Conduction and displacement currents for a spherical solid

[Huan Nguyen]

Homework Statement

Show that the conduction and displacement currents cancel each other for a spherical radioactive solid emitting charged particles radially outwards

Homework Equations

Maxwell's equations
Current density (j)
Displacement current density (jd)

The Attempt at a Solution

I haven't made it very far into the problem. I think "the conduction" means i=dq/dt and "displacement current" id=jd*S with jd= ∂D/∂t. However, I am still struggling to understand the problem.

 

[TSny]

I am still struggling to understand the problem.

Hello, Huan. Welcome to PF!

Every volume element of the sphere will be continually emitting charged particles due to the radioactive decay. The emitted particles are assumed (unrealistically) to stream out radially without being reabsorbed anywhere in the sphere. So, as the sphere emits particles the sphere builds up charge. Charge must be conserved locally and this can be used to relate the radial conduction current density, ##j_r##, to the charge density, ##\rho##, of the sphere. There is a well known formula for local charge conservation.

As you noted, the displacement current, ##j_d##, is related to ##\dot{\mathbf{D}}##. You'll need to relate ##\mathbf{D}## to ##\rho##.