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A plane electromagnetic wave moves in the +X direction with wavelength 3 mm. The electric field oscillates along the Y direction with amplitude 66 V/m. Which set correctly gives the electric and magnetic field equations as functions of x and t?

1. E_y = 66 cos[2*pi*10¹¹ (t - x/c)]; B_z = 2.2*10⁻⁷ cos[2*pi*10¹¹ (t - x/c)]
2. E_y = 33 cos[pi*10¹¹ (t - x/c)]; B_z = 1.1*10⁻⁷ cos[pi*10¹¹ (t - x/c)]
3. E_y = 11 cos[2*pi*10¹¹ (t - x/c)]; B_z = 1.1*10⁻⁷ cos[2*pi*10¹¹ (t - x/c)]
4. Eₓ = 33 cos[pi*10¹¹ (t - x/c)]; B_z = 1.1*10⁻⁷ cos[pi*10¹¹ (t - x/c)]

Correct answer: E_y = 66 cos[2*pi*10¹¹ (t - x/c)]; B_z = 2.2*10⁻⁷ cos[2*pi*10¹¹ (t - x/c)]

Solution

The amplitude of E must remain 66 V/m as given. omega = 2*pi*f = 2*pi*c/lambda = 2*pi*(3*10⁸)/(3*10⁻³) = 2*pi*10¹¹ rad/s. The magnetic amplitude B0 = E0/c = 66/(3*10⁸) = 2.2*10⁻⁷ T. Since the wave travels along +X with E along Y, B is along Z. Hence E_y = 66 cos[2*pi*10¹¹ (t - x/c)] and B_z = 2.2*10⁻⁷ cos[2*pi*10¹¹ (t - x/c)].

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