5.1.6 Pulling it all together
The electric and magnetic fields given by Equations 7.21 and 7.23 can satisfy all four of Maxwell's equations in empty space. Gauss's law and the no-monopole law are immediately satisfied because the fields are transverse. Faraday's law and the Ampère–Maxwell law will also be satisfied if we can find electric and magnetic fields that obey Equations 7.24 and 7.26.
We are looking for wave-like solutions, so it is sensible to try
which is a typical expression for a monochromatic plane wave propagating in the z-direction. In this equation, E0 is the maximum value of the electric field: this is the amplitude of the wave. At any fixed time, λ is the distance between successive wave crests: this is the wavelength of the wave. At any fixed position, T is the time between successive wave crests: this is the period of the wave. Because there is only one wavelength associated with the wave, it is said to be monochromatic. Figure 9 shows the progression of the wave at times t = 0, T / 4, T / 2, 3T / 4 and T. The sinusoidal shape travels undistorted in the positive z-direction at the constant speed c = λ / T.
Equation 7.27 is more commonly written in the form
where k = 2/λ is the wavenumber of the wave and ω = 2/T is the angular frequency (not to be confused with the frequency f = 1/T). The speed of the wave is then given by
Substituting this expression for the electric field into Equation 7.24 (a consequence of Faraday's law) we obtain
This equation can be integrated to give
where K (x,y,z) is any time-independent function. Time-independent fields such as K can always exist, but they obviously play no part in the propagation of electromagnetic waves. It is therefore sensible to set K = 0. Remembering that the speed of the wave is given by c = ω/k, we can write
Figure 10 shows how the electric and magnetic fields are related to one another. The electric and magnetic waves have similar shapes and are exactly in phase with one another. At all times E = cB, and both waves travel through empty space at the speed c.
Finally, we impose the condition given in Equation 7.26 (a consequence of the Ampère–Maxwell law). Rearranging this equation and inserting our expressions for the electric and magnetic fields (Equations 7.28 and 7.29), we obtain
where we have used ω = ck and E0 = cB0 in the final line.
We therefore conclude that, in empty space, electromagnetic waves propagate at the fixed speed
Now for the moment of truth. The constants ε0 and μ0 can be found by measuring electrostatic and magnetostatic forces. In fact, the proportionality constant in Coulomb's law is
and the proportionality constant in the Biot–Savart law is
The speed of electromagnetic waves in empty space is the square root of the ratio of these proportionality constants:
To a fanfare of trumpets, we note that this is numerically the same as the measured speed of light in a vacuum. In 1865, Maxwell wrote:
‘This velocity is so nearly that of light that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.’
Maxwell's ‘strong reason’ was irresistible – it is now fully accepted that light is an electromagnetic wave, with frequencies in the narrow band that our eyes can detect. Optics has become a branch of electromagnetism.
Maxwell also hinted that other electromagnetic waves, with frequencies beyond the visible range, might exist, but he suggested no mechanism for producing these waves. The problem was not just to generate the waves, but also to detect them and measure their properties. In 1887, Heinrich Hertz embarked on a magnificent series of experiments which succeeded in doing all of this (Figure 11). Feeding an oscillating current into a circuit containing two metal spheres, he created an oscillating electric dipole. This generated electromagnetic waves with wavelengths more than 107 times greater than the wavelength of visible light. The electric field of these waves was detected by the spark it produced across a narrow gap in a conducting metal loop. Using this primitive equipment, Hertz measured the speed of the waves and confirmed that it agreed with the known speed of light. He showed that the waves are transverse rather than longitudinal, and he observed refraction, reflection and focusing of the waves. Everything was similar to visible light, but on a much larger length-scale and a much more leisurely time-scale.
Hertz's work had a dual effect. It provided vital confirmation of Maxwell's theory, and it also led to rapid technological developments. In 1895 a radio signal was transmitted a distance of one mile; by 1900, the range had increased to 200 miles, and in 1901 a signal crossed the Atlantic. The first broadcasting radio station opened in Pittsburgh in 1920. The rest, as they say, is history. Society has been totally transformed by broadcast radio and television, satellite communication, mobile phones and wireless internet connection.
Today, the known electromagnetic spectrum extends over at least 20 orders of magnitude, from gamma rays to very low-frequency radio waves. There is no reason to believe that it does not stretch further, but there are practical difficulties in producing significant amounts of electromagnetic radiation at the extremes of frequency. Figure 12 shows the entire spectrum, with named regions characterised by their wavelength and frequency. The visible part of the spectrum occupies only a tiny fraction of the whole – from 4 × 1014 Hz for red light to 8 × 1014 Hz for violet light.
Click to view a larger version of figure 12.
An electromagnetic wave is incident on a filter which absorbs all the electric field. Describe the magnetic field beyond the filter.
The electric wave does not exist beyond the filter, so its curl is equal to zero there. There can be no curl due to electrostatic fields either because electrostatic fields have zero curl. Faraday's law, curl E = −∂B/∂t, therefore shows that the magnetic field must be independent of time beyond the filter. There is no magnetic wave beyond the filter.
How many cycles of orange light pass a given point in 1.0 × 10−14 s? (Orange light has a wavelength 600 nm.)
In time Δt, a wave crest moves a distance Δz = cΔt. If n cycles of the wave pass the given point in this time, nλ = cΔt so
A moving charged particle travels at speed v in the same direction as an electromagnetic wave. What is the ratio of the magnitudes of the electric and magnetic forces exerted on the particle by the electromagnetic wave? Under what conditions do these two force magnitudes become comparable?
The magnitude of the electric force is qE. Because the magnetic wave is transverse, perpendicular to the velocity of the particle, the magnitude of the magnetic force is qvB. In an electromagnetic wave, E = cB, so the ratio of the force magnitudes is
The magnetic force is much smaller than the electric force for non-relativistic particles, but the two forces become comparable for a charged particle that travels close to the speed of light.