Planar Pierce Gun.
(Taken from the footnotes of test data file test2d12.dat).
The potential outside the beam is given by (for example see P. H. Hawkes and E. Kasper, Principles of Electron Optics, 1989, Academic Press):
V = C*r**(4/3)*cos(4*phi/3),
where
x = a + r*sin(phi), z = r*cos(phi),
C = (2.25*j/epsilon0)**(2/3)*(0.5*m/e)**(1/3),
and where 2*a is the width of the beam (and ** signifies 'to the power of').
The theoretical Childs Law current density is w*(V**1.5)/L**2, where w = .0023340 mA/mm**2.