\Lecture 3 Maxwell’s Equations
' Div[ 0 E ] = - Div[P] + Div[ µ0 H ] = - Div[ µ0 M ] 0 0 0 0 Curl[E] = - / t ( µ0 H ) - / t ( µ0 M ) Plane Waves, Phasors, 0 0 0 0 Curl[H] = / t ( 0 E ) + / t ( P ) + J % -7 Refractive Index and Dispersion µ0 = 4 . 10 in SI units -9 % 0 = 10 /(36 ) in SI units
Wave Equation Transverse Waves
For a linear, isotropic, charge-free medium Del2[ H ]] - 02/0t2 ( µH ) = 0 Del2[ H ]] - 02/0t2 ( µH ) - 0/0t ( µ)H ) = 0 Del2[ E ]] - 02/0t2 ( µE ) = 0 Del2[ E ]] - 02/0t2 ( µE ) - 0/0t ( µ)E ) = 0 The most general plane wave solutions of the above The third term in the above equation represents the are dissipation E = x [ f+ (z - vt) + f- (z + vt) ] 1/2 For a linear, isotropic, charge-free, non-conducting H = ( /µ ) y [ f+ (z - vt) + f- (z+vt) ] medium ( ) = 0 ) the wave does not dissipate Note that E and H are perpendicular to each other and A Vacuum is a particular case of the above!! to the direction of propagation - z Monochromatic Waves Velocity In a Medium
7 1 3 3 E = E0 x cos( k . r ± t + ) If e, m are # 0, then the phase speed is reduced 1 is an arbitrary phase Phase speed becomes c/n 7 ½ is the angular frequency n = ( / 0 . µ/µ0 ) k defines both the direction of propagation (DOP) and n is the “refractive index” and is > 1 (mostly) % 3 3 the wave vector |k| = 2 / = k If either of e, m are dependent upon frequency, then k . r ± 7t defines a surface of constant phase which is the medium has “dispersion” displaced in time at a speed 7/k which is called the This means that if n is frequency-dependent, then the “phase speed” medium is dispersive
Complex Numbers Complex Exponentials - Phasors
Complex numbers for physicists are a “convenience” If C = cos(a) + i sin(a) for solving problems Then it is shown in all the best text books that you can Rely on the real-world fiction that -1 can be write C = exp(ia) represented by i (the engineers use j) In fact any complex number C can be written as A complex number can be written as C = a + ib C = |C| exp(i arg(C)) The conjugate complex is C* = a - ib Since manipulation of exponentials in integration and The modulus is |C| = (a2 + b2 ) differentiation is very easy, it is tempting to try to The argument is Arg(C) = tan-1 (b/a) manipulate any wave equation into this format Complex Superposition Differentiation wrt Time
7 1 7 If E0 x cos (k . r - t + ) is a solution For any quantity C exp(-i t) where C is independent of 7 1 Then so is E0 x sin (k . r - t + ) time 7 1 0 0 7 7 7 And so is i . E0 x sin (k . r - t + ) / t (C exp(-i t) ) = -i C exp(-i t) By the principle of linear superposition, the sum of the Similarly Int[C exp(-i7t) ] = -1/i7 C exp(-i7t) above solutions is also a solution which can be What does this do to Maxwell’s equations? ' expressed as Div[ 0 E ] = - Div[P] + - unchanged 7 1 E0 x exp i(k . r - t + ) Div[ µ0 H ] = - Div[ µ0 M ] - unchanged 7 0 0 0 0 or x . E . exp(-i t) where E is complex and contains all Curl[E] = - / t ( µ0 H ) - / t ( µ0 M ) becomes 7 7 spatial and phase components Curl[E] = i µ0 H + i µ0 M 0 0 0 0 Curl[H] = / t ( 0 E ) + / t ( P ) + J becomes 7 7 Curl[H] = - i 0 E - i P + J
ME Monochromatic Wave Eqn Dispersion Relation
Del2[ H ]] + 72µH + i7µ)H = 0 Putting all these components into the wave equation, Del2[ E ]] + 72µE + i7µ)E = 0 we find that % -7 2 72 µ0 = 4 . 10 in SI units k = µ for a linear, isotropic, charge-free medium -9 % 0 = 10 /(36 ) in SI units Remember that is generally complex and therefore If we now REDEFINE as being + i)/7, we can so is k automatically incorporate conducting media in the If we define refractive index to be complex following , solutions with a complex dielectric constant then This implies that the effect of conductance is the the k2 = 72 /c2 . n2 polarisation P is still proportional to the field E, but This is the “dispersion relationship” and measures how there is a phase lag between the field and the the wave vector k, varies with frequency polarisation We can also look at the same arguments for the spatial differential Del2[] and find that in a charge-free medium Del2[X] = -k2 X where k is the wave vector n and Bringing It Back to Reality
7 1 Most materials are non-magnetic, assume µ=µ0 The wave propagates like exp i( k . r - t + ) 2 = r + i i = n 0 Suppose it propagates along the z axis -> kz So if n = nr + i ni The wavelength in free space is v 2 2 Then r / 0 = nr - ni and the wavevector is kv And i / 0 = 2 nr ni Look at the spatial part only What’s the point? exp(ikz) = exp{i ( nr + i ni ) kv z } We can measure the refractive index very easily exp(ikz) = exp (i nr kv z) exp(- ni kv z) therefore we can relate the macroscopic n to the First term is phase factor, second is a decay term microscopic, complex Measure the phase speed - gives nr (historically n) Measure the decay - gives ni (historically )
Refractive Indexes
Vacuum = 1.00 (surprise, surprise) Glasses = 1.57-1.77 Sapphire = 1.77, Diamond = 2.42 Silicon = 3.8 + 0.4i Silver = 2.3 + 3.8i -6 ' ' Air = 1.0 + 290 . 10 ( / stp)