Potential due to a Line Charge of Infinite Length

Consider a line charge of infinite length along x-axis. Let the charge per unit length be λ. P is a point at a perpendicular distance y from the line charge.
The electric field strength at P
            = Ey = λ /2π ε₀y                                … (1)

The electric Potential at this point is give by
            Vy = ∫ - E dy = - ∫ λ/2π ε₀y
            = λ/2πε₀ loge (∞/y) = ∞                            … (2)

Thus the electric potential due to an infinite line charge is infinity everywhere.
The potential difference between two points at distance y1 and y2
-    ∫E dy = ∫ - λ/2πε₀y dy = λ/2πε₀ (loge y₂ – loge y1)
. :    V₁ – V₂ = λ/2π ε₀ loge (y₂/y₁)                                … (3)

Eq. (3) give the potential difference between two points.



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