Use Gauss' law to derive the expression for the electric field `(vecE)`due to a straight uniformly charged infinite line of charge density λ C/m.
Field due to an infinitely long straight uniformly charged wire
Consider a thin, infinitely long straight line charge of linear charge density λ.
Let P be the point at a distance a from the line. To find the electric field at point P, draw a cylindrical surface of radius ‘a’ and length l.
If E is the magnitude of electric field at point P, then electric flux through the Gaussian surface,
Φ = E × Area of the curved surface of a cylinder of radius r and length l
As electric lines of force are parallel to end faces (circular caps) of the cylinder, there is no component of the field along the normal to the end faces.
Φ = E × 2πal … (i)
According to Gauss's Theorem,
`phi = q/epsilon_0`
`∵ q = lambdal`
`:. phi = (lambdal)/epsilon_0` ...(ii)
From equations (i) and (ii), we get:
`E xx 2pial = (lambdal)/epsilon_0`