis.

Explanation

i. Inside surface : Consider a cylinder surface within
the metal. Since E inside the conducting shell is zero,
the total charge inside the Gaussian surface must be
zero, so the `charge// length` on the inner surface of the
cylinder is `-lambda`.
Outer surface: The total charge on the `metal//
cylinder is
`2lambdal = q_(in) + q_(out)` or `q_(out) = 2lambdal - q_(in)`,
where `q_(in) = -lambdal`
`q_(out) = 2 lambdal + lambdal = 3lambdal`
So the outside `charge//length` is `q_(out)//l = 3 lambda`.
ii. Electric field out side the cylinder `E = (lambda_(net))/(2piepsilon_0r) = (3lambda)/(2piepsilon_0r)`
radially outward.