EM-1 Electromagnetic Waves Last semester, we studied Classical Mechanics. The fundamental laws (axioms) of Classical Mechanics are called Newton's Laws. This semester, we are studying a subject called Classical Electromagnetism. There are four fundamental laws of electromagnetism, called Maxwell's Equations (after the Scottish physicist James Clerk Maxwell). (1) Gauss's Law enclosed0qEda⋅=ε∫KKv ( E-fields are caused by charges. ) (2) Faraday's Law: SdEd Bdadt⎛⎞⎟⎜⎟⎜⋅ = −⋅⎟⎜⎟⎟⎜⎝⎠∫∫KKKKAvL (E-fields are also caused by changing B-fields.) (3) Ampere-Maxwell Law: 0enclosed 0 0(A) AdBd I Edadtµµε⎛⎞⎟⎜⎟⎜⋅ =+ ⋅⎟⎜⎟⎟⎜⎝⎠∫∫KKKKAvL (B-fields are caused both by currents and by changing E-fields.) (4) Gauss's Law for B-fields: Bda 0⋅ =∫KKv (There are no magnetic monopoles). Except for the last term in equation (3), all four of these laws had been discovered experimentally before Maxwell started his research in the 1850's. So why do we call them Maxwell's Equations? Maxwell realized that Ampere's Law, , was incomplete. He noticed that there are situations in which Ampere's Law fails to give the correct answer. For instance, if a capacitor is being charged up by a steady current, then there must be a B-field around the capacitor, caused by the nearby currents. But according to the original form of Ampere's law, if we consider an imaginary loop circling the capacitor (diagram below), the current through this loop is zero. So Ampere's Law predicts that the B-field is along that loop is zero (since I0 enclosed(A)Bd Iµ⋅ =∫KKAvLthru = 0). Maxwell noticed that although there is no current through the loop, there is a changing E-field flux through the loop. He saw I I E (increasing) Last update: 11/16/2009 Dubson Phys1120 Notes, ©University of ColoradoEM-2 that he could fix the problem by modifying Ampere's Law with the addition of a new term. The changing electic flux in the capacitor leads to a quantity that has the dimensions of current: 0AdEdadtε⎛⎞⎟⎜⎟⎜⋅⎟⎜⎟⎟⎜⎝⎠∫KK . Notice that, from Gauss's Law, the quantity has the dimensions of charge. So, 0AEdaε ⋅∫KK0AdEdadtε⎛⎞⎟⎜⎟⎜⋅⎟⎜⎟⎟⎜⎝⎠∫KK has the dimensions of current. Maxwell called this new quantity the displacement current. By replacing the current I in Ampere's Law 0AdIEdtε⎛⎞⎟⎜⎟⎜+ ⋅⎟⎜⎟⎟⎜⎝⎠∫KKda, he was able to resolve the problem. This new form of Ampere's Law (now called the Ampere-Maxwell Law) appealed to Maxwell's sense of aesthetics. There was now a pleasing symmetry in the equations: • changing B-fields create E-fields (Faraday's Law) • changing E-fields create B-fields (Ampere-Maxwell Law) Maxwell realized that because of this symmetry, the equations predicted a peculiar kind of self-sustaining interaction between E and B fields. Maxwell thought: Suppose you have a charge q and you shake it, back and forth. The q creates an E-field, but when you shake the charge, you are changing the E-field in the space around it. This changing E-field creates a B-field. But now you just created a B-field where there was none before, so you have a changing B-field. This changing B-field will create an E-field, and that newly created E-field will create a B-field, which will create an E, which will create a B, which will … (the process will go on, forever). Maxwell showed that the equations predicted the existence of an electromagnetic wave which travels outward from the shaking charge: E E E E E E B B B B Ithru = 0, B = 0 ? I I imaginary loop L Last update: 11/16/2009 Dubson Phys1120 Notes, ©University of ColoradoEM-3 Maxwell computed the speed of this strange, new electromagnetic wave and found that the speed was given by a simple formula: 8oo1speed v = c 3.0 10 m/s==×ε µ. This number is the same as the speed of light! Maxwell had shown that light was an electromagnetic wave! Before Maxwell, scientists had no clear idea what light is. This was a great synthesis, a bringing together of previously separate fields of physics: electricity, magnetism, and optics. Before Maxwell, no one knew what light was. It was known that light was some kind of wave (we will see the evidence for this later), but no one knew what kind of wave it was. Maxwell figured it out. Light is an electromagnetic wave which is created by accelerating electric charge. Wave speed is distance 1 wavelengthvtime time for 1 to go by Tλ== =λ vfTλ==λ For light waves, speed v = c, this is written cf= λ EM waves are transverse waves: the E- and B-field vectors are both perpendicular to the direction of the wave. Drawing an EM wave in space is quite difficult; the E and B-fields are everywhere and intimately mixed. The figure here shows the E-field along a particular line, at a moment in time. wavelength λ E (up and down) speed c B (in and out) All EM radiation is caused by shaking (accelerating) electric charge. The more rapidly the charge is shaken (the higher the frequency of the shake), the shorter the wavelength of the light, since cfλ =. Now we can understand why all things glow (give off light) when they get hot. Last update: 11/16/2009 Dubson Phys1120 Notes, ©University of ColoradoEM-4 When something is very hot, its atoms are jiggling furiously. Atoms are made of charges (electrons and protons), and the jiggling charges emit EM radiation. Different wavelength ranges are given names: Wavelength λ Name Use/occurrence < ≈ 0.01 nm Gamma-rays Radioactivity ≈ 0.01 nm → ≈ nm X-ray medical ≈ nm → 400 nm Ultraviolet(UV) Sunburns, "black" lights 400nm → 700 nm Visible Human seeing 700nm → ≈ 1mm Infrared (IR) "Heat rays" ≈ cm microwave Communications, microwave ovens ≈ m → km → ∞ radio Radio, TV Electromagnetic radiation (light) can have any wavelength. But our eyes are sensitive only to a narrow range of wavelengths between 400 nm and 700 nm. Different wavelengths in this range of visible light correspond to different colors. Wavelength = 700 nm light appears red to us, 400 nm light appears violet, and the wavelengths in between correspond to all the colors of the rainbow (ROYGBIV). All wavelengths outside this narrow band are invisible to human eyes. Some important facts about EM waves: • EM waves are transverse: The E and B-field are perpendicular to each other are each perpendicular to the
View Full Document
Unlocking...