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Something has been bothering me about deriving the wave equation for a plane EM wave. We were showed this derivation in class and had to reproduce it but something is not making sense to me...

The derivation is as follows:

Suppose you have a plane EM wave (in a vaccuum) travelling in the x-direction. The E-field is in the y-direction and the B-field in the z-direction.

Take a rectangular loop in the xy-plane with length [itex]dx[/itex] and height [itex]l[/itex].

Using the maxwell equation:

[tex]\oint \vec{E} \cdot \vec{dl} = - \frac{ d \phi_B}{dt}[/tex]

[tex]E(x+dx,t)l - E(x,t)l = -\frac{d}{dt} B(x,t) l dx[/tex]

[tex]\frac{E(x+dx,t) - E(x,t)}{dx} = - \frac{d}{dt} B(x,t)[/tex]

[tex]\frac{ \partial E(x,t)}{\partial x} = - \frac{ \partial B(x,t)}{\partial t}[/tex]

And similar for B.

Now what bothers me is the calculation of [itex]\phi_B[/itex]... I am told that we are assuming B to be constant over the infinitesemal distance dx. However, we are not assuming the same for E? We are using the change in E-field explicitly to arrive at the partial derivative for E, but we ignore it for B, isn't that strange?

Also, similarly, since we are advancing x with dx, shouldn't we advance t with dt also?

I guess ignoring this is a safe 'approximation' to make since dt = dx / c which is even more small...

What is going on here? Why can we do this?

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# EM waves - wave equation derivation

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