The Nature of Electromagnetic Radiation

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1 II The Nature of Electromagnetic Radiation The Sun s energy has traveled across space as electromagnetic radiation, and that is the form in which it arrives on Earth. It is this radiation that determines the effect of the Sun s energy on the Earth and its climate. Infrared radiation, radio waves, visible light, and ultraviolet rays are all forms of electromagnetic radiation. One of the best ways to understand the production of this type of energy is to consider how it is emitted by atoms, in particular the hydrogen atom. Energy Levels of Atoms Allowed Energies of the Electron Since before the turn of the century, it has been known that an individual atom is made up of a nucleus (composed of protons and neutrons) and electrons bound to the nucleus, and that the electrons (and hence the atom) have very well defined, discrete amounts of energy. The simplest atom, hydrogen, is composed of a proton (its nucleus) and an electron bound to it by the electrical force of attraction. (Electrons have a negative charge and protons a positive one.) The electron may have only certain values of energy when it is bound to the nucleus in this way. The lowest energy level, in which the electron is closest to the nucleus, is called the ground level. The next level is the first excited level, and so on (see Figure 4). There are various ways the electron may be moved to higher levels, and one of those ways is by receiving energy from electromagnetic radiation. What determines the amount of energy electromagnetic radiation may have? Energy of Electromagnetic Radiation In many situations electromagnetic radiation may be described as having a wave-like nature. Three important features of waves of any sort are the wavelength (the distance between adjacent crests), the frequency (how fast the crests move up and down), and the speed (how fast the crests move forward). There is a basic relationship between these features. If we multiply the wavelength (symbolized by λ, the Greek letter lambda) by the frequency (f), we obtain the speed of the wave, v. The mathematical formula is λf = v When the electromagnetic radiation is moving through space or another vacuum, regardless of its wavelength or frequency, it travels at the speed of light, c. Because c is constant, the product of λ and f is always the same, so if one gets larger, the other gets smaller. Electromagnetic radiation also, under certain conditions, exhibits a particle-like nature. The particles are called photons, and it is helpful to think of them as energy packets having a welldefined wavelength and frequency. In the early part of this century, Albert Einstein demonstrated that the energy of these photons, E, is directly proportional to their frequency: E = hf where h is a constant called Planck s constant (see Appendix III). The frequency of electromagnetic radiation is inversely proportional to its wavelength, so its energy is, too. Radiation with a long wavelength has less energy than short-wavelength radiation. 7

2 THE SUN-EARTH SYSTEM Absorption and Emission of Electromagnetic Radiation Absorption Suppose that electromagnetic radiation of a given frequency strikes a hydrogen atom and that the frequency is such that the energy of the radiation equals the difference in the energy of the ground and first excited levels of the atom. Then an electron in the ground level may be raised to the first excited level. This process is called absorption. Because the atom has a unique set of energy levels, each will absorb radiation over a particular set of wavelengths. The pattern of wavelengths absorbed is called the absorption spectrum of the atom or molecule. The radiation has vanished (been absorbed), but the total energy of the radiation and the atom energy is conserved. The atom now has more energy than it did. Emission Line Emission Just as a ball kicked up a hill will roll back down to the bottom, the electron very quickly returns to its lowest possible energy level (usually within a hundred-millionth of a second). On the way back to its ground level it must release energy, and it emits the energy as radiation that has the same wavelength as the radiation that first hit the atom. So if the atom is being given energy by some means (such as radiation or collisions with other particles), we can expect to find it emitting electromagnetic radiation at wavelengths governed by the difference in these energy levels. This kind of radiation is called line emission, because when the individual wavelengths are measured with an instrument called a spectrograph, the results show up as lines on a photographic plate. Figure 4. Energy-level diagram for hydrogen. Arrows indicate direction of electron transitions. When the electron moves from a higher to a lower energy level, a photon is emitted; the atom emits energy. When a photon of the right energy strikes the hydrogen atom, the electron moves from a lower to a higher energy level. The atom absorbs energy. Figure 5. Relative amounts of energy emitted by the sun at different wavelengths. Note that most of this energy occurs in the visible region, which extends from about 400 nm to 750 nm. 8

