Electric field due to infinite long straight Charged wire

 

1)Suppose an infinite long straight charged wire has linear charge density λ

2)Now we have to calculate electric field intensity at a distance r from the wire.



 3)Now we draw a cylindrical closed surface as shown in below, where the wire is on the axis, and cylinder has radius r.


 

4)Now there are 3 types of surfaces A, B & C. As shown in the above diagram.

5)Electric flux through the surface A & C is zero. Because electric field E making 90 degree angle with the surface 



6)now for the surface B, which has area


7) now the total charge inside this close surface is


therefore from Gauss’s law:



But 

So



Now from 1.21 and 1.31 we get  and q value



On solving the above we get,




Thus we get electric field intensity at a distance r from the wire.

Note to be remember: the electric field intensity is independent from the length of the wire. It only depends on the distance r from the wire.