Two long straight wires with equal cross-sectional radii a are located parallel to each other in air. The distance between their axes equals b. Find the mutual capacitance of the wires per unit length under the condition b >> a.

At a distance of 'x'  from the linear charge the is given by 

$\mathrm{E}=\frac{\mathrm{\lambda }}{2{\mathrm{\pi ϵ}}_0\mathrm{x}}$

Potential difference = ${\int }_{\mathrm{a}}^{\mathrm{b}-\mathrm{a}}\mathrm{Edx}=\frac{\mathrm{\lambda }}{2{\mathrm{\pi ϵ}}_0}\ln \frac{\mathrm{b}-\mathrm{a}}{\mathrm{a}}$

since b >> a. $\mathrm{V}=\frac{\mathrm{\lambda }}{2{\mathrm{\pi ϵ}}_0}\ln \frac{\mathrm{b}}{\mathrm{a}}$

capacitance, per unit length is given by 

${\mathrm{C}}_{\mathrm{per} \mathrm{length}}=\frac{\mathrm{Charge} \mathrm{per} \mathrm{length} }{\mathrm{Potential} \mathrm{difference}}=\frac{\mathrm{\lambda }}{\frac{\mathrm{\lambda }}{2{\mathrm{\pi ϵ}}_0}lnb/\mathrm{a}}$

$\mathrm{C}=\frac{2{\mathrm{\pi ϵ}}_0}{lnb/\mathrm{a}}.$