Question

Derive an expression for the electric field due to a long straight thin uniformly charged wire (linear charge density λ) at a point lying at a distance r from the wire.

Answer 1

Electric field due to an infinitely long straight uniformly charged wire:

charge density = λ

P is a point where the electric field is to be calculated.

r = distance of point P from the wire

E = electric field at the point P

A cylinder of length l, radius r, closed at each end, is imagined as a Gaussian surface.

ds = a very small area on the Gaussian surface

The electric field will be directed radially outward and have the same magnitude at every location on the cylinder's curving surface due to symmetry.

E and ds are along the same direction.

The electric flux (Φ) through a curved surface = `oint E d s cos theta`

= `oint E d s`    ...[∵ θ = 0°]

= E(r π l)

Q = λ l = the net charge enclosed by the Gaussian surface is:

∴ By Gauss’s law,

Φ = `Q/epsilon_0`

Or, E(2r π l) = `Q/epsilon_0`

= `(lambda l)/epsilon_0`

∴ E = `lambda/(2 pi r epsilon_0)`