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Chapter 16 ELECTROMAGNETIC WAVES IN MATTER • Introduction • Energy and momentum in electromagnetic waves • Angular distribution and polarization • Scattering of electromagnetic waves • Electromagnetic waves in matter • EM waves at boundaries • Summary 2 2 = 2 2 INTRODUCTION The complete description of electromagnetism requires the four Maxwell equations plus the Lorentz force relation that defines the electric and magnetic fields. That is: Φ = I → − − → 1 E · S = 0 Φ = I I → − − → E · l = − I → − − → B · dl = 0 = (2) Equations 1 and 2 describe a system of coupled electric and magnetic fields where an oscillating field generates a travelling wave and an oscillating field generates a travelling wave. Also it was shown that the non-zero derivatives of and with respect to the z direction do correspond to a travelling wave in the z direction by derivation of the following wave equations − Z Z Inserting the known values = 8854×10−12 2 2 and = 410−7 2 gives that the velocity of an electromagnetic wave in vacuum is = 299792458 × 108 These waves are transverse waves comprising perpendicular electric and magnetic oscillatory fields that are in phase and perpendicular to the direction of propagation. The changing E field induces the perpendicular B field, while this changing B field induces the corresponding E field. The amplitudes of the E and B fields are related in that 0 = 0 and the wave travels → − − → in the E × B direction. The prediction of transverse electromagnetic waves is a general feature of Maxwell’s equations and was demonstrated. −→ → B − · S → − − → → E − ) · S ( j + 0 − → → → − − → F = ( E + − v × B) It was shown in chapter 15 that Maxwell’s equations for vacuum that is, with zero current and charge → − densities, j = = 0 led to the fact that the oscillating electric and magnetic fields of a plane wave must be transverse to the direction of propagation of the wave. Moreover, Maxwell’s equations lead to the following relations for a plane electromagnetic wave travelling in the z direction. =− 2 2 = (4) 2 2 These are examples of the general wave equation in one dimension 1 2Ψ 2Ψ = 2 2 2 where both the electric and magnetic waves travel at a common velocity in vacuum, called c: 1 ≡= √ → → − − B · S = 0 Z (3) (1) ENERGY AND MOMENTUM IN ELECTROMAGNETIC WAVES A remarkable feature of electromagnetic waves is that energy is transmitted through vacuum, that is the travelling electric and magnetic fields carry energy. The existence of life on earth is a direct result of the transmission of energy from the Sun to the Earth via electromagnetic waves in vacuum. The energy transmitted via an electromagnetic wave can be related to the and fields in the wave. 119 Figure 1 An electromagnetic wave incident upon a point charge that initially is at rest. The instantaneous power transmitted by the wave usually is written as the Poynting vector S where: − → → − → 1− E×B S ≡ 0 → − The Poynting vector S gives the power, in watt/m2 → − − → radiated in the direction E × B Consider a harmonic EM wave of the form: → − → − E = 0bi sin( − ) and B = 0bj sin( − ) Figure 2 The radiation pattern of the Poynting vector for an electric dipole radiator. Inserting these into the Poynting vector and using the fact that the average of a sin2 function equals 1/2, gives: |S| = 1 1 1 0 0 = 2 0 0 0 0 where = √ and = √ 2 2 An electromagnetic wave carries momentum as well as energy. The momentum carried by an electromagnetic wave can be seen by considering the force on a charge in the EM wave and remembering that force is rate of change of momentum. A charge q, mass −→ m, will experience a force F = bi in the electric − → → field leading to a velocity − v = E at a time t. The moving charge then will experience a magnetic force 2 − → − → → − −−→ → v × B = E × B That is, the net force F = − is in the direction of the electromagnetic field. This force per unit area is called radiation pressure. ANGULAR DISTRIBUTION AND POLARIZATION For an electric dipole radiator it is obvious that the radiated electric field vector Ē will point along the axis in figure 2 while the magnetic field vector B̄ vector will be in the − plane circulating around the → − → − axis. Moreover, both the transverse E and B fields will be zero for a wave propagating along the axis with = = 0 As a consequence the Poynting vector of the radiated field has an angular distribution of the form → sin2 − S ∝ 2 120 Figure 3 Polarization analyzers. where is the angle with respect to the axis. Thus → − the electromagnetic wave is polarized with the E vector oriented parallel to the axis. This behaviour was demonstrated by the microwave demonstration which emitted linearly-polarized waves, that is, waves with the electric field vector parallel to the axis of the electric dipole radiator. If a grid comprising parallel wires of a good conductor, is inserted between