Question

The space between two metallic coaxial cylinders of length L and radii R1 and R2 is completely filled with a material having a resistivity rho. The figure below shows a cross section of the configuration. Derive an expression for the resistance R of this cylindrical configuration in terms of the given parameters L, 1R

1 2R2, and rho. Include the relevant formulas and steps in your explanation.

Answer 1

The resistance R of a cylindrical material between two coaxial cylinders is found by integrating the differential resistance of infinitesimally thin rings, leading to the formula R = (rho / (2 * pi * L)) * ln(R2 / R1), where rho is the resistivity, and R1 and R2 are the radii of the cylinders.

Step-by-step explanation:

The student is asking for the derivation of an expression for the electrical resistance of a configuration consisting of two coaxial cylinders with radii R1 and R2, filled with a material of resistivity rho, and length L.

To derive this expression, we utilize the formula for resistance R of a cylindrical object, which is R = rho * (L / A), where A is the cross-sectional area.

To find the resistance across the material filling the space between the coaxial cylinders, we consider a differential area dA at a radius r, which is an infinitesimally thin ring with thickness dr.

The resistance dR of this thin ring is given by dR = rho * (dr / (2 * pi * r * L)).

To find the total resistance, we integrate dR from R1 to R2, resulting in the expression R = (rho / (2 * pi * L)) * ln(R2 / R1).