Confusion regarding continuity equation in electrodynamics

[ppoonamk]

Suppose I have two charged particles with charge densities ρ1(r,t) and ρ2 (r,t) with corresponding velocity fields V1(r,t) and V2(r,t). Can I write continuity equation for the combined system? Wouldn't charges moving with different velocities would contribute differently to the current which will violate the continuity equation?

 

[Ken G]

The continuity equation is linear in the charge density-- that means the sum of any two solutions is also a solution.

 

[Ken G]

Perhaps the key point to stress is that if two different charge distributions are associated with two different velocity distributions, you can add the charge densities to get the local charge density, but you don't add the velocities-- there is no local velocity, you'd have to define a charge-weighted velocity to be able to use it like an independent local quantity in the continuity equation. You can think in terms of the charge flux density vector, which is the product of the charge density and the velocity, and then that quantity can be superimposed if you have two separate solutions, but the velocities of the two separate components cannot be meaningfully added the way the charge densities can.