## Monochromatic Plane Waves In Vacuum

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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/c

^{2}∂

^{2}E/∂t

^{2}, ∇

^{2}B = 1/c

^{2}∂

^{2}B/∂t

^{2}… (1)

Here, c = 1/√ϵ

_{0}μ

_{0}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) = E

_{0}e

^{i(kx – ωt)}, B(x,t) = B

_{0}e

^{i(kx – ωt) }… (2)

K is called the wave number.

**E**

_{0}and

**B**

_{0}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

(E

_{0})x = (B

_{0})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(E

_{0})

_{z}= ω(B

_{0})

_{y}k(E

_{0})

_{y}= ω(B

_{0})

_{z }Or B

_{0}= k (I x E

_{0})/ω

E and B are in phase and mutually perpendicular. The real amplitudes of E and B are related by

B

_{0}= k E

_{0}/ω = 1/c E

_{0}

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