Use Gauss Law to find the electric field inside and outside

Use Gauss Law to find the electric field inside and outside an infinitely long cylinder of radius R. The change density in the cylinder depends only on the radial coordinate s, with rho = rho_0 R/S.

Solution

First let us find the electric field inside the infinitely long cylinder.

For that ,imagine a Gaussian surface inside the given cylinder.Let the length of the cylinder which is considered as Gaussian surface is L, and the radius of Gaussian surface is \'s\' . Since the cylinder is infinitely long the edges of the cylinder has no contribution in this case.

Then by Gauss\'s law , E.da = Q_enclosed/ €₀

Here da,which is the area of Gaussian surface = 2pisL

Q_enclosed = dv =₀^2pi ₀^L ₀^s (1/s) (₀ R )s ds d d z = 2pi L ₀ R s

E 2 pi s L = 2pi sLR ₀ / €₀

Einside = R ₀ /€₀

B) to find electric field outside, imagine a Gaussian surface ( cylinder ) of length L and radius s , out side the infinitely long cylinder.

In this case enclosed charge Q = ₀^2pi ₀^L ₀^R R ₀ (1/s) s ds d d z = 2 pi L R² ₀

E 2pi s L = 2 pi L ₀ R²

E outside = R²₀ /€₀ s