Understanding the Link Between Electric and Magnetic Fields in Light - Prof. David F. Catt, Study notes of Physics

Download Understanding the Link Between Electric and Magnetic Fields in Light - Prof. David F. Catt and more Study notes Physics in PDF only on Docsity! p. 76 Chapter 24 Electromagnetic Waves Consider a chain made of copper and iron links so that every link of iron is adjacent to a link of copper (and vice versa). Connect an AC source to the leftmost copper link and an AC ammeter to the rightmost copper link. Assume that the links do not make physical contact. The oscillating electric current in the leftmost copper link induces an oscillating magnetic field in the adjacent iron link to its right. This oscillating magnetic field in turn induces an oscillating current in the next copper link to the right, and so on. The AC ammeter on the far right registers an oscillating current in the rightmost copper link due to the oscillating current in the leftmost copper link despite the fact that there is no physical connection between the two. Now suppose that we remove the iron links: An oscillating magnetic flux still links all the copper links and an electric current is still induced in the three copper links on the right, causing the AC ammeter on the far right to register an oscillating current. Now suppose that we remove the two copper links in the center: As we saw before, an electric field is induced in the region of a changing magnetic field even if no physical conductor is present to carry a current: Rev. 1/23/08 p. 76 The AC ammeter on the far right still registers an oscillating current, even if there is no matter present between the leftmost and rightmost conducting links! Now let us add the magnetic field lines (shown in perspective) to the above drawing: Note how the planes of the electric and magnetic field lines in the space between the copper links make right angles with one another. Now suppose that the AC source on the far left is off. We now turn it on. The electric and magnetic field lines in the space between the copper links do not appear at the same instant we turn on the AC source. First the leftmost oscillating magnetic field appears, then the leftmost oscillating electric field appears, then the oscillating magnetic field in the middle appears, then the next oscillating electric field on the right appears and so on. The oscillating electric and magnetic field lines advance to the right, forming an electromagnetic wave in space. The advancing wavefront moves at the speed of light: 3 x 108 m/s. This wavefront spreads out in many different directions from the copper link on the left, so that if we take the copper link on the right (still connected to the AC ammeter) and shift it up, down, in or out of the diagram the ammeter still registers an oscillating current. (See the drawings in The Restless Universe by Max Born1.) Maxwell showed his equations for electricity and magnetism predict the existence of electromagnetic waves and that the speed with which these waves move is the speed of light. Maxwell concluded that visible light, along with the other radiations known at the time, infrared and ultraviolet radiation, are electromagnetic waves. Spurred on by these results, the German physicist Heinrich Hertz later discovered what we now call radio waves and verified that they move through the laboratory at the same speed as visible light. We now know that electromagnetic radiation has a very wide spectrum, consisting of waves with wavelengths as small as 10-16 m and as large as 108 m (and larger). Visible light occupies only a very small portion of this spectrum, with wavelengths in air from 400 nm (violet light) to 700 nm (red light). Recall from Physics 111 that the frequency, wavelength and speed of a wave are related by f = c 1Born, Max, The Restless Universe, N. Y., New York, Dover, 1951 The Speed of Light Maxwell showed that his four equations of electromagnetism predict the existence of electromagnetic waves; they also predict the speed of these waves to be 0 0 1 c    where 0 0 is the permittivity of free (empty) space and is the permeability of free space.  