Current density constant in time implies charge density linear in time?

[davethecello]

Does current density being constant in time and the charge density NOT being constant in time imply that the charge density has to be a linear function of time? i.e of the form
$$p(r,t) = p(r,0) + \dot{p}(r,0)t }$$
Can this be derived using the continuity equation alone?

 

[Meir Achuz]

Yes. For j to be constant in time requires dp/dt to be constant in time.

 

[astrokreng]

I must clarify that current density and charge density are two separate physical quantities and cannot be equated. Current density refers to the flow of electric charge per unit area, while charge density refers to the amount of electric charge per unit volume.

In general, the continuity equation states that the change in charge density over time is equal to the negative divergence of the current density. This means that if the current density is constant in time, the change in charge density over time will also be constant. However, this does not necessarily imply that the charge density itself is linear in time.

In fact, the charge density can vary in a nonlinear manner even when the current density is constant. This is because the continuity equation does not provide information about the initial conditions of the system, which can greatly affect the behavior of the charge density over time.

Therefore, it is not possible to derive the specific form of the charge density (i.e. p(r,t) = p(r,0) + \dot{p}(r,0)t) using the continuity equation alone. Other physical laws and equations, such as Ohm's law, may be necessary to fully describe the relationship between current density and charge density.