Two concentric spherical conducting shells of radii R1 and R2 have potentials V1 and V2. Find V(r) b/w shells?

Two concentric spherical conducting shells of radii R1 and R2 have potentials V1 and V2. Find V(r) between the shells.

Two concentric spherical conducting shells of radii R1 and R2 have potentials V1 and V2. Find V(r) b/w shells?
If V2 is for the outer shell and V1 for the inner shell, then whatever is between the shells is V2 - V1 (V(r)). The following integral is from ra to rb (inner radius a and inner radius b)



V(r) = - ∫ E dr (vector E, electric field and vector dr, infinitesimally small radius. It's dr instead of dl since your dealing with radii rather than "regular lengths")



= - ∫ E * dr * cos α (dot product definition)

= - ∫ E * dr * cos0° (the angle between the electric field and dr is 0°)



= - ∫ (2kλ)/r * dr (E = (2kλ)/r since that's the formula got the electric field of a rod (cylinder))



= -2kλ [ ln r ] (evaluated at rb and ra)



= -2kλ ln rb - ln ra

= -2kλ ln (rb / ra)



and that's it