91Ó°ÊÓ

The conductivity of the material,$\mathrm{σ}\left(s\right)=\frac{k}{s}$

Here k is the constant.

The current is l.

The equation to calculate the surface current density is given as follows.

$\mathbf{J}\left(s\right)\mathbf{=}\frac{\mathbf{l}}{\mathbf{A}}$ …… (1)

Here, A is the area of surface perpendicular to the current.

The equation to calculate the area of the surface perpendicular to the current is given as follows.

$\mathbf{A}\mathbf{=}\mathbf{2}\mathbf{Ï„Ï„²¹³¢}$ (2)

Here, is the radius of the cylinder and is the length of cylinder.

The surface current density also given as follows.

$\mathbf{J}\left(s\right)\mathbf{=}\mathbf{·¡Ïƒ}$ …… (3)

Here, E is the electric field intensity.

The equation to calculate the potential difference between the cylinder is given as follows.

$\mathbf{V}\mathbf{=}\mathbf{-}{\mathbf{∫}}_{\mathbf{b}}^{\mathbf{a}}\mathbf{E}\mathbf{.}\mathbf{dl}$ …… (4)

The equation to calculate the resistance between the cylinder is given as follows.

$\mathbf{R}\mathbf{=}\frac{\mathbf{V}}{\mathbf{l}}$ …… (5)

Consider the gaussian cylinder having the radius and length .

Equate the equation (1), equation (2) and (3),

$E\mathrm{σ}=\frac{l}{A}$

Substitute for A and $\frac{k}{s}$for $\mathrm{σ}$into above equation.

$$\begin{gathered} E×\frac{k}{s}=\frac{l}{2\mathrm{Ï€²õ³¢}} \\ E×\frac{k}{s}=\frac{l}{2\mathrm{Ï€³¢}} \\ E=\frac{l}{2\mathrm{Ï€°ì³¢}} \end{gathered}$$

Calculate the potential difference between the cylinder.

Substitute $\frac{l}{2\mathrm{Ï€°ì³¢}}$for E into equation (4).

$$\begin{gathered} V=-{∫}_b^a\frac{l}{2\mathrm{Ï€°ì³¢}}.dl \\ V=-\frac{l}{2\mathrm{Ï€°ì³¢}}{∫}_b^{a}dl \\ V=-\frac{l}{2\mathrm{Ï€°ì³¢}}\left(a-b\right) \\ V=\frac{l}{2\mathrm{Ï€°ì³¢}}\left(b-a\right) \end{gathered}$$

Calculate the expression for the resistance of the cylinder.

Substitute$\frac{l}{2\mathrm{Ï€°ì³¢}}\left(b-a\right)$ for V into equation (5).

$R=\frac{{\displaystyle \frac{l}{2\mathrm{Ï€°ì³¢}}}\left(b-a\right)}{l}$

role="math" localid="1657699971864" $R=\frac{l}{2\mathrm{Ï€°ì³¢}}\left(b-a\right)$

Hence the resistance between the cylinder is$\frac{l}{2\mathrm{Ï€°ì³¢}}\left(b-a\right)$ .