8.3.2. Nature of electromagnetic waves

It can be shown from Maxwell’s equations that electric and magnetic fields in an electromagnetic wave are perpendicular to each other, and to the direction of propagation.

Maxwell’s equations establish that in an electromagnetic wave, the electric field (E) and the magnetic field (B) are always perpendicular to each other and also to the direction the wave travels. This means if the wave is moving in one direction, the electric and magnetic fields are oriented in two other directions, forming a right-angle relationship. This feature is crucial as it defines how electromagnetic waves propagate through space.

Imagine a flag waving in the wind. The direction the flag is waving (horizontal) is like the direction of the wave propagation, while the up-and-down movement of the flag represents the electric field, and a side-to-side movement represents the magnetic field. The flag cannot wave in a different direction than the wind pushes it, just as the electric and magnetic fields cannot exist in the same direction as the wave is traveling.

In Fig. 8.3, we show a typical example of a plane electromagnetic wave propagating along the z direction (the fields are shown as a function of the z coordinate, at a given time t). The electric field E is along the x-axis, and varies sinusoidally with z, at a given time. The magnetic field B is along the y-axis, and again varies sinusoidally with z.

In a typical electromagnetic wave, as it moves through space, both the electric field and the magnetic field change in a sinusoidal manner over space and time. This means that if you were to observe a wave at a fixed point, the intensity of the electric field would go up and down in a smooth, wave-like pattern, similar to how ocean waves rise and fall. The same happens for the magnetic field but at 90 degrees to the electric field, retaining the perpendicularity.

Think of sound waves traveling through the air, where the compressions and rarefactions of the wave occur in a pattern. Just as sound waves create regions of high and low pressure, electromagnetic waves create peaks and troughs in electric and magnetic fields at various points in space as they propagate, but without needing any medium to travel through.

We can write E and B as follows: E = E₀ sin(kz - ωt) [8.7(a)] B = B₀ sin(kz - ωt) [8.7(b)]

These equations express how the electric field (E) and magnetic field (B) are mathematically connected to the wave's progress. Here, E₀ and B₀ are the maximum strengths of the respective fields, 'k' is the wave number (related to wavelength), 'ω' is the angular frequency, 'z' represents the position, and 't' represents time. These equations illustrate that both fields oscillate at the same frequency and are in sync as they propagate through space.

Imagine a swing set where each swing moves back and forth: as one swing (electric field) rises, so does another (magnetic field), both following the same rhythmic pattern. Just as the swings reach their highest point at the same time, the electric and magnetic fields reach their peak strengths together in the wave equation.

Using Eqs. [8.7(a) and (b)] for E and B and Maxwell’s equations, one finds that ω = ck, where, c = 1/√(μ₀ε₀).

This equation reveals a foundational connection in wave dynamics; it states that the angular frequency (ω) of the wave is directly proportional to its wave number (k), which in turn is related to the speed of light (c). In simple terms, it shows how changes in frequency influence wavelengths and vice versa, underscoring the wave nature of electromagnetic phenomena. The speed of electromagnetic waves in a vacuum is constant, elucidating that all electromagnetic waves, regardless of their specific frequency, travel at this speed.

Think of a concert where the band plays different tunes. The faster the tempo (frequency), the shorter the individual notes (wavelengths). Regardless of the tune, the concert keeps the same pace of play (speed of light), demonstrating that as we change one aspect (frequency of the tune), the other aspect (length of notes) must adjust while maintaining the overall rhythm.

It is also seen from Maxwell’s equations that the magnitude of the electric and the magnetic fields in an electromagnetic wave are related as B = (E₀/c),

This relationship illustrates that for a given electromagnetic wave, the strength of the magnetic field (B) is directly dependent on the strength of the electric field (E₀) when measured at their peak values, with 'c' being the speed of light. This signifies not only a direct connection but also helps in predicting one field based on the other, thus emphasizing the interdependence of electric and magnetic phenomena in these waves.

Think of a dance routine where every dance move (magnetic field) corresponds to a specific position or gesture (electric field). The intensity of the dance performance reflects the electric fields, while the synchronization of the dancers showcases the magnetic fields. If one dancer moves more dynamically, the other must follow suit to match the choreography, hence illustrating their interconnectedness.

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