A Time derivative jump of the electric/magnetic field

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Summary
I want to know if the time derivative of the electric and or magnetic field jumps over a material interface.
So I just wanted to see if anyone could offer some suggestions. So in my mind this seems impossible, in the case of electric field a jump in time derivative of that field would indicated in my mind that electric charge was either introduced or removed from the system instantaneously which would violate conversation of electric charge. If you prescribe to the quantum mechanic perspective that magnetic charge exists, the same argument would hold the magnetic field, since magnetic charge would also have to be conserved. If you just assume gausses law of magnetism holds, then again, this would mean a magnetic monopole would be introduced to the system, which is also problematic.
I'm not sure if there is a rigorous mathematical argument for this though - or if this is just true based on empirical observation (similarly how gausses law of magnetism is derived)
I would very much appreciate some comments on this subject.
 
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Maxwell's Equations include equations for the time derivatives of both the electric and magnetic fields. What do they imply about your question?
 
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Coincidentally I have a similar doubt but probably it comes from a different perspective. The boundary conditions proposed for electromagnetic fields consider only a spatial normal term. But I want to know what happens with the jump in an arbitrary spacetime surface. Mathematically, we know what happens when we dot and cross a purely spatial vector with the jump of the electromagnetic field, but what happens when we dot and cross with a time vector? For instance, we ask: for the electric field at a given instant to another instant, what is the jump between a media that changes its properties at that point (maybe a nonlinear material)? In the formalism of Clifford Algebra (See John W. Arthur - Understanding Geometric Algebra for Electromagnetic Theory), I ask

$$ \vec{e}_t (E_2 - E_1) + \vec{e}_t (B_2 - B_1) = ? $$

Where ## E_2 ## and ## B_2 ## are the electromagnetic fields at the same spatial point of ## E_1 ## and ## B_1 ## but at an infinitesimal instant after. This is analogous to the jump between two material medium in space, but with the normal being a time vector.

Maybe we can derive something like this by applying the integral laws and considering a jump in ## \varepsilon ## and ## \mu ## from a given instant to another. Of course, this implies changing the constitutive laws of the material. In a similar spirit of the space normal, probably it will be a function of a charge and a current surface density (which in our spatial notion should appear as a sudden volumetric change of current or charge).

I physically think it's obvious that in a medium free of charges, the fields should be continuous, changing only speed of propagation. But maybe we need a more rigorous treatment just to be sure. Considering that charges cannot disappear, the situation where ## \rho ## would peak at a given volume would be physically impossible, but maybe a change in current?

Maybe this can give a new perspective to your problem.
 

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