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Monochromatic Plane Waves in Vacuum

The electric and magnetic field vectors E and B in empty space, satisfy the three-dimensional wave equation
        ∇2 E = 1/c22E/∂t2, ∇2B = 1/c22B/∂t2                     … (1)
Here, c = 1/√ϵ0μ0 is the speed of light in vacuum.

We consider sinusoidal waves of frequency ω. Such waves are called monochromatic. Suppose the waves are travelling in the x-direction and have no y – or z-dependence These are called plane waves, because the fields are uniform over every plane perpendicular to the direction of propagation.

    E(x,t) = E0 ei(kx – ωt), B(x,t) = B0ei(kx – ωt)                    … (2)
K is called the wave number.

E0 and B0 are the complex amplitudes of the electric and magnetic fields. The physical fields are the real parts of E and B.

Since ∇.E = 0 and ∇.B = 0, it follows that
        (E0)x  = (B0)x = 0
This implies that electromagnetic waves are transverse.
The electric and magnetic fields are perpendicular to the direction of propagation.

Faraday’s law, ∇ x E = - ∂B/∂t, implies a relation between the electric and magnetic amplitudes.
-    K(E0)z = ω(B0)y         k(E0)y = ω(B0)z
Or    B0 = k (I x E0)/ω

E and B are in phase and mutually perpendicular. The real amplitudes of E and B are related by
            B0 = k E0/ω = 1/c E0
Monochromatic Plane Waves in Vacuum


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