Consider the following ``spherical cow'' model of a battery connected to an external circuit. A sphere of radius a and conductivity σ is embedded in a uniform medium of conductivity σ′. Inside the sphere there is a uniform (chemical) force in the z direction acting on the charge carriers; its strength as an effective electric field entering Ohm's law is F. In the steady state, electric fields exist inside and outside the sphere and surface charge resides on its surface.
(a) Find the electric field (in addition to F) and current density everywhere in space. Determine the surface-charge density and show that the electric dipole moment of the sphere is p=4πϵ0σa3F/(σ+2σ′).
(b) Show that the total current flowing out through the upper hemisphere of the sphere is
I=σ+2σ′2σσ′⋅πa2F
Calculate the total power dissipation outside the sphere. Using the lumped circuit relations, P=I2Re=IVe, find the effective external resistance Re and voltage Ve.
(c) Find the power dissipated within the sphere and deduce the effective internal resistance Ri and voltage Vi.
(d) Define the total voltage through the relation, Vt=(Re+Ri)I and show that Vt=4aF/3, as well as Ve+Vi=Vt. Show that IVt is the power supplied by the ``chemical'' force.
Reference: W. M. Saslow, Am. J. Phys. 62, 495--501 (1994).