Electric field due to an infinitely long straight charged wire:
Consider a thin infinitely long straight wire having a uniform linear charge density λ C/m. The field Ē of the line charge is directed radially outwards and its magnitude is the same at all points equidistance from the line charge, we choose a cylindrical gaussian surface of radius r, length l and with the axis along with with the line charge. It has curved surface S1 and flat S2 and S3.
So only the curved surface contributes towards the total flux.
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| Cylindrical Gaussian surface for a line charge |
ϕE=∮.SE⃗ .dS−→=∫S1E⃗ .dS1−→−+∫S2E⃗ .dS2−→−+∫S3E⃗ .dS3−→−
=∫S1E.dS1cos00+∫S2E.dS2cos900+∫S3E.dS3cos900
=E ∫dS1+0+0
= E x area of the curved surface
ϕE=E×2πrl
Charge enclosed by the Gaussian surface, q=λl
Using Gauss's theorem ϕE=q/ε0
so, E.2πrl=λl/εo or
E=λ2πϵ0r
Thus the electric field of a line charge is inversely proportional to the distance from the line charge.