State Gauss theorem in electrostatics. Derive the expression for electric field at a point due to an infinitely long straight charged conductor.

Explanation

Statement:
The total electric flux through a closed surface in free space is equal to `(1)/(epsi_0) `times the net charge enclosed by the surface.
`phi = (q)/( epsi_0)`
Derivation :

In the above fig.
AB is the infinitely long wire
E is the electric field
P is a point at a distance r from the wire
. r. is the radius of Gaussian cylinder
.l is the length of the Gaussian cylinder
Let .q. be the charge enclosed by the Gaussian cylinder. Let .`lamda `. be the linear charge density on the wire.
Flux through the end faces is zero because there are no components of electric field along the normal to the end faces.
`phi = ` flux tħrough curved surface
`phi = E xx `area of curved surface `(phi =E xx` area)
`phi = E xx 2pi rl to (1)`
From Gauss.s theorem
`phi =(q)/(epsi_0) to (2)`
but ` lamda = (q)/(I) = q= lamda I`
`(2) = phi =(lamda I)/( epsi_0 ) to ( 3) `
On comparing (1) and (3), we get
`E xx 2 pi rI =(lamda I )/( epsi_0)`
` E= ( lamda )/(2 pi r epsi_0)`
` E=(1)/( 2 pi r epsi_0 ) (lamda )/(r)`
The direction of .E. is perpendicular to the wire and directed away from the wire.