By symmetry, the electric potential of an infinite line charge has no z -dependence. The line charge λ λ is the charge per unit length in the z -direction, and has units of (charge/length). If the radius ρ ρ of the observation point is greater than the radius ρ ρ ′ of the line charge, we may factor out ρ ρ 2 Φ Φ ( ρ ρ , θ θ ) = − − λ λ 4 π π ϵ ϵ { 2 ln ⁡ ⁡ ρ ρ + ln ⁡ ⁡ ( 1 − − ρ ρ ′ ′ ρ ρ e i ( θ θ − − θ θ ′ ′ ) ) ( 1 − − ρ ρ ′ ′ ρ ρ e − − i ( θ θ − − θ θ ′ ′ ) ) } and expand the logarithms in powers of ( ρ ρ ′ / ρ ρ ) < 1 Φ Φ ( ρ ρ , θ θ ) = − − λ λ 2 π π ϵ ϵ { ln ⁡ ⁡ ρ ρ − − ∑ ∑ k = 1 ∞ ∞ 1 k ( ρ ρ ′ ρ ρ ) k [ cos ⁡ ⁡ k θ θ cos ⁡ ⁡ k θ θ ′ + sin ⁡ ⁡ k θ θ sin ⁡ ⁡ k θ θ ′ ] } which may be written as Φ Φ ( ρ ρ , θ θ ) = − − Q 2 π π ϵ ϵ ln ⁡ ⁡ ρ ρ + 1 2 π π ϵ ϵ ∑ ∑ k = 1 ∞ ∞ C k cos ⁡ ⁡ k θ θ + S k sin ⁡ ⁡ k θ θ ρ ρ k where the multipole moments are defined as Q = λ λ , C k = λ λ k ( ρ ρ ′ ) k cos ⁡ ⁡ k θ θ ′ , S k = λ λ k ( ρ ρ ′ ) k sin ⁡ ⁡ k θ θ ′ .