1.12 Electromagnetic waves and Photons
Electromagnetic radiation is so named due to the electric and magnetic properties of the wave. These two properties propagate at right angles to one another in a sinusoidal manner at the "velocity of light" c. The waves obey the general form:
c = vλ
c is the speed of light essentially a constant 2.997 925 *108 ms-1
v is the frequency of the radiation
λ is the wavelength of the radiation, that is the distance between successive troughs or peaks on the wave.
Although it is most often useful to think of electromagnetic radiation as a wave it is also useful to remember that it demonstrates particulate nature as well. By using particle theory electromagnetic radiation can be examined as if it were composed of small particles of energy called quanta. The energy of each quanta or photon can therefore be given by:
Q = hv
Q is the energy of a quantum Joules (j)
h is the Planck constant 6.626 * 10-34 J sec
v is the frequency.
By combining the two equations it becomes clear how the principles apply to one another.
It can clearly be seen that the energy of each quanta of light is inversely proportional to the wavelength. The longer the wavelength the smaller the energy content. Energy is emitted by all materials above absolute zero in temperature with low energy (cool objects) emitting in the long wavelengths with high energy (hot objects) emitting in increasingly shorter wavelengths. The extreme case of this is the Universe whose energy started at near infinite near the Big Bang and has cooled until the only evidence is the background radiation detected by Arno Penzias and Robert Wilson at the Bell Telephone Laboratories in New Jersey in 1965 and mapped by COBE. The radiation has been shifted so far into the microwave range with a corresponding temperature of 2.73 K. This phenomenon of wavelengths to temperature was known by experiment in the 19th century where the strengths of the various colours in a continous spectrum depended on the temperature of the radiating body. (Roy and Clarke 1988 (a)) By making quantative measurements of the strength of the emitted radiation at different wavelengths it was possible to build up an energy-wavelength distribution curve. In more usual terms we use the Stefan-Boltzmann law to examine how energy is radiated by a body of given temperature. This law built on the work of Kirchhoff who was fascinated by black body radiation. That is any object that absorbs and emits radiation without favouring particular frequencies. Kirchoff's law states that for radiation of the same wavelength and for matter at the same temperature, the energy emitted and the fraction of the incident radiation absorbed were in the same ratio no matter the material used to construct the black body. Stefan in 1879 was able to formulate a quantative law concerning the total energy density, the radiation energy per unit volume υ, in equilibrium with a black body at a temperature T:
υ α T4
Lillesand and Kiefer 1994
A more useful version of this is to know the excitance M, that is the amount of energy emitted per unit area. Since the emitted energy is proportional to the energy density we also have:
M α T4
This is most normally written as:
M =σ T4
The constant σ is the Stefan-Boltzmann constant and is independent of the material examined. The generally accepted value for σ is 5.67*10-8 W m-2 K-4. This means that for each 1cm2 of a black body heated to 1000 K radiates approximately 5.7 W. Stefan came to his conclusions based on observations of heated platinum wire. Boltzmann meanwhile worked on purely thermodynamic reasoning. Wein in 1894 provided further useful quantative information in the form of the displacement law. Just as the total energy emitted by an object varies with temperature the spectral distribution of the emitted energy also varies. As temperature increases there is a shift towards the shorter wavelengths in the peak of a black body radiation. The dominant wavelength is what Wein's displacement law describes:
λm T = constant
λm is the wavelength of maximum spectral radiant excitance
T is the temperature
constant (A) is 2898 μm K
Wein's displacement law was proven experimentally by Lummer and Pringsheim however it was noted that there were discrepancies at long wavelengths low frequencies. Rayleigh and then Jeans tried to overcome this in around 1900 with the formulation of a law first by Rayleigh then modified by Jean. The law takes the form:
dυ = (8 π k / λ4) dλ
For very low frequencies long wavelengths the Rayleigh Jeans law is strikingly close to that of the observed experimental curve but at medium and high frequencies the curves produced by this law are at best absurd. It was the introduction of Planck's work that led to a resolution of the two discrepencies. The interpolation is equivalent to the assumption that the energy of an oscillator of frequency v is not continuously variable, but is restricted to integral multiples of the quantity hv, where h is a constant. This was at first contradictory to the seeming analogue nature of waves and wave like motion but led to the formation of quantum mechanics and the wave particle duality of light. (Atkins 1983). By using the quantum concept Planck showed that the energy envelope of a black body could be represented by a curve given by:
Bvb = 2hv3/c2 * 1/exp (hv/kT) -1 dv
Planck,s constant is now generally accepted as 6.63*10-34 Js.
