Question

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 $$b > > a$$.

Answer 1

Let us suppose that linear charge density of the wires be $$\lambda$$ then, the potential difference, $$\varphi_{+} - \varphi_{-} = \varphi - (-\varphi) = 2\varphi$$. The intensity of the electric field created by one of the wires at a distance $$x$$ from its axis can be easily found with the help of the Gauss's theorem,
$$E = \dfrac {\lambda}{2\pi \epsilon_{0} x}$$
Then, $$\varphi = \int_{a}^{b - a} E\ dx = \dfrac {\lambda}{2\pi \epsilon_{0}} ln \dfrac {b - a}{a}$$
Hence, capacitance, per unit length,
$$\dfrac {\lambda}{\varphi_{+} - \varphi_{-}} = \dfrac {2\pi \epsilon_{0}}{ln b/a}$$.