/sci/ - Science & Math

now, consider maxwell's equation for the curl of H
[eqn]\nabla \times \vec{H} = \vec{J}_e +\frac{\partial \vec{D}}{\partial t}[/eqn]
take the divergence
[eqn]\nabla \cdot (\nabla \times \vec{H}) = \nabla \cdot \vec{J}_e + \frac{\partial}{\partial t} \nabla \cdot \vec{D}[/eqn]
note that the divergence of a curl is zero and insert maxwell's equation for the divergence of D
[eqn]0 = \nabla \cdot \vec{J}_e + \frac{\partial \rho_e}{\partial t}[/eqn]
this is the continuity equation. it is a consequence of maxwell's equations, and not independent of them.
rewrite the equation as
[eqn]\nabla \cdot \vec{J}_e = - \frac{\partial \rho_e}{\partial t}[/eqn]
put it into integral form
[eqn]\oint_S \vec{J}_e \cdot d\vec{S} = - \frac{\partial}{\partial t} \int_V \rho_e dV[/eqn]
rewrite the last integral as total electric charge
[eqn]\oint_S \vec{J}_e \cdot d\vec{S} = - \frac{\partial Q_e}{\partial t}[/eqn]
assume the net charge in the circuit is constant (this is mostly a good approximation for everyday electronics). being constant, the time derivative is zero
[eqn]\oint_S \vec{J}_e \cdot d\vec{S} = 0[/eqn]
in words, if you track all of the current going into and out of a volume/node, it must sum to zero. this is kirchoff's current law.

you aren't going to win the nobel prize by finding holes in kirchoff's laws because they make assumptions to simplify maxwell's equations to make them easier to work with when designing circuits. of course, if you fall outside those assumptions, be prepared to do full wave simulations to understand what's going on. debate the validity of maxwell's laws all you want (even they aren't "reality") but i'm out