Are Maxwell's Equations More Complex Than We Thought?

[lichen]

Easy question for those who know, I expect. It would help me understand though.

Generally, wherever I look for information the equation

curl E = - dB/dt

is given, but in some areas I see the equation

curl E(r,t) = j*omega*u0*H(r,t)

where j is the imaginary unit, omega is angular freq., u0 is permeability of free space.

B=u0*H so that's ok, but does this imply that dB/dt = -j*omega*B ?

Also, curl H(r,t) = -j*omega*e0*e*E(r,t) is given where e0 is permittivity of free space, e is a dielectric tensor. I assume this can be explained in the same way.

I'm missing a chunk of understanding as you probably notice ;)

 

[Andy Resnick]

Usually, when trying to write down solutions to Mawell's equations, people assume two things about the field:

1) The spatial part and the temporal part can be separated: E(r,t) = R(r)T(t).
2) The temporal part can be written as T(t) = Ae$^{i \omega t}$.

The first assumption is fairly basic to solving differential equations, and separable functions are a very important class of solutions- I can't give a consise reason why, but for now, it makes it possible to analytically solve the equations.

The second assumption just means that the temperal part oscillates like a sine wave. It's written that way to be more general (and actually, the full expression is T(t) = Ae$^{i \omega t}$ + BAe$^{-i \omega t}$ ). There's good reasons for this assumption as well, which I don't need to get into now.

Anyhow, hopefully you can see where the j$\omega$ comes from now- electrical engineers use 'j' instead of 'i' becasue 'i' is current density.

The other part is the conversion of E to D, and B to H. But you seem to have a handle on that part.

 

[lichen]

Ah, yes I see now.

Thanks very much!