the source and detector with → − the wires parallel to the E field vector then the wave is reflected by the current induced in the wires. Whereas, if the orientation of the wires is perpendicular to the electric field vector then the wave is transmitted as demonstrated and illustrated in figure 3. The grid serves as a polarization analyzer and is analogous to use of polaroid material to analyze the polarization of light. SCATTERING OF EM WAVES Electromagnetic waves are scattered by free or bound charges. These charges oscillate along the direction of the electric field vector of the incident electromagnetic wave. These oscillating charges reradiate like small electric dipoles. The coherent oscillations of free elec- ELECTROMAGNETIC WAVES IN MATTER In a dielectric the total real charge density = + where is the applied free charge density and is the polarization charge in the dielectric. In the discussion of dielectrics it was shown that the polarization charges can be absorbed into the dielectric constant allowing an alternate expression of Gauss’s Law as: I Z → → − − 1 E · S = Figure 4 Scattering of sunlight in the upper atmosphere. This can be written as: I Z → → − − E · S = trons in a metal or plasma lead to the reflection of the incident wave at frequencies below the plasma oscillation frequency. Above the plasma oscillation frequency the electromagnetic radiation is transmitted and scattered by reradiation from charges oscillating in the incident electromagnetic field. A nice example of scattering of electromagnetic waves is scattering of sunlight by the atmosphere. The bound electrons in the atoms have resonances in the ultraviolet region. Because the eye is sensitive only to light, that spans frequencies just below the resonance frequency for these bound electrons, it can be shown that the power radiated by these reradiating oscillating bound electrons, charge q and mass m, is: − → 4 4 sin2 b S ∝ 2 02 2 r ( 0 − 2 )2 2 where 0 is the resonance frequency of the bound electrons, the incident frequency, and is the angle with respect to the electric field vector of the incident wave. Note that the radiation peaks perpendicular to the oscillating electric dipoles and that the reradiated power increases as 4 As a consequence, blue light is scattered much more than red light. This is the reason that the sky looks blue and the setting sun looks red because the blue component has been scattered away due to the long path length travelled by the sunlight in the atmosphere. Sunlight transmitted through the upper atmosphere is reradiated downwards preferentially for the horizontal component of the incident light wave, thus the scattered light is plane polarized as illustrated in figure 4. The demonstration illustrates this phenomena. The atmosphere becomes opaque near resonances such as the UV region which is helpful in reducing the risk of skin cancer. At high frequencies, well above the UV, the scattering becomes small. This formula can be simplified by defining the permittivity ≡ Note that only the applied charge density need be known when applying this alternate form of Gauss’s Law. Similarly the total current density in magnetic ma→ − −−→ −−→ → − terials can be split as j = j + j where j → − is the applied current density and j is the atomic bound current density. It was shown that the eﬀects of magnetisation can be absorbed into the relative permeability allowing Ampère’s Law to be written as: − → Z −−→ − − → B → · dl = 0 j · S I This can be rewritten as: Z → − I −−→ − → − → B · dl = j · S 0 This can be simplified by defining permeability : = Two other terms must be added to the current density, − → namely Maxwell’s displacement current j and the −−→ polarization current density j that occurs in a dielectric when the bound polarization charges are flowing as the polarization is changing. It can be shown that the polarization current can be absorbed into the dielectric constant giving an alternate expression of the Ampère-Maxwell equation: I → − − → Z −−→ E − − → B → ( j + · dl = ) · S Thus Maxwell’s equations, can be expressed in a form where both the bound charges and bound currents were absorbed implicity into the permittivity 121 and permeability giving an alternate expression of these laws. Maxwell’s laws-Alternate form I Z → → − − Φ = E · S = Φ = I I → − − → E · l = − I → − − → B · S = 0 Z Figure 5 Schematic illustration of the frequency response of the dielectric constant of a typical dielectric. The dielectric constant equals the square of the refractive index, n. −→ → B − · S − → → − Z −−→ E − − → B → ( j + · dl = ) · S These equations are similar to those for waves in vacuum except that 0 and 0 are replaced by and As shown in chapter 15, the two flux equations require that electromagnetic waves be transverse. Following the same procedure as used for waves in vacuum it is easy to see that for a plane wave propagating in the