Great Calculations: A Surprising Look Behind 50 Scientific Inquiries
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The image of a star will not be a point but a bright circle [he should really say disc] surrounded by a series of bright rings. The angular diameters of these will depend on nothing but the aperture of the telescope, and will be inversely as the aperture.11
Airy goes on to calculate what is now known as the Airy diffraction pattern and to discuss how it limits the details that may be seen using a telescope. In modern terms, the intensity of light I at a point a distance r from the center of the pattern is given by
In equation (9.4) I0 is a constant, R is the aperture radius, f is the lens focal length, and λ is the wavelength of light. (J1 is the Bessel function of order one. There will be more on Bessel functions in chapter 12.) The result is plotted in figure 9.9 where an example of the appearance of an Airy pattern is also shown (with the outer rings somewhat enhanced).
This might have remained an astronomical specialty, but in fact this diffraction effect is the limiting factor in all imaging systems, so it is also vitally important for microscopes and for imaging systems such as the human eye.
Figure 9.9. A plot of the intensity of light in the Airy diffraction pattern and an experimental example. Reprinted with permission of John Wiley & Sons, from O. S. Heavens and R. W. Ditchburn, Insight into Optics (Hoboken, NJ: John Wiley & Sons, 1991).
9.3.2 Resolution of Image Details
If images are blurred by diffraction effects, we need to ask what level of detail in an object can still be resolved in its image. Using Rayleigh's criterion (named after its originator Lord Rayleigh) is a widely accepted way to proceed. Rayleigh suggested that two points in an image may be said to be just resolved if the Airy disc center of one of them falls on the first zero radius for the second one, as shown in figure 9.10. Images of points closer than that look like one blurred big point, whereas at that separation or greater, two distinct image peaks may be discerned. In terms of a resolved angle βr, equation (9.4) leads to the famous formula in terms of the wavelength of light involved and the aperture diameter D:
Figure 9.10. Diffraction patterns (Airy discs) for two point sources imaged with varying closeness. Reprinted with permission of John Wiley & Sons, from D. Halliday and R. Resnick, Fundamentals of Physics, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 1991).
For the human eye in bright light (so D is about 2 mm) and light of wavelength 500 nm, the angle of resolution is around one minute of arc. The optics of the eye is quite complicated, but the pupil does provide an aperture in the visual system. (The book by Wandell gives experimental data for the form of the blurred image on the retina; see chapter 2, figure 2.5.) Because there is a resolution limit forced by the imaging system, it would be pointless to have an arrangement of the photoreceptor cells in the retina of the eye (known as the photoreceptor mosaic) to sample the image beyond the level of detail present as shown in figure 9.10. Thus the matching of optics and image-sampling photoreceptor density can be observed as discussed by Denny and McFadzean (see bibliography).
In section 6.4.3, we saw that George Biddell Airy was often portrayed as the villain in the story of planet Neptune's discovery. However, most people would know him for the Airy disc and the ideas about limits to the details that may be extracted from an image. He has redeemed himself then with his part in the highly significant calculation 33, diffraction and the limit to vision. It is the fact that light behaves like waves, and there is diffraction from apertures, that limits our visual capacity.
9.4 FROM OPTICAL WAVES TO ELECTROMAGNETIC WAVES
The wave theory of light was enormously successful for describing a large range of optical phenomena. However, there remained the underlying mystery: Physically, what are these waves? It was clear that the surface of water moves up and down as a water wave travels along that surface, and sound waves are related to pressure variations in the air. For many people, the existence of a strange medium—the ether—was required to give some sort of mechanical picture for light waves. The story of the ether and its eventual banishment is an intriguing part of the history of physics. A major step forward was made by James Clerk Maxwell in the second half of the nineteenth century.
Maxwell (1831–1879) was the greatest theoretical physicist of the nineteenth century, making major contributions to many areas of physics. (There are several biographies of Maxwell—I particularly like those by Harmon and Mahon, and Segrè gives a good introduction to his work.) Maxwell was born in Edinburgh and educated at Cambridge University. He would go on to become a professor of physics, first at Aberdeen and then at Kings College, London. Maxwell had a lifelong interest in electricity and magnetism and the part that a hypothetical ether might play in it. His magnificent 1873 Treatise was the first comprehensive treatment of the subject.
9.4.1 The Phenomena of Electricity and Magnetism
The work of André-Marie Ampère, Charles-Augustin de Coulomb, Michael Faraday, Hans Christian Oersted, and others gradually built up knowledge of electrical forces, magnetic phenomena, and interactions between electric and magnetic elements. The force between charges acts over a distance and was shown to follow an inverse square law. Faraday showed that moving a magnet in and out of a loop of wire generated an electric current in that wire; this is magnetic induction and the origin of the electric generator. Conversely, a current flowing in a wire moves a magnetic needle brought close to that wire. Faraday's experiments with iron filings showed that a magnet can influence objects around it but not in contact with it. It was clear that electric and magnetic effects could occur without the need for electric charges or magnets to be in contact. Thus the influence of charges and magnets act over a distance, and constants were introduced to characterize the nature of the intervening medium, which might be air, water, or glass, for example. These constants are the dielectric constant (or permittivity) ε for electrical effects and the magnetic permeability μ for magnetic effects. In free space, these are denoted by ε0 and μ0. For example, the force between charges q1 and q2 separated by a distance r has magnitude (q1 q2)/εr2.
