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TheJfactors

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When I go to back solve for the strength of the electric field in the x, y, and z directions after generically taking the curl of the electric field's components and setting it equal to the negative of the partial derivative with respect to time of a circular magnetic field (arbitrarily made the magnetic field change linearly with time to make the math easier), I find myself with a differential equation that I sort of just simplified with assumptions that I don't think are valid. Assuming the E-fields X and Y components are zero ensures the cross product on the left hand side ∇ x E still remains a valid solution to the partial derivative -dB/dt on the right hand side of the equation. However this is sort of just what I'm hoping will happen and it's not exactly a valid method of solving differential equations last time I checked haha.

Anyone want to shed some light on how they've done this problem in the past? I guess I am unsure whether I am just missing some first principle physics or if the lack of Z component in the magnetic field allows some assumptions about the X and Y direction of the of the E-field to be formed. Or if I just have a lot of dif eq to do... :)

Thanks in advance for any help!