Electric and magnetic field relationship

AI Thread Summary
The electric field E(x, y, z, t) and magnetic field B(x, y, z, t) presented are invalid because they do not satisfy the requirements of Maxwell's equations. Specifically, both fields must propagate in the same direction, necessitating that they share the same phase term, either (kz-wt) or (kz+wt). The differing phase terms indicate a violation of the relationship between electric and magnetic fields in electromagnetic waves. Additionally, the cross product of these fields results in a standing wave, which is inconsistent with the behavior of propagating electromagnetic waves. Therefore, the fields cannot coexist as valid solutions in classical electromagnetism.
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Homework Statement


Why is the following set of electric and magnetic fields invalid?
##E(x, y, z, t)= E_0 \sin{(kz-wt)} \hat{i}##

##B(x, y, z, t) = B_0 \sin{(kz+wt)} \hat{j}##

Homework Equations

The Attempt at a Solution


So, I understand that either both should have (kz-wt) or both should have (kz+wt). Either than the fact that the cross product of electric and magnetic fields is a scalar times a vectors that does not depend on time, whereas here the cross product gives me something like a standing wave in the z direction, I'm not sure why both should have (kz-wt) or both should have (kz+wt). Please help.
 
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Have you studied Maxwell's equations?
 
Yes
 
If you can show that anyone of Maxwell's equations is violated at some point of space at some instant of time, then the fields are invalid.
 
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Thank you! Got it.
 
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