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In free space at t = 0 the electric field of a plane-polarized electromagnetic wave is E(x,z) = 10 j cos(6x + 8z). The corresponding magnetic field B(x,z,t) is (c = speed of light):

1. (1/c)(6 k - 8 i) cos(6x + 8z - 10ct)
2. (1/c)(6 k + 8 i) cos(6x - 8z + 10ct)
3. (1/c)(6 k + 8 i) cos(6x + 8z - 10ct)
4. (1/c)(6 k - 8 i) cos(6x + 8z + 10ct)

Correct answer: (1/c)(6 k - 8 i) cos(6x + 8z - 10ct)

Solution

Wave vector k_vec = 6 i + 8 k, |k| = sqrt(36+64) = 10, so omega = c|k| = 10c and the propagating phase is (6x + 8z - 10ct). Direction of propagation k_hat = (6 i + 8 k)/10. With E along j, B is along k_hat x j: (6 i + 8 k)/10 x j = (6 (i x j) + 8 (k x j))/10 = (6 k - 8 i)/10. Its magnitude is E0/c = 10/c, so B = (10/c)*(6 k - 8 i)/10 * cos(...) = (1/c)(6 k - 8 i) cos(6x + 8z - 10ct).

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