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The electric field of a standing electromagnetic wave in space is given by E = E0*sin(k*y)*sin(omega*t) in the z-direction (k-hat). What is the corresponding magnetic field B? (c = speed of light in vacuum)

1. B = (E0/c)*sin(k*y)*sin(omega*t) in the x-direction (i-hat)
2. B = -(E0/c)*sin(k*y)*sin(omega*t) in the x-direction (i-hat)
3. B = (E0/c)*cos(k*y)*cos(omega*t) in the x-direction (i-hat)
4. B = -(E0/c)*cos(k*y)*cos(omega*t) in the x-direction (i-hat)

Correct answer: B = (E0/c)*cos(k*y)*cos(omega*t) in the x-direction (i-hat)

Solution

From Faraday's law: curl E = -dB/dt. With E = E0*sin(ky)*sin(omega*t)*k-hat, only dEz/dy is nonzero: curl E = E0*k*cos(ky)*sin(omega*t)*i-hat. So -dB/dt = E0*k*cos(ky)*sin(omega*t)*i-hat. Integrating: B = (E0*k/omega)*cos(ky)*cos(omega*t)*i-hat = (E0/c)*cos(ky)*cos(omega*t)*i-hat (since k/omega = 1/c).

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