Question

State Gauss’s law. Obtain an expression for electric field due to line charge (or infinite long wire) using this law.

Or

Using Gauss’s law, determine the electric field intensity due to a long thin wire of uniform linear charge density.

Answer 1

Gauss’s law states that total electric flux over the closed surface S in vacuum is \(\frac{1}{ɛ_o}\) times the total charge contained inside surface S.

\(ϕ_E= ∮ \vec E . \vec {dS} = \frac{q}{ɛ_o}\)

Electric field intensity due to a line charge.

Consider a thin charged rod with uniform linear charge density λ. We wish to find an expression for electric intensity at point P at a perpendicular distance r, from the rod. Consider a right circular cylinder of radius r and length l with the infinite long line of charge as its axis.

In surface I and III, \(\vec E\) and d \(\vec S\)are ⊥ to each other. In cases of surface II, \(\vec E\)and \(d \vec S\)are parallel to each other and hence θ = 0°.

From Gauss's theorem, we have