Electromagnetic (EM) waves can transport energy across empty space. The equations of electrodynamics are Maxwell's equations.
Maxwell's equations.
| (1) |  |
| (2) |  |
| (3) |  |
| (4) |  |
Electromagnetic waves are solutions to Maxwell's equations. Equation 2 (Faraday's law) tells us that changing magnetic fields can produce electric fields. The circulation of the electric field around any closed loop G is proportional to the rate of change of the magnetic flux through the loop.
Equation 4 tells us that changing electric fields can produce magnetic fields. Magnetic fields are produced by currents, but also by changing electric fields. The circulation of the magnetic field around any closed loop G is equal to the sum of m0Ithrough_G and 1/c2 times the rate of change of the electric flux through the loop. We have
m0e0 = 1/c2.
In a space free of charges and currents, we still can have electric and magnetic fields. They are changing electric and magnetic fields, carrying energy through space. EM waves require no medium, they can travel through empty space. Sinusoidal plane waves are one type of electromagnetic waves. Not all EM waves are sinusoidal plane waves, but all electromagnetic waves can be viewed as a linear superposition of sinusoidal plane waves traveling in arbitrary directions. A plane EM wave traveling in the x-direction is of the form
E(x,t) = Emaxcos(kx-wt+f),
B(x,t) = Bmaxcos(kx-wt+f).
E is the electric field vector and B is the magnetic field vector of the EM wave. For electromagnetic waves E and B are always perpendicular to each other, and perpendicular to the direction of propagation. The direction of propagation is the direction of E´B.
If, for a wave traveling in the x-direction E = Ej, then B = Bk and j´k = i. Electromagnetic waves are transverse waves.
The wave vector k points into the direction of propagation, and its magnitude k = 2p/l, where l is the wavelength of the wave. The frequency f of the wave is f = w/2p. w is the angular frequency. The speed of any sinusoidal wave is the product of its wavelength and frequency.
v = lf =w/k.
Maxwell's equations require that v = c = 3*108 m/s for any electromagnetic wave in free space. The speed of any electromagnetic waves in free space is the speed of light c. Periodic electromagnetic waves in free space can have any wavelength l or frequency f as long as lf = c. When an electromagnetic wave travels through free space, Maxwell's equations require that at every instant and at any point the ratio of the electric to the magnetic field in SI units is equal to the speed of light, E/B = c.
When electromagnetic waves travel through a medium. the speed of the waves in the medium is
v = c/n(lfree),
where n(lfree) is the index of refraction of the medium. The index of refraction n is a properties of the medium, and it depends on the wavelength lfree of the EM wave. If the medium absorbs some of the energy transported by the wave, then n(lfree) is a complex number. For air n is nearly equal to 1 for all wavelengths.
When an EM wave travels from one medium with index of refraction n1 into another medium with a different index of refraction n2, then its frequency remains the same, but its speed and wavelength change.
Electromagnetic waves are categorized according to their frequency f or, equivalently, according to their wavelength l = c/f. Visible light has a wavelength range from ~400 nm to ~700 nm. Violet light has a wavelength of ~400 nm, and a frequency of ~7.5´1014 Hz. Red light has a wavelength of ~700 nm, and a frequency of ~4.3´1014 Hz.
Visible light makes up just a small part of the full electromagnetic spectrum. Electromagnetic waves with shorter wavelengths and higher frequencies include ultraviolet light, x-rays, and gamma rays. Electromagnetic waves with longer wavelengths and lower frequencies include infrared light, microwaves, and radio and television waves.
Type of Radiation |
Frequency Range (Hz) |
Wavelength Range |
| gamma-rays | 1020-1024 | <10-12 m |
| x-rays | 1017-1020 | 1 nm-1 pm |
| ultraviolet | 1015-1017 | 400 nm-1 nm |
| visible | 4-7.5´1014 | 750 nm-400 nm |
| near-infrared | 1´1014-4´1014 | 2.5 mm-750 nm |
| infrared | 1013-1014 | 25 mm-2.5 mm |
| microwaves | 3´1011-1013 | 1 mm-25 mm |
| radio waves | <3´1011 | >1 mm |
Electromagnetic waves transport energy through space. In free space this energy is transported by the wave with speed c. The magnitude of the energy flux S is the amount of energy that crosses a unit area perpendicular to the direction of propagation of the wave per unit time. It is given by
S = EB/(m0) = E2/(m0c),
since for electromagnetic waves B = E/c. The units of S are J/(m2s). m0 is a constant called the permeability of free space, m0 = 4p´10-7 N/A2.
The Poynting vector is the energy flux vector. It is named after John Henry Poynting. Its direction is the direction of propagation of the wave, i.e. the direction in which the energy is transported.
S = (1/m0)E´B.
Energy per unit area per unit time is power per unit area. S represents the power per unit area in an electromagnetic wave. If an electromagnetic wave falls onto an area A where it is absorbed, then the power delivered to that area is P = S×A.
The time average of the magnitude of the Poynting vector, <S>, is called the irradiance or intensity. The irradiance is the average energy per unit area per unit time. <S> = <E2>/(m0c) = Emax2/(2m0c).
Electromagnetic waves transport energy. EM wave also transport momentum. The momentum flux is S/c. S/c is the amount of momentum that crosses a unit area perpendicular to the direction of propagation of the wave per unit time. If an electromagnetic wave falls onto an area A where it is absorbed, the momentum delivered to that area in a direction perpendicular to the area per unit time is dp^/dt = (1/c)S×A.
The momentum of the object absorbing the radiation therefore changes. The rate of change is dp^/dt = (1/c)SA^, where A^ is the cross-sectional area of the object perpendicular to the direction of propagation of the electromagnetic wave. The momentum of an object changes if a force is acting on it.
F^ = dp^/dt = (1/c)SA^
is the force exerted by the radiation on the object that is absorbing the radiation. Dividing both sides of this equation by A^, we find the radiation pressure (force per unit area) P=(1/c)S. If the radiation is reflected instead of absorbed, then its momentum changes direction. The radiation pressure on an object that reflects the radiation is therefore twice the radiation pressure on an object that absorbs the radiation.
 | How much electromagnetic energy per cubic meter is contained in sunlight
if the intensity of the sunlight at the earth's surface under a fairly clear
sky is 1000 W/m2?
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| A plane electromagnetic wave of intensity 6 W/m2 strikes a small
pocket mirror of area 40 cm2 held perpendicular to the approaching wave. (a) What momentum does the wave transfer to the mirror each second? (b) Find the force that the wave exerts on the mirror.
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