The expression for the resistance (R) between two spherical conducting shells of radii a and b (with b>a) separated by a material with conductivity is 1/4πσ( 1/a-1/b), where σ is the conductivity of the material.
The expression for the resistance (R) between two spherical conducting shells of radii a and b (with b>a) separated by a material with conductivity can be derived based on the formula for the resistance of a cylindrical conductor. In this case, the cylindrical conductor is represented by the material between the two shells.
The resistance of the cylindrical section is given by R= ρL/A, where ρ is the resistivity, L is the length, and A is the cross-sectional area. Considering a thin cylindrical shell between the two spheres, the length (L) is the radial separation between the shells, and the cross-sectional area (A) is 2πrdr, where r is the radial distance from the center.
Integrating this expression over the radial distance from a to b provides the total resistance (R). Utilizing the conductivity (σ= 1/ρ), the final expression for the resistance between the two spherical conducting shells is 1/4πσ( 1/a-1/b). This expression encapsulates the influence of the material's conductivity and the geometrical configuration of the two spherical shells on the overall resistance between them.