The electrodes of a spherical capacitor have radii a and b > a. The inner electrode is grounded; the outer electrode is held at potential V. In vacuum diode mode, the thermionic current which flows from the inner cathode to the outer anode increases with temperature until the electric field due to space charge produced between the concentric electrodes compensates the voltage-induced field and ∂φ/∂r|_{r = a} = 0 on the surface of the cathode. Assume that the initial velocity of the thermally emitted electrons is zero.
(a) Show that the maximum current I between the electrodes is described by the expression 

The function y(x) is the solution to a differential equation which cannot be integrated in closed form. 

(b) Find the maximum current in the limit when b >> a.

Transcribed Image Text:

I -√√ = 2e -4π €0 y(x) 3/2 x = ln(b/a).