Answer to Question #98454 in Electricity and Magnetism for Muhammad sani

Question

the long cylindrical conductor with radius (a) is surrounded by coaxial cylindrical conducting shell with inner radius (b) .determine the capacitance per unit length possist capacitance,assuming the medium between the two concentric cylinder is vacuum.

Answer 1

Lets find electric field of cylindrical conductor:

$\lambda$ - charge per unit length

From Gauss Law, electric flux over coaxial cylindrical surface with length L and radius a < r < b:

$$\Phi = E2\pi r L = \frac{L\lambda}{\varepsilon_0}$$

Therefore,

$$E = \frac{\lambda}{2\pi\varepsilon_0 r}$$

Voltage difference between two conducting cylinders:

$$V=\int_a^b E(r)dr=\int_a^b\frac{\lambda dr}{2\pi\varepsilon_0 r} = \frac{\lambda}{2\pi\varepsilon_0} \log{\frac{b}{a}}$$

By definition, capacitance per unit length is

$$C = \frac{\lambda}{V} = \frac{2\pi\varepsilon_0}{\log{\frac{b}{a}}}$$