3 THE NATURE OF ELECTROMAGNETIC RADIATION Continuous Emission When the density of atoms in a given area is sufficiently high, the radiation that ultimately leaves the area is smeared into a continuous distribution of wavelengths made up of the many separate wavelengths that the individual atoms emit. This is called continuous emission. The radiation we receive from the Sun is continuous radiation. Figure 5 shows a graph of the relative amount of energy of different wavelengths that the Earth receives from the Sun. Blackbody Radiation The type of continuous radiation the Sun emits is often called blackbody radiation. There are many special features of this type of radiation that allow us to determine various properties of the objects emitting it. One is that the total amount of energy the blackbody emits is determined solely by its temperature. Specifically, the amount of energy that a body emits increases as the fourth power of the temperature. The mathematical expression for this is the Stefan-Boltzmann law: E = σ T 4 where s is a constant (the Stefan-Boltzmann constant; see Appendix III). When the temperature of a blackbody doubles, for example, the amount of energy emitted increases by a factor of two to the fourth power, or 16 (two multiplied by itself four times: 2x2x2x2). A blackbody whose temperature is 4,000 K emits 16 times as much energy as one at 2,000 K. The energy emitted by a blackbody always peaks at some wavelength and decreases toward longer and shorter wavelengths. Figure 5 shows the energy curve for an object at 5,800 K, the approximate temperature of the surface of the Sun. In fact, there is a simple equation, called Wien s displacement law, to determine the temperature of an object by measuring this peak in the energy curve. The equation is: λ max T = x 10 6 where wavelength is in nanometers and temperature is in kelvins. (Note that the constant is very close to 3.0 x 10 6, which is useful for rough calculations.) Wien s law and the Stefan- Boltzmann law are central to understanding the greenhouse effect, discussed in Section III. The Electromagnetic Spectrum The continuous emission spectrum an object radiates is a display of the amount of energy it emits at all wavelengths. The entire electromagnetic spectrum covers an enormous range of wavelengths, divided into regions. Going from the shortest wavelengths to the longest, there are: gamma rays, x-rays, ultraviolet radiation, visible light, infrared radiation, and radio waves. Figure 6 shows the various regions of the spectrum and their approximate wavelength ranges. The visible region occupies only a small portion of the entire spectrum. As Figure 5 shows, the vast majority of our Sun s energy is emitted in the visible, ultraviolet, and infrared ranges. In fact, about 41% of the energy emitted from the Sun lies in the visible bands alone, between 390 nm and 750 nm. Since the surface temperature of the Sun is about 5,800 K, we can calculate from Wien s displacement law that the maximum amount of energy is emitted at about 500 nm, right in the middle of the visible spectrum. About 50% of the energy the Sun emits lies in the infrared and radio regions, above 750 nm, and only about 9% is in the ultraviolet, x-ray, and gamma-ray regions, below 390 nm. It is this electromagnetic radiation, of all wavelengths and frequencies, that determines what effect the Sun s energy has on the Earth and its climate. We have seen that one of the ways electromagnetic radiation interacts with atoms (and molecules) is by being absorbed by them, and that very soon afterward it is emitted again. The energy is taken away but then given back. Does this mean, then, that there is no net 9

4 THE SUN-EARTH SYSTEM effect on the amount of radiation traversing the atmosphere? Indeed, it does not. The radiation being absorbed by an atom is moving in a given direction (for example, from the Sun to the Earth). The radiation that the atom emits may go in any direction with about equal probability. So at the wavelength being absorbed, only an insignificant fraction of the original radiation will continue moving in the original direction, and there will be a net loss of energy in the direction the radiation was originally traveling. Therefore, how much energy, at various wavelengths, is lost from the beam of radiation depends on what atoms and molecules (and how many of them per unit volume) are in its path. So it is the makeup of our atmosphere that determines how much of the Sun s energy gets to the Earth s surface (as well as how much leaves the Earth). Figure 6. The electromagnetic spectrum consists of many regions. Each differs in wavelength, frequency, and energy. 10

5 THE NATURE OF ELECTROMAGNETIC RADIATION THE STEFAN-BOLTZMANN LAW AND THE EARTH S TEMPERATURE Using the Stefan-Boltzmann law, we should be able to determine the approximate average temperature of the Earth, if we assume that the Earth emits the same amount of energy each second that it absorbs from the Sun; that is, that the Earth, on average, is neither heating up nor cooling down. The Stefan-Boltzmann law tells us how much radiant energy per second per unit area an object at a given temperature emits. We know the Sun s temperature, so we can calculate this value for the sun. Multiplying this number by the Sun s surface area gives us the total energy the Sun emits each second. To figure out what fraction of this the Earth receives, imagine, as we did in earlier, a huge sphere with a radius equal to the distance between the Sun and Earth. Imagine a round spot the size of a cross section of Earth pasted on the surface of that sphere. If we divide the cross sectional area of the Earth (the size of the spot) by the surface area of the imaginary sphere, and multiply this fraction by the total energy emitted by the Sun, we have the amount of energy per second the Earth receives from the Sun. Assume the Earth absorbs all of this energy. It must lose the same amount as it gains each second if it is to remain at a constant temperature. We know from the above calculation how much energy the Earth receives and loses. Now we can use the Stefan-Boltzmann law to calculate what temperature the Earth must be to radiate away that amount of energy. It works out at 279 K. To get that number, we simplified some things; we assumed that all the Sun s radiation made it to the Earth s surface and that all the Earth s radiation left unimpeded. The actual average temperature of the Earth is about 288 K. To obtain this number we must take into account the natural greenhouse warming and the fact that about 30% of the Sun s radiation is reflected away and thus does not heat the Earth. These effects will be discussed later. Problems 1. In calculating the temperature of the Earth, we assumed that all of the Sun s energy was absorbed by the Earth and its atmosphere. In fact, only about 70% of this energy is absorbed. The rest is reflected and scattered by the atmosphere and Earth s surface. (This is another way of saying that the albedo of the Earth is about 30%.) What would the Earth s temperature be if its albedo were only 10%? 2. How much solar energy would be collected by a 3-foot by 5-foot solar collector on a clear day if the collector absorbs 95% of this energy, and if 70% of the solar radiation penetrates the atmosphere? 3. What is the energy difference in the two levels of hydrogen that absorb radiation at nm? This line is called the H α line and is one of the most important spectral lines in astrophysics. It is the first line of the Balmer series, and the two energy levels involved are the first and second excited levels. 4. At what wavelength does the maximum radiation occur for an O-type star with a temperature of 30,000 K? This type of star appears blue, since much more radiation is emitted at shorter, blue, than longer, red, wavelengths. 5. How much more energy per second per square meter does the star in Problem 4 emit than the Sun? 6. What is the temperature of an M-type star whose maximum radiation is at 1,000 nm? 11

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