A homogeneous poorly conducting medium of resistivity ρ 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 I. Neglecting the edge effects, find the resistance of the medium between the cylinders.

Solution

A poorly conducting medium of resistivity ρ fills the space between two thin coaxial ideally conducting cylindrs.
Radius of inner cylinder = a
Radius of outer cylinder = b

The current will be conducted radially outwards from the inner conductor (say) to the outer.
The area of cross-section for the conduction of the current is, therefore, the area of an elementary cylindrical shell and which varies with radius.

The length of the conducting shell is measured radially from radius a to radius b.

Consider an elementary cylindrical shell of radius r and thickness dr.
Its area of cross- section (normal to flow of current) = 2πrl
Length of the cylindrical shell = dr.
Hence, the resistance of the elementary cylindrical shell of the medium is given using the formula,

R = ρ \frac{l}{A}

i.e.,

dR = \frac{ρdr}{2 πrl} = \frac{ρ}{2 πl} [\frac{dr}{r}]

The resistance of the medium is obtained by integrating for r from a to b.
Hence, required resistance is

R = \frac{ρ}{2 πl} \int_{a}^{b} \frac{dr}{r} = \frac{ρ}{2 πl} {[\log_{e} r]}_{a}^{b} = (\frac{ρ}{2 πl}) \log_{e} \frac{b}{a}

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