Consider an uniformly charged wire of infinite length having a constant linear charge density λ (charge per unit length). Let P be a point at a distance r from the wire (Fig.) and E be the electric field at the point P. A cylinder of length l, radius r, closed at each end by plane caps normal to the axis is chosen as Gaussian surface.
Consider a very small area ds on the Gaussian surface.
Fig. Infinitely long straight charged wire
By symmetry, the magnitude of the electric field will be the same at
all points on the curved surface of the cylinder and directed radially
outward.
Vector E and ds are along the same direction.
The electric flux (φ) through curved surface = ∮ E ds cos θ
φ = ∮ E ds [∵θ = 0; cosθ = 1]
φ = E (2πrl) (∵ The surface area of the curved part is 2π rl)
∴ Total flux through the Gaussian surface, φ = E. (2πrl)
The net charge enclosed by Gaussian surface is, q = λl
∴ By Gauss’s law,
E (2πrl) = λl/εo
or E = λ/2 π rεo
The direction of electric field E is radially outward, if line charge
is positive and inward, if the line charge is negative.
Note :
At the microscopic level, charge distribution is discontinuous, because they are discrete charges separated by intervening space where there is no charge.
