I have a question about the derivation of the boundary conditions at surfaces of electromagnetic fields. These conditions say, that the tangential component of the electric and the normal component of the magnetic field are continuous at surfaces.
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/electrica...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?
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/electrica...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?