direction the two circulation equations lead to the relations =− (1) = (2) − and from these one obtains the wave equations: 2 2 = 2 2 2 2 = 2 2 Therefore the velocity v of the EM wave is: (3) (4) 1 1 =√ =√ =√ √ 0 0 Note that the dielectric constant always is greater than unity while the relative permeability is essentially unity except for ferromagnetic materials where it is a large number. Thus the velocity ≤ Refractive index In the discussion of optics the refractive index n is defined as: √ ≡ = For most transparent materials = 1 and thus the √ refractive index = The refractive index of a material depends on the frequency of the radiation as illustrated schematically 122 in figure 5. As an example, consider water. For a constant electric field, the dielectric constant of water is 81, implying a refractive index of 9. However, this large dielectric constant is due to alignment of the large polar water molecule that has a large moment of inertia. Above a moderately low frequency, the slowly moving polar molecules cannot move fast enough to follow the oscillating electric field and thus the alignment of the polar molecule does not contribute to the oscillating polarization electric field in the dielectric. As a consequence, the dielectric constant decreases considerably. At a higher frequency the ionic polarization of the molecules ceases to contribute to the polarization, reducing the dielectric constant further, when the molecular polarization cannot follow the oscillating electric field. As a consequence, in the visible region of the electromagnetic spectrum, the refractive index of water equals 133, that is the dielectric constant is reduced to 177. At still higher frequencies the atomic polarization of the more weakly-bound electrons cease to contribute again because they cannot follow the rapid oscillations. Eventually, for hard X rays and gamma rays, even the atomic polarization eﬀects become negligible and the refractive index approaches unity. The frequency dependence of the refractive index is further complicated in that the eﬀects are enhanced when the electromagnetic frequency approaches resonant frequencies of bound electrons resulting in localized frequency regions where the refractive index increases as illustrated in figure 4. For example, there are many resonances of bound atomic electrons in the ultra-violet region of the EM spectrum. These cause the dielectric constant, and thus, the refractive index, to increase with frequency from the red to the blue regions of the spectrum. As a consequence, blue light travels slower than red light in glass or water. This frequency dependence of the refractive index is called dispersion since it leads to dispersion of the diﬀerent frequencies in a glass prism or rainbow as will be discussed later. This discussion is overly simple in that Figure 6 Boundary conditions for electric and magnetic fields at a plane surface. it ignores phase eﬀects and damping. EM WAVE AT A BOUNDARY Wavelength Electromagnetic waves in material travel at a velocity = and the wavelength and frequency are related by = = At a boundary, the frequencies of the waves in the two materials must be the same. Thus if the refractive indices of the materials are diﬀerent, so are the wavelengths in these materials. That is, = = 1 1 2 2 thus: 1 1 = 2 2 Figure 7 Electromagnetic wave incident normally upon a plane interface between two materials of refractive index n1 and n2 travelling to the right. Assume that the incident , reflected , and transmitted electric vectors point upward, then using the boundary condition that k is continuous implies that: + = (1) Assuming that the electric field vector points upwards implies that the magnetic field points out of the page for the incident and transmitted waves which are travelling to the right, and into the page for the left-going reflected wave. Requiring that k be continuous across the boundary implies: 1 1 ( − ) = 1 2 The relation between and for a wave is = Thus: = = The wavelength is shorter in materials with a large refractive index. That is the wave does not travel as far in a given time because the velocity is lower. Using this to rewrite the boundary condition for gives: 2 1 ( − ) = (2) 1 2 Boundary conditions Solving equations 1 and 2 gives the electric field vector for the reflected wave: Remember that polarization of a dielectric causes surface charge density that partially cancels the normal component of E in the dielectric but leaves the parallel component unchanged, as shown in figure 6. By contrast, the surface currents due to magnetization of matter enhances the parallel component of the B field inside the material but leaves the normal component of B unchanged. The use of boundary conditions allows detailed calculation of the behaviour of EM waves at boundaries. As an example, consider the simplest possible case of an EM wave incident normally to a boundary