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Their derivation goes as follows: To derive them for the electric field,

one starts with the Maxwell equation $\mbox{rot } E=- \frac{1}{c} \frac{\partial B}{\partial t}$ and uses Stokes theorem for a line integral of $E$ across the boundary, as it is depicted at http://ocw.mit.edu/courses/electric...applications-fall-2005/lecture-notes/lec2.pdf. The derivation for the magnetic field goes similar, but now we have to use the equation $\mbox{div} B=0$.

My question now is: Why can't we use some other Maxwell equations as well, to obtain further conditions? We still have the Maxwell equation $\mbox{rot} B= \frac{4 \pi}{c}j+ \frac{1}{c} \frac{\partial E}{\partial t}$. Shouldn't we get something out of it at least for $j=0$? What is the reason that there isn't something similar, that works for other Maxwell equations?