Maxwell introduced the electric field E to describe the influence of charges in the space around them and the magnetic field H for similar magnetic effects. These fields are vector quantities and depend on the position in space and change with time as the charges and magnets move, so we should write E(x, y, z, t) and H(x, y, z, t). Taken together, we refer to the electromagnetic field. Maxwell then demonstrated his genius by writing down a set of equations showing how electric and magnetic fields are linked and how they are generated by charges and electric currents. These are the theoretical equivalent of the experimental picture built up by Faraday. It is the form of these equations in a region of space devoid of charges and currents that concerns us here. In a uniform medium (with the dielectric constant the same everywhere) the electromagnetic fields must satisfy the following equations:
These are known as Maxwell's equations. It is not necessary to understand the mathematics in detail, but you should appreciate that these wonderful equations do tell us how the electromagnetic fields can vary in space and time. You can also see how E and H are mixed together in the same equation to give Faraday's magnetic induction. (The books by Heavens and Ditchburn, Lipson and Lipson, and Segrè give further details with particular reference to optics.)
9.4.2 Maxwell's Miraculous Calculation
Maxwell manipulated his equations (see the aforementioned references) to show that the electromagnetic fields in a charge-free and magnet-free uniform isotropic medium must satisfy
Again, you need not appreciate the mathematical details of these equations, but you should recognize their form: these are wave equations. Maxwell had shown that the electromagnetic field could take the form of waves in space; electromagnetic disturbances, or fields, can propagate in space in the form of waves.
Faraday had speculated about the links between light and electromagnetism. For example, in what is now called the Faraday effect, he showed that a magnetic field could change the polarization of a light wave. He also wondered about how ra
diation and vibrations of lines of force might be connected. Faraday clearly had the ideas about a link between optics and electromagnetism, but he lacked the mathematical background to establish the full picture. It was Maxwell who was ready and able to take the definitive step; it involved showing that his electromagnetic waves are the same as light waves. According to the standard wave equation, the waves described by equations (9.4) have speed V given by
However, Maxwell knew that there was another quantity with the dimensions of a velocity but which was related to definitions of electrostatic and electromagnetic units. He called this quantity v. Most importantly, v could be found experimentally using electric circuits (see Segrè appendix 6, or Lipson and Lipson section 4.2.2, for a simple introduction). The electromagnetic theory for those circuits gave , exactly the same form as he had derived for the wave speed V. I will let Maxwell tell the amazing story (as it is written in his Treatise), and you should note that he is referring to the case of free space or waves and the electronic circuits in air.
On the theory that light is an electromagnetic disturbance, propagated in the same medium through which other electromagnetic actions are transmitted, V must be the velocity of light, a quantity the value of which has been estimated by several methods. On the other hand, v is the number of electrostatic units of electricity in one electromagnetic unit, and the methods of determining this quantity have been described in the last chapter. [See the references given above.] They are quite independent methods of finding the velocity of light. Hence the agreement or disagreement of the values of V and of v furnishes a test of the electromagnetic theory of light.
In the following table the principal results of direct observation of the velocity of light, either through the air or through the planetary spaces, are compared with the principal results of the comparison of the electric units:
Velocity of Light (mètres per second). Ratio of Electric Units. (metres per second)
Fizeau…………………………………314000000 Weber……………………310740000
Aberration, etc and Sun's Parallax…308000000 Maxwell…………………288000000
Foucault………………………………298360000 Thomson…………………282000000
It is manifest that the velocity of light and the ratio of the units are quantities of the same order of magnitude.12
(In the version of Maxwell's Treatise cited in the bibliography, there is an extended table of V and v values that were available in 1889; the equality of V and v is apparent.)
Of course, Maxwell has now been fully vindicated, and it is generally accepted that light waves are indeed electromagnetic waves, and Maxwell's equations are the appropriate vehicle for theoretically investigating their behavior. Maxwell's calculation and comparison of those velocities obtained from such wildly different methods ranks as one of the greatest achievements in science and no list of important calculations would be complete without calculation 34, light and electromagnetism.
9.4.3 Triumphs and New Mysteries
If light is an electromagnetic wave, we have an immediate explanation of things like the Faraday Effect (a magnet influencing a light wave). Also, as Maxwell pointed out in his Treatise, the form of the velocity V in equation (9.5) tells us how light behaves in different materials (through their values of ε and μ) and leads to the introduction of the refractive index. As mentioned in the previous section, light has a polarization, meaning that its waves are transverse. Using Maxwell's equations (9.3), it is easy to show that electromagnetic waves must have their E and H perpendicular to the direction of propagation; electromagnetic waves are transverse waves and thus explain the polarization of light.