as shown in figure 7. For normal incidence, only the parallel components of and are involved. In the left-hand medium, refractive index 1 one has two waves, the incident wave moving to the right and the reflected wave moving to the left. In the right-hand medium, refractive index 2 there is only the transmitted wave ( 1 − = 11 ( + 1 2 2 ) 2 2 ) ≈ 1 − 2 1 + 2 and the electric field vector for the transmitted wave: 2 1 21 = 1 1 2 ≈ ( + ) 1 + 2 1 2 where the approximation applies since typically 1 = 1 2 2 = 0 Note that is positive if 1 2 while it 1 2 is negative if This is analogous to reflections 1 2 of waves on a rope, as illustrated in figure 8. When the right-hand rope is heavier that the left-hand rope, it does not move as easily as the left-hand rope so the amplitude of the reflected wave is opposite in sign to the amplitude of the incident wave leading to partially cancellation of the motion at the joint between the 123 Conductive materials Figure 8 Reflection and transmission at the interface between ropes of diﬀerent mass per unit length. two ropes and a smaller amplitude for the transmitted wave. On the other hand, when the right-hand rope is lighter than the left-hand rope, the reflected wave and incident wave have the same phase at the junction leading to an amplitude of the transmitted wave that is the sum of the reflected and incident amplitudes. The intensity of the reflected and transmitted waves → − is obtained by evaluating the Poynting vectors S = → − → − 1 E × B for the waves, where is the permeability of the material. Substituting = in the Poynting vector gives for the magnitude: = 2 This gives that the reflectivity of the surface, defined as the ratio of reflected and incident Poynting vectors, is: ¯ ¯2 ¯¯ 1 ¯ ¯ ¯ 2 ¯2 ¯ 1 − 2 ¯2 ¯¯ ¯¯ ¯ ( 1 − 2 ) ¯ ¯ ¯ = =¯ = ¯ 1 ¯ ≈¯ ¯ ( + 2 ) ¯ ¯ 1 + 2 ¯ 1 2 Similarly, the transmission , is given by: = 2 1 = 2 1 Note that: ¯ ¯ 4 1 2 ¯ ¯2 41 2 1 2 ¯ ¯ = ≈ 2 ¯ ¯ ( 1 + 2 )2 ( 1 + 2 ) 1 2 +=1 As an example, consider 1 = 1 and 2 = 15 typical of glass. Then this gives = 4% and = 96%. For a typical 4−element camera lens there are 8 surfaces implying 32% of the light can be reflected which can seriously degrade the contrast if anti-reflection coatings are not used. When the two media have the same value of the reflectivity is zero and transmission 100%. The reason for discussing the derivation of the transmission and reflection coeﬃcients for normal incidence, is to show that electromagnetism explains the details of reflection and refraction. One can derive more complicated relations relating the incident, reflected and transmitted waves for non-normal incidence using the same boundary conditions. 124 A good conductor is an equipotential for constant electric fields implying that the electric field is zero inside the good conductor. This implies that a low frequency electromagnetic wave is reflected and not transmitted by a good conductor. The electric field in conductors continues to be zero below the plasma frequency. For a good metallic conductor the plasma frequency, which depends on the density of charge carriers, is in the high ultraviolet region while it is between the AM and FM radio bands for the earth’s ionosphere. This explains why AM radio waves are reflected by the ionosphere and bounce around the earth because they are below the ionosphere plasma oscillation frequency. However, FM, TV, light waves, and higher frequencies are above the plasma frequency and thus such waves are transmitted by the ionosphere. The much higher plasma frequency in metals explains why radio waves and even light waves are reflected well by a good conductor. Radio and microwaves penetrate only a very small distance into a good conductor, for example, microwaves penetrate only the order of a micrometer in aluminum, thus even a thin layer of kitchen foil is suﬃcient to reflect microwaves or light as demonstrated. SUMMARY This is the completion of the discussion of electromagnetism. We see that the laws of electricity and magnetism lead to Maxwell’s equations, which predict electromagnetic radiation ranging from radio waves to light, X rays to rays. They predict that electromagnetic waves in vacuum travel at one fundamental speed, . Maxwell’s equations also contain the Einstein’s Theory of Special Relativity, although it took Einstein to realize this crucial fact. Considerable stress has been placed on electromagnetism in this course because it underlies much of science, medicine and technology. Electrical forces bind atoms, molecules such as DNA. Electromagnetic radiation transmits the energy from the sun allowing the existence of life on earth. Electromagnetic devices such as computers, TV, electrical power generation and distribution are an essential part of life. Nerve cell, the senses of smell, touch, vision all are ultimately electromagnetic in origin. Reading assignment: Giancoli, Chapters 32