0
$\begingroup$

A quick question. Given that charge density is the charge per unit length/area/volume, I feel that it should be possible to write current density using vector notation as $ \frac{\partial}{\partial t}\nabla Q(\vec r) $ where Q is the charge. I haven't see this expression around however. Is this correct, or am I overlooking something?

$\endgroup$
0
$\begingroup$

No, that's not correct. For one, it doesn't have the correct units - if $Q$ is a charge, then $\partial_t\nabla Q$ is a charge per unit time per unit length, instead of per unit area. More importantly, it is perfectly possible to have a whole region (both in a line, as a surface, and in bulk space) where the charge density is zero and the current density is nonzero; to see examples, choose any current-carrying wire around you.

You can relate the current and charge densities through the continuity equation, $$ \nabla\cdot\mathbf j+\frac{\partial \rho}{\partial t}=0, $$ with analogues in lower dimensions, but other than that there's no other universal ways to connect them.

$\endgroup$
  • $\begingroup$ Yes you're quite right! $\endgroup$ – Nonsematter Apr 21 '17 at 8:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.