We should also note that Maxwell's theory covers all electromagnetic waves—not just those we know as light, but also things like x-rays and radio waves.
Maxwell's electromagnetic theory of light together with Newton's laws of motion form the basis for classical physics, which reached its triumphal peak at the end of the nineteenth century. But what about that old mystery: What exactly is light? The answer—it is an electromagnetic wave—only brings out the obvious question: Physically, what is an electromagnetic wave? And there we remain stuck. (A wonderful discussion by the matchless expositor Richard Feynman can be found in section 20-3, “Scientific Imagination,” of The Feynman Lectures on Physics.) One approach to the conceptual difficulty encountered when discussing electromagnetic waves is to just use Maxwell's theory to calculate the results of observations and experiments. In effect, we simply follow the opinion of Heinrich Hertz (the man who first demonstrated the existence of radio waves): “Maxwell's theory is Maxwell's system of equations.”13
9.5 BACK TO PARTICLES?
We have just seen that the combination of Newton's laws of motion and Maxwell's equations for electromagnetism covers what today we call classical physics. Some people thought that they covered virtually all of physics. Here is the famous statement made early in the twentieth century by the eminent experimental physicist Albert Abraham Michelson (1852–1931):
The more important laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote. Nevertheless, it has been found that there are apparent exceptions to most of these laws, and this is particularly true when the observations are pushed to a limit…. Many other instances might be cited, but these will suffice to justify the statement that “our future discoveries must be looked for in the sixth place of decimals.”14
It is ironic that Michelson's experiments designed to detect the ether played a part in Einstein's thinking about the theory of relativity, which, together with the introduction of quantum theory, revolutionized physics in the twentieth century.
Serious problems began to arise when physicists tried to account for the spread over wavelengths of radiation coming from very hot bodies. In trying to derive the observed radiation law using Maxwell's theory and classical statistics (which had been successful in deriving the properties of gases—see the next chapter), scientists like Sir James Jeans and Lord Rayleigh ran into unexpected difficulties. In some cases, they were getting infinite answers for the (obviously finite) amount of radiation emitted—hardly a problem “in the sixth place of decimals”! People spoke of the ultraviolet catastrophe. (For a very readable introduction to this, see Gamow's lovely book. The book by Pais is also a wonderful account of these events and Einstein's contributions. See Rigden for a briefer introduction.)
The leading figure in the field of radiation was Max Planck who looked at radiating “black bodies” in terms of discrete oscillators rather than continuous waves. In this way, he was able to bring order back to the field. The whole area of radiation theory and optics was completely changed by Albert Einstein's 1905 paper “On a Heuristic Point of View about the Creation and Conversion of Light.” Einstein wrote that
according to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of energy quanta that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole.15
Later, these “energy quanta” came to be called photons, and the quantum theory of light was born. Light of wavelength λ has frequency ν given by frequency = speed of light divided by wavelength ν = c/ λ. For this light, each photon carries the same energy and momentum:
Equation (9.6) contains h, the newly introduced quantum constant, now known as Planck's constant. The modern value for h is 1.0546 × 10–27 erg-sec, and its remarkably small size tells us that we have entered the quantum world of very small things.
Einstein quite rightly used the term “very revolutionary”16 to describe his 1905 paper launching the quantum theory of light. But the very small value of h means that everyday light involves an enormous number of photons, and, generally, we are quite
unaware of this discrete nature of light. The question inevitably arises: How do we know that photons actually exist? The answer requires us to once again appreciate that vital link between theory and experiment, and it is the calculations involved that form my choice of calculation 35, photons exist.
9.5.1 Einstein Knocks Out Electrons with Photons
In his 1905 paper, Einstein considered three topics which could profitably be discussed using his new quantum theory of light. The most convincing for many people was Einstein's theory of the photoelectric effect. (The other two topics were Stokes's rule in photoluminescence and the ionization of gases by ultraviolet light.) The photoelectric effect was discovered in 1887 by Heinrich Hertz (of radio-waves fame) and subsequently investigated by Philipp Lenard in 1903 and many other experimentalists. It refers to the emission of electrons from a metal surface when light is incident on it. There are three key observations for the photoelectric effect:
The energy of the individual photoelectrons is independent of the intensity of the light.
The number of emitted electrons (the photoelectric current) is proportional to the intensity of the light.
For a given metal, electrons are emitted only when some threshold frequency of light is exceeded, and then their energy increases linearly as frequency is increased.
These results are shown graphically in figure 9.11.
Figure 9.11. Results for the photoelectric effect. Number of photoelectrons emitted from a metal surface versus light intensity, and emitted electron maximum energy versus light frequency. No electrons are emitted when the frequency is less than the threshold frequency νc. Figure created by Annabelle Boag.
Einstein did a simple calculation to show a likely origin for these facts. He assumed that a given metal had a characteristic work P that must be done on its electrons in order to displace them from the metal surface. In the light of frequency ν incident upon the metal, there will be photons with energy hν. If these photons hit the electrons, they can give them energy to overcome the required work P and have them emitted with energy E according to