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The electric field of a plane polarised electromagnetic wave in free space at t = 0 is given by E(x, z) = 10 * j_hat * cos(6x + 8z). Find the magnetic field B(x, z, t), where c is the speed of light.

1. (1/c) * (6*k_hat + 8*i_hat) * cos[(6x - 8z + 10ct)]
2. (1/c) * (6*k_hat - 8*i_hat) * cos[(6x + 8z - 10ct)]
3. (1/c) * (6*k_hat + 8*i_hat) * cos[(6x + 8z - 10ct)]
4. (1/c) * (6*k_hat - 8*i_hat) * cos[(6x + 8z + 10ct)]

Correct answer: (1/c) * (6*k_hat - 8*i_hat) * cos[(6x + 8z - 10ct)]

Solution

Wave vector k_vec = 6i + 8k, |k_vec| = 10, so omega = 10c. Unit vector k_hat = (6i+8k)/10. B_hat = k_hat x E_hat = ((6i+8k)/10) x j = (6(i x j) + 8(k x j))/10 = (6k - 8i)/10. Magnitude |B| = |E|/c = 10/c. So B = (10/c)*(6k-8i)/10 * cos(6x+8z-10ct) = (1/c)*(6k-8i)*cos(6x+8z-10ct).

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