Q43P - Introduction to Electrodynamics

Question

Find the capacitance per unit length of two coaxial metal cylindrical tubes, of radi${}^{\mathbf{a}}$and${}^{\mathbf{b}}$.

Step-by-Step Solution

Answer

The capacitance per unit length is${}^{\frac{C}{I}=\frac{2\pi {\epsilon }_0}{In\left({\displaystyle \frac{b}{a}}\right)}}$

Step 1: Determine the expression for the electric field.

Write the expression for electric filed,

$\mathit{E}\mathbf{=}\frac{\mathbf{q}}{\mathbf{2}\mathbf{\pi }\mathbf{s}\mathbf{I}\mathbf{ }{\mathbf{\epsilon }}_{\mathbf{0}}}\hat{\mathbf{s}} for \mathit{a}\mathbf{ }\mathbf{<}\mathit{s}\mathbf{ }\mathbf{<}\mathit{b}$

Here, ${}^q$ is the charge, ${}^E$is the electric filed.

Consider the given figure.

Now consider${}^I$ is the length of the tube.

Let’s give the charge${}^{+q}$to the inner cylinder and${}^{-q}$ to the outer cylinder.

Step 2: Determine the expression for potential difference

Now finding the potential difference between inner and outer faces of cylinder and it is given by,

$$\begin{gathered} \mathit{V}\left(a\right)\mathbf{-}\mathit{V}\left(b\right)\mathbf{=}\mathbf{-}{\mathbf{\int }}_{\mathbf{b}}^{\mathbf{a}}\left(\frac{q}{2\pi sI {\epsilon }_0}\right)\mathit{d}\mathit{s} \\ \mathit{V}\left(a\right)\mathbf{-}\mathit{V}\left(b\right)\mathbf{=}\mathit{V} \end{gathered}$$

Solve further as,

$$\begin{gathered} V=-{\int }_b^a\left(\frac{q}{2\pi sI{\epsilon }_0}\right)ds \\ =-\frac{q}{2\pi {\epsilon }_{0}I}\left(\log a-\log b\right) \\ =\frac{q}{2\pi {\epsilon }_{0}I}In\left(\frac{b}{a}\right) \end{gathered}$$

Step 3: Determine Capacitance per unit length

$\frac{C}{I}=\frac{2\pi {\epsilon }_0}{In\left({\displaystyle \frac{b}{a}}\right)}$Consider the expression for the potential.

Now comparing the above equation with $V\frac{q}{C}$, here ${}^C$ is the capacitance. Then,

$\frac{C}{I}=\frac{2\pi {\epsilon }_0}{In\left({\displaystyle \frac{b}{a}}\right)}$

${}^{\frac{C}{I}=\frac{2\pi {\epsilon }_0}{In\left({\displaystyle \frac{b}{a}}\right)}}$

Hence, the capacitance per unit length is given by,