Bvb is the luminance or specific intensity of radiation radiated per projected unit area per unit solid angle per unit time by a black body in the frequency range v to v +dv.
The quanta of light are known as photons and though they can demonstrate particulate properties they are believed to be massless. However, though photons do not have mass, they do have momentum. The proper, general equation to use is E2 = m2c4 + p2c2 So in the case of a photon, m=0 so E = pc or p = E/c. On the other hand, for a particle with mass m at rest (i.e., p = 0), you get back the famous E = mc2.
This equation often enters theoretical work in X-ray and Gamma-ray astrophysics, for example in Compton scattering where photons are treated as particles colliding with electrons. Whether or not light (or more accurately photons, the indivisible units in which light can be emitted or absorbed) has mass, and how it is affected by gravity, puzzled scientists for many, many years. One of Albert Einstein’s most famous works was on the description of this paradox.
Back in the 1700s, scientists were still struggling to understand which theory of light was correct: was it composed of particles or was it made of waves? Under the theory that light is waves, it was not clear how it would respond to gravity. But if light was composed of particles, it would be expected that they would be affected by gravity in the same way apples and planets are. This expectation grew when it was discovered that light did not travel infinitely fast, but with a finite measurable velocity.
Using these facts a paper was published in 1783 by John Michell, in which he pointed out that a sufficiently massive compact star would possess a strong enough gravitational field that light could not escape any light emitted from the star's surface would be pulled back by the star's gravity before it could get very far given that the escape velocity would be higher than the speed of light. The French scientist Laplace came to a similar conclusion at roughly the same time. The size of body required to exert this level of force would have been something of the order of 500 times the diameter of the Sun. Not much was done over the next hundred years or so with the ideas of Michell and Laplace. This was mostly true because during that time, the wave theory of light became the more accepted one and no one understood how light, as a wave, could be affected by gravity.
The major advance in this field was down to Albert Einstein who in 1915 proposed the theory of general relativity. General relativity explained, in a consistent way, how gravity affects light. We now knew that while photons have no mass, they do possess momentum. We also knew that photons are affected by gravitational fields not because photons have mass, but because gravitational fields change the shape of space-time. The photons are responding to the curvature in space-time, not directly to the gravitational field. Space-time is the four-dimensional "space" we live in; there are 3 spatial dimensions (think of X,Y, and Z) and one time dimension.
Let us relate this to light travelling near a star. The strong gravitational field of the star changes the paths of light rays in space-time from what they would have been had the star not been present. Specifically, the path of the light is bent slightly inward toward the surface of the star.
Figure Showing the Deflection of Light From a Star About the Sun. Figure from Bergström and Goobar 1999
We see this effect all the time when we observe distant stars in our Universe. As a star contracts, the gravitational field at its surface gets stronger, thus bending the light more. This makes it more and more difficult for light from the star to escape, thus it appears to us that the star is dimmer. Eventually, if the star shrinks to a certain critical radius, the gravitational field at the surface becomes so strong that the path of the light is bent so severely inward so that it returns to the star itself. The light can no longer escape. According to the theory of relativity, nothing can travel faster than light. Thus, if light cannot escape, neither can anything else. Everything is dragged back by the gravitational field. We call the region of space for which this condition is true a black hole a term first used by American scientist John Wheeler in 1969. In 1936 Einstein discussed the creation of multiple images from a single source due to lensing effects caused by stars. Einstein concluded that the effect was rare and below a threshold that would be resolvable by Earth based telescopes. It was in the next year Zwicky pointed out that if the lensing was caused not by a star but by a galaxy the effects would be observable.
Whether photons have mass or not has yet to be resolved but thus far, general relativity has withstood every test possible and it seems likely that photons have no rest mass. Even though we can not state that the photon has no mass it is possible and significant to attribute upper limits to the mass it can have. These upper limits are determined by the sensitivity of the experiment used to try measure the mass of the photon. At present the constraining limit for the photon is that if it does demonstrate mass it must be equal to or less than 4 x 10-48 grams. For comparison, the electron has a mass of 9 x 10-28 grams.
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