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Application of Gauss's theorem:1)electric field due to infinitely long straight conductor. 2)electric field due to a spherical conductor at a point on the surface
ans : The Gauss's theorem relates the electric charge distribution and corresponding electric field.
1) Electric field due to an infinitely long conductor

                Consider  an infinite long straight conductor having uniform linear charge density (?) and a point P located at a perpendicular distance r from the linear charge distribution. 
According to Gauss’ law, the electric field (E) intensity at point P on an infinitely long straight charged line is     
                               
where ,
? = linear charge density
?0 = electrical permittivity  of free space
r = radius
? = unit vector in the direction of radius.

2) Electric field due to a spherical conductor at a point on the surface

Consider an uniformly charged sphere having volume charge density ? and radius R. ?  is the charge per unit length of the wire.The electric field resulting from such a sphere is radial and hence electric field intensity is calculated for a point lying inside and outside the spherical shell.
The charge in the Gaussian surface is ? times the length. Since E is at right angles to the wire, the component of E normal to the end faces is zero. surface area of the Gaussian surface is 2(pi)rl. Then 
 


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