Use of potentials (electrodynamics, classical)

AI Thread Summary
The discussion focuses on the derivation of a wave equation using the vector potential A in the context of a time-varying current sheet located in the xy plane. The user expresses confusion specifically regarding the transition from equation 6.46 to 6.47, suspecting it may involve a straightforward application of the chain rule. They seek clarification on the boundary conditions related to this problem, as they find jump conditions challenging. The user is looking for assistance to understand this step, which is causing frustration. Overall, the thread highlights the complexities of applying electrodynamics concepts to specific problems in classical physics.
bman!!
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hello, I've just gone through a derivation using the vector potential A, and using the equation derived (a wave equation) which arises from using the lorentz gauge, it is applied to the problem of a current sheet in the xy plane, located at z=0 with current in the x direction only. the current varies with time so we are looking for wave solutions propagating in the + and - z directions.

Ive attached the relevant page (its not long at all), and basically my problem is simply: i understand pretty much everything up until equation 6.46, however, i cannot for the life of me understand how he gets to 6.47.

i know its probably something really simple, probably like a nifty application of the chain rule, but Id appreciate if someone could point out the elephant to me becuase i really can't see it and its annoying the hell out of me.

cheers

(im sure the answer will make me feel like an Rtard)
 

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is the picture not working or something? its just a simple boundary (jump?) conditions problem. i say simple, I'm pretty terrible with jump conditions, so any help is really appreciated. this one step is driving me mad!
 
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