Question

A homogeneous poorly conducting medium of resistivity $\rho$ fills up the space between two thin coaxial ideally conducting cylinders. The radii of the cylinders are equal to $a$ and $b$, with $a<b$, the length of each cylinder is $l$. Neglecting the edge effects, find the resistance of the medium between the cylinders.

Answer 1

Let us mentally isolate a thin cylindrical layer of inner and outer radii $r$ and $r+dr$ respectively. As lines of current at all the points of this layer are perpendicular to it, such a layer can be treated as a cylindrical conductor of thickness $dr$ and cross-sectional area $2\pi rl$. So, we have,
$dR=\rho \frac{dr}{S\left(r\right)}=\rho \frac{dr}{2\pi rl}$ and integrating between the limits, we get,
$R=\frac{\rho }{2\pi l}ln\frac{b}{a}$.