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.

 

[raining Pete]

a) To show that the net current J(r) is zero, we can use the definition of current as the rate of flow of particles per unit area. In this case, since the flux is isotropic, there is an equal number of particles moving in every direction around the point r(vector). This means that the particles moving in one direction will be balanced out by the particles moving in the exact opposite direction, resulting in a net current of zero.

b) To show that the magnitude of the current in any direction is one quarter of the scalar flux, we can use the fact that the scalar flux is defined as the total number of particles passing through a unit area per unit time. In this case, since the flux is isotropic, the total number of particles passing through a unit area in any direction will be equal to the scalar flux divided by the total solid angle (4π). Therefore, the magnitude of the current in any direction will be equal to one quarter of the scalar flux.