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An electromagnetic wave has electric field E = 4.8*10² * cos(2*10⁷ * z + 6*10¹⁵ * t) * (-i_hat + 2*j_hat) V/m. Find the associated magnetic field B.

1. B = 1.6*10^(-6) * cos(2*10⁷*z + 6*10¹⁵*t) * (2*i_hat - j_hat) Wb/m²
2. B = 1.6*10^(-6) * cos(2*10⁷*z + 6*10¹⁵*t) * (-i_hat + 2*j_hat) Wb/m²
3. B = 1.6*10^(-6) * cos(2*10⁷*z + 6*10¹⁵*t) * (2*i_hat + j_hat) Wb/m²
4. B = 1.6*10^(-6) * cos(2*10⁷*z + 6*10¹⁵*t) * (2*i_hat - j_hat) Wb/m²

Correct answer: B = 1.6*10^(-6) * cos(2*10⁷*z + 6*10¹⁵*t) * (2*i_hat + j_hat) Wb/m²

Solution

The wave travels in the -z direction (positive kz with positive omega*t indicates -z propagation). B = (k_hat x E)/c = (-z_hat x E)/c. Computing: -z_hat x (-i+2j) = z_hat x (i-2j) = z_hat x i - 2*(z_hat x j) = j_hat - 2*(-i_hat) = 2i+j. Magnitude: |B| = |E|/c = 4.8*10² * sqrt(5) / (3*10⁸ * sqrt(5)) = 1.6*10⁻⁶. So B = 1.6*10⁻⁶*(2i+j)*cos(...).

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