Transport fundamentals question

[nucleargt]

Consider a point in space r(vector) where the flux is isotropic; i.e., equal numbers of neutrons move into solid angles d(omega) about every direction omega.

a) Show that the net current J(r) is zero
b) Show that the magnitude of the current in any direction is just one quarter of the scalar flux.

 

[Astronuc]

It's been many moons since I had to show those.

a) Isotropic flux implies that this it is the same in all directions, so at a point the current in the + direction (J⁺) equals the current in negative direction (J^-), so the two cancel in all orientations of the 4$\pi$ solid angle.

b) One shows that the current is integrated over the surface area of a sphere 4$\pi$r², but the scalar flux applies to a disc shape area or the projected area of the sphere, which is just $\pi$r². Think of the definitions of the current and flux.