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The electric field of an electromagnetic wave in vacuum is given by E = (3.1 N/C) cos[(1.8 rad/m) y + (5.4 × 10⁸ rad/s) t] i. (a) In which direction does the wave travel? (b) Find the wavelength λ. (c) Find the frequency ν. (d) Find the amplitude of the magnetic field B0. (e) Write the expression for the magnetic field of the wave.

1. Propagation along -y; λ = 3.49 m; ν = 8.6 × 10⁷ Hz; B0 = 1.03 × 10⁻⁸ T; B = (1.03 × 10⁻⁸ T) cos[(1.8 rad/m) y + (5.4 × 10⁸ rad/s) t] k
2. Propagation along +y; λ = 1.75 m; ν = 4.3 × 10⁷ Hz; B0 = 2.06 × 10⁻⁸ T; B = (2.06 × 10⁻⁸ T) cos[...] k
3. Propagation along +x; λ = 3.49 m; ν = 8.6 × 10⁷ Hz; B0 = 1.03 × 10⁻⁸ T; B = (1.03 × 10⁻⁸ T) cos[...] j
4. Propagation along -z; λ = 3.49 m; ν = 1.72 × 10⁸ Hz; B0 = 0.52 × 10⁻⁸ T; B = (0.52 × 10⁻⁸ T) cos[...] k

Correct answer: Propagation along -y; λ = 3.49 m; ν = 8.6 × 10⁷ Hz; B0 = 1.03 × 10⁻⁸ T; B = (1.03 × 10⁻⁸ T) cos[(1.8 rad/m) y + (5.4 × 10⁸ rad/s) t] k

Solution

The phase (ky + ωt) increases with t for fixed phase requires y decreasing -> wave travels in -y. k = 1.8 rad/m so λ = 2*pi/1.8 = 3.49 m. ω = 5.4×10⁸ so ν = ω/(2*pi) = 5.4×10⁸/6.283 = 8.6×10⁷ Hz. B0 = E0/c = 3.1/(3×10⁸) = 1.03×10⁻⁸ T. With E along i and propagation along -j, B must be along k (so that E×B ∝ i×k = -j, the propagation direction). Hence B = (1.03×10⁻⁸) cos[(1.8)y + (5.4×10⁸)t] k.

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