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A plane electromagnetic wave travels along the direction (i + j)/sqrt(2) and is polarized (E field) along k. Which expression correctly gives the magnetic field of the wave (B0 is a suitable constant)?

1. B0*(i - j)/sqrt(2) * cos(wt - k(i + j)/sqrt(2))
2. B0*(i + j)/sqrt(2) * cos(wt - k(i + j)/sqrt(2))
3. B0*k * cos(wt - k(i + j)/sqrt(2))
4. B0*(j - i)/sqrt(2) * cos(wt + k(i + j)/sqrt(2))

Correct answer: B0*(i - j)/sqrt(2) * cos(wt - k(i + j)/sqrt(2))

Solution

For an EM wave, the direction of B satisfies n = E_hat x B_hat (right-handed). With E along k and n along (i+j)/sqrt2, B must be perpendicular to both. Computing B_hat from n = E_hat x B_hat gives B along (i - j)/sqrt2. The phase (wt - k*n.r) shows propagation along +n. Hence option with B0*(i - j)/sqrt2 and (wt - k(i+j)/sqrt2).

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