Question

Suppose that the electric field amplitude of an electromagnetic wave is \(E_0 = 120 \ N/C\) and that its frequency is \(ν = 50.0 \ MHz\). 

1. Determine, B0, ω, k, and λ. 
2. Find expressions for E and B.

Answer 1

Electric field amplitude, \(E_0 = 120\  N/C\)
Frequency of source, \(ν = 50.0 \ MHz = 50 × 10^6 Hz\)
Speed of light, \(c = 3 × 10^8 m/s\)

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(a) Magnitude of magnetic field strength is given as:

\(B_o =\frac { E_o}{c}\)

\(Bo = \frac {120}{3\times 10^8}\)

\(Bo = 4 \times 10^{-7} T\)

\(B_o = 400 \ nT\)

Angular frequency of source is given as:
\(ω = 2\piν = 2\pi × 50 × 10^6\)
\(ω = 3.14 × 10^8 rad/s\)
Propagation constant is given as:

\(k = \frac {ω}{c}\)

\(k = \frac {3.14 \times 10^8}{3\times10^8}\)

\(k = 1.05 \ rad/m\)

Wavelength of wave is given as:

\(λ = \frac cv\)

\(λ =\frac { 3 \times 10^8}{50\times 10^6}\)

\(λ = 6.0 \ m\)

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(b) Suppose the wave is propagating in the positive x direction. Then, the electric field vector will be in the positive y direction and the magnetic field vector will be in the positive z direction. This is because all three vectors are mutually perpendicular.
Equation of electric field vector is given as:
\(\vec E = E_o sin \ (kx-ωt)\hat j\)
\(\vec E= 120 sin\  (1.05x - 3.14 x 10^8t)\hat j\)
And, magnetic field vector is given as:
\(\vec B = B_o sin \ (kx-ωt)\hat k\)
\(\vec B = (4\times 10^{-7}) sin \ (1.05x - 3.14 \times 10^8t)\hat k\)