Question

A long straight wire is located parallel to an infinite conducting plate. The wire cross-sectional radius is equal to $$a$$, the distance between the axis of the wire and the plane equals $$b$$. Find the mutual capacitance of this system per unit length of the wire under the condition $$a < < b$$.

Answer 1

The field in the region between the conducting plane and the wire can be obtained by using an oppositely charged wire as an image on the other side.
Then the potential difference between the wire and the plane,
$$\triangle \varphi = \int_{b} \vec {E} \cdot d\vec {r}$$
$$= \int_{a}^{b} \left [\dfrac {\lambda}{2\pi \epsilon_{0} r} + \dfrac {\lambda}{2\pi \epsilon_{0} (2b - r)}\right ] dt$$
$$= \dfrac {\lambda}{2\pi \epsilon_{0}} ln \dfrac {b}{a} - \dfrac {\lambda}{2\pi \epsilon_{0}} ln \dfrac {b}{2b - a}$$
$$= \dfrac {\lambda}{2\pi \epsilon_{0}} ln \dfrac {2b - a}{a}$$
$$= \dfrac {\lambda}{2\pi \epsilon_{0}} ln \dfrac {2b}{a}$$, as $$b > > a$$
Hence, the sought mutual capacitance of the system per unit length of the wire
$$= \dfrac {\lambda}{\triangle \varphi} = \dfrac {2\pi \epsilon_{0}}{ln 2b/a}$$.