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A plane electromagnetic wave with frequency nu = 23.9 GHz propagates in the +z direction in free space. The peak electric field is 60 V/m. Which of the following correctly represents the magnetic field component of this wave?

1. B = 2 * 10⁻⁷ sin(0.5 * 10³ * z + 1.5 * 10¹¹ * t) i-hat
2. B = 2 * 10⁻⁷ sin(1.5 * 10² * x + 0.5 * 10¹¹ * t) j-hat
3. B = 2 * 10⁻⁷ sin(0.5 * 10³ * z - 1.5 * 10¹¹ * t) i-hat
4. B = 60 sin(0.5 * 10³ * x + 1.5 * 10¹¹ * t) k-hat

Correct answer: B = 2 * 10⁻⁷ sin(0.5 * 10³ * z - 1.5 * 10¹¹ * t) i-hat

Solution

Given: nu = 23.9 GHz, E0 = 60 V/m, propagation along +z. B0 = E0/c = 60/(3*10⁸) = 2*10⁻⁷ T. omega = 2*pi*23.9*10⁹ ≈ 1.5*10¹¹ rad/s. k = omega/c = 1.5*10¹¹/(3*10⁸) = 500 m⁻¹ = 0.5*10³ m⁻¹. For +z propagation, phase is (kz - omega*t). B must be perpendicular to E and to propagation direction z. If E is along j-hat (y), then B is along i-hat (x) [since E cross B points in +z]. Option C has: B0=2*10⁻⁷, kz-omega*t, direction i-hat — all correct. Option A has +sign (wrong direction of travel). Option B has x-dependence (wrong axis). Option D has wrong amplitude (60 instead of 2*10⁻⁷) and wrong axis.

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