If one substitutes the measured values of 0 and 0 one obtains the correct value of the speed of light in a vacuum. This is a remarkable result, since 0 and 0 were unrelated to wave phenomena: 0 is a constant in Coulomb’s law and serves as a measure of the relative strength of the electrostatic force in a vacuum. Similarly, 0 is a constant that appears in Ampere’s law and serves as a measure of the relative strength of the magnetic force in a vacuum. The fact that Maxwell was able to bring these two fundamental constants together to correctly predict an aspect of what appeared to be a completely unrelated phenomenon (the speed of light) was a great triumph for Maxwell’s theory and classical physics. For not only do Maxwell’s four equations describe all electromagnetic phenomena, they contain all the laws of optics as well! We will study some of the laws of optics in the chapters that follow. The fact that light travels at a finite speed means that there is time lapse between when light leaves an object and when it strikes the retina of a person’s eye. What the person sees is the object as it was a short time ago, not the way it is when the light strikes the retina. Because of the enormous speed of light, this time lapse is not large enough to have a significant effect in our daily lives when we deal with nearby objects. However, when one views astronomical objects (either with an unaided eye or a telescope), one sees the objects as they were in the past. How far in the past depends on the distance of the astronomical object. Examples The mean earth - moon distance is 83.85 10 m . How far back in time is the image of the moon when viewed from earth? 8 8 Use formulas from Physics 111: 3.85 10 m or ; ; ; 1.28 s 3.00 10 m/s d d vt d ct t t t c        The mean earth - sun distance is 111.50 10 m . How far back in time is the image of the sun when viewed from earth? 11 8 Use formulas from Physics 111: 1.50 10 m or ; ; ; 500 s or 8.33 minutes 3.00 10 m/s d d vt d ct t t t t c         Astronomical instruments such as the Hubble space telescope can image objects at enormous distances. A light- year is the distance light travels in one year (in a vacuum). Hubble has taken pictures of objects billions of light years away. Because of the time lapse between the instant the light left those objects and the instant the light arrived on earth, those pictures show those objects as they were in the past, several billion years ago! June 10, 1996 Photo No.: STScI-PRC96-23a A huge, billowing pair of gas and dust clouds is captured in this stunning NASA Hubble Space Telescope image of the supermassive star Eta Carinae. Using a combination of image processing techniques (dithering, subsampling and deconvolution), astronomers created one of the highest resolution images of an extended object ever produced by Hubble Space Telescope. The resulting picture reveals astonishing detail. Even though Eta Carinae is more than 8,000 light-years away, structures only 10 billion miles across (about the diameter of our solar system) can be distinguished. Dust lanes, tiny condensations, and strange radial streaks all appear with unprecedented clarity. The event observed in this Hubble photograph occurred 8,000 years ago. Energy Carried by Electromagnetic Waves All traveling waves transport energy from one location to another. The energy density (energy per unit volume) of an electromagnetic wave can be found from the formulas for the energy densities of the electric and magnetic fields that comprise the wave: Electric Energy Density: 2E 0 1 , where is the electric field amplitude 2 u E E  . Magnetic Energy Density: 2 B 0 1 , where is the magnetic field amplitude 2 u B B  . Total energy density: 2 2 E B 0 0 1 1 2 2 u u u E B      It can be shown that for an electromagnetic wave propagating through the vacuum or air the electric and magnetic field amplitudes are related by E cB If we square both sides of the equation we get 2 2 2 0 0 2 2 0 0 1 or, using , 1 E c B c E B        0which gives, after multiplying by and dividing by 2, 2 2 0 0 1 1 2 2 E B   The expression on the left is the energy density of the electric field; the expression on the right is the energy density of the magnetic field. These two energy densities are equal in an electromagnetic wave. The total energy density can then be expressed as either 2 2 0 0 1 or .u E u B   Example The rms electric field striking the top of earth’s atmosphere is rms 720 N/CE  . Find the average total energy density of the electromagnetic wave.     22 12 2 2 0 rms 2 2 6 6 6 2 2 2 3 6 3 ; 8.85 10 C / N m 720 N/C C N N N m 4.6 10 4.6 10 4.6 10 N m C m m J 4.6 10 m u E u u u                        Intensity Suppose that a portion of a wave passes through a rectangular area (such as a window) in a direction perpendicular to the area. Since a wave carries energy, some of the wave’s energy has passed through the area. Since the wave travels with a finite speed, it will take some time (call it t) for this energy E to pass through. If the rectangular region has area A the intensity of the wave is defined as 2 / watts or . Units: m E t E S S A t A    If the wave is a light wave traveling in air, the distance the wave travels in time t is ct; the volume that it sweeps out as it passes through the area A is Act. Since the energy density of the wave is u, we have E = uAct and or uAct S S cu tA   . Using the formulas for energy density this can be written as 2 2 0 0 or c S c E S B   