Section 3 Electromagnetic Waves 1 EM waves in vacuum In regions of space where there are no charges and currents Maxwell equations read E 0 3 1 B 0 3 2 E B t B 0 0 E t 3 3 3 4 They are a set of coupled first order partial differential equations for E and B They can be decoupled by applying curl to eqs 3 3 and 3 4 E E 2 E B 2E B 0 0 2 t t t E 2B B B 2 B 0 0 E 0 0 0 0 t t t 2 3 5 3 6 Since E 0 and B 0 we have 2 E 0 0 2E t 2 3 7 2 B 0 0 2B t 2 3 8 We now have separate equations for E and B but they are of second order that s the price you pay for decoupling them In vacuum then each Cartesian component of E and B satisfies the three dimensional wave equation 2 f 2 f 0 0 2 3 9 t The solution of this equation is a wave So Maxwell s equations imply that empty space supports the propagation of electromagnetic waves traveling at a speed c 1 0 0 3 00 108 m s 3 10 which happens to be precisely the velocity of light c The implication is astounding light is an electromagnetic wave Of course this conclusion does not surprise anyone today but imagine what a revelation it was in Maxwell s time Remember how 0 and 0 came into the theory in the first place they were constants in Coulomb s law and the Biot Savart law respectively You measure them in experiments involving charged pith balls batteries and wires experiments having nothing whatever to do with light And yet according to Maxwell s theory you can calculate c from these two numbers Notice the crucial role played by Maxwell s contribution to Ampere s law without it the wave equation would not emerge and there would be no electromagnetic theory of light 1 Monochromatic plane waves Since different frequencies in the visible range correspond to different colors such waves are called monochromatic Suppose that the waves are traveling in the z direction and have no x or y dependence these are called plane waves because the fields are uniform over every plane perpendicular to the direction of propagation We are interested then in fields of the form E z t E0 ei kz t 3 11 B z t B0 ei kz t 3 12 where E0 and B0 are the complex amplitudes the physical fields of course are the real parts of E and B Substituting eqs 3 11 and 3 12 to eqs 3 7 and 3 8 respectively we find that c k 3 13 Here k is the wave number which is related to the wavelength of the wave by the equation 2 k 3 14 and is the angular frequency of EM wave Fig 1 This is the paradigm for a monochromatic plane wave The wave is polarized in the x direction by convention we use the direction of E to specify the polarization of an electromagnetic wave Now the wave equations for E and B were derived from Maxwell s equations However whereas every solution to Maxwell s equations in empty space must obey the wave equation the converse is not true Maxwell s equations impose extra constraints on E0 and B0 In particular since E 0 and B 0 it follows that E0 z B0 z 0 3 15 That is electromagnetic waves are transverse the electric and magnetic fields are perpendicular to the B direction of propagation Moreover Faraday s law E implies a relation between the electric t and magnetic amplitudes E E0 e x ikE 0y i kz t E 0 e i kz t E 0 e i kz t y ikE0 x ei kz t x i B0 x y i B0 y ei kz t which results in 2 x E0 x 0 y E0 y 0 z 0 ei kz t ik 3 16 kE0 y B0 x kE0 x B0 y 3 17 or more compactly B0 k 1 c z E0 z E0 3 18 Evidently E and B are in phase and mutually perpendicular their real amplitudes are related by B0 The fourth of Maxwell s equations B 0 0 1 E0 E0 c k 3 19 E does not yield an independent condition it simply t reproduces Eq 3 16 There is nothing special about the z direction of course we can easily generalize to monochromatic plane waves traveling in an arbitrary direction The notation is facilitated by the introduction of the wave vector k pointing in the direction of propagation whose magnitude is the wave number k The scalar product k r is the appropriate generalization of kz so E r t E0eei k r t 3 20 1 1 B r t E0 n e ei k r t n E c c 3 21 k is the unit vector in the direction of propagation of the EM wave and e is the k polarization vector Because E is transverse Where vector n n e 0 3 22 Linear and circular polarizations The plane wave 3 20 and 3 21 is a wave with its electric field vector always in the direction e Such a wave is said to be linearly polarized with polarization vector e1 e Evidently there exists another wave which is linearly polarized with polarization vector e2 e1 and is linearly independent of the first Thus the two waves are E1 r t E1e1ei k r t 3 23 E2 r t E2e 2 ei k r t 1 with B1 2 n E1 2 c They can be combined to give the most general homogeneous plane wave propagating in the direction k kn E r t E1 r t E2 r t E1e1 E2e 2 ei k r t 3 24 The amplitudes E1 and E2 are complex numbers to allow the possibility of a phase difference between waves of different linear polarization If E1 and E2 have the same phase wave 3 24 represents a linearly polarized wave with its polarization vector making an angle tan E2 E1 with respect to e1 and a magnitude E E2 2 E12 as shown in Fig 2a 3 Fig 2 a Electric field of a linearly polarized wave b Electric field of a circularly polarized wave If E1 and E2 have different phases the wave 3 24 is elliptically polarized To understand what this means let us consider the simplest case circular polarization Then E1 and E2 have the same magnitude but differ in phase by 90 The wave 3 24 becomes E r t E0 e1 ie2 ei k r t 3 25 with E0 the common real amplitude We imagine axes chosen so that the wave is propagating in the positive z direction while e1 and e2 are in the x and y directions respectively Then the components of the actual electric field obtained by taking the real part …