Electromagnetic Waves

Time-dependence couples electricity and magnetism .D = r  x E = -∂B/∂t .B = 0  x H = J + ∂D/∂t Maxwell’s equations for Electromagnetism B = mH D = eE J = sE We won’t learn equations for s, e, m These require quantum mechanics/statistical mechanics’ Sometimes equations interdependent eg. e = e(E), r = r(E) Much of physics research involves solving both together.

Maxwell’s equations in free space (r = J = 0)  x E = - ∂B/∂t .B = 0  x B = me∂E/∂t  x ( x E) = - ∂( x B)/∂t

Maxwell’s equations in free space (r = J = 0)  x E = - ∂B/∂t .B = 0  x B = me∂E/∂t (.E) - 2E = - ∂( x B)/∂t

Maxwell’s equations in free space (r = J = 0)  x E = - ∂B/∂t .B = 0  x B = me∂E/∂t 2E = me∂2E/∂t2 Wave equation Solution: Any function F(r ± vt)  Traveling waves me = 1/v2 In free space m0e0 = 1/c2, c = 3 x 108m/s

.. And let there be light!

Plane waves in free space 2E = me∂2E/∂t2 Wave equation One set of Solutions: Plane waves E(r,t) = E0ejwt-jb . r Exponent derivatives are products! ∂/∂t  jw   -jb b = mew = w/v Substituting in wave equation:

Plane waves in free space .B = 0  x B = me∂E/∂t .E = 0  x E = - ∂B/∂t Plane w: E(r,t) = E0ejwt-jb . r  -jb /t  jw

Characteristic Impedance ! Plane waves in free space b.E = 0 b x E = jwB b.B = 0 b x B = - wmeE .B = 0  x B = me∂E/∂t Characteristic Impedance ! (b  E) (B  E  b) B0 = bE0/w = meE0 H0 = B0/m = E0/Z0 Why Impedance? ∫H0.dl = ∫E0.dl/Z0 I = V/Z0 Z0 = √(m0/e0) = 377 W

.. Nature of plane waves Oscillating E and B fields E, B and b form a right-handed triad

E curls because it changes along x Ey = E0e-jbx ( x E)z = -∂Ey/∂x = jbEy E rotates in the x-y plane to switch sign, so its curl is along the z axis, perpendicular to propagation y x

Not necessary in bound space with surface charges and currents z y x y z E must have no tangential component along metal plates… So choose one set of plates to be perpendicular to Along other direction, keep E parallel, but make E drop to zero as they reach the plate Since this gives an in-plane ‘rotation’ of E, with ∂Ey/ ∂ z non-zero, this implies curl(E) and thus B now has a component ALONG x, the propagation direction !! These are TE modes

TE vs TM modes http://sub.allaboutcircuits.com/images/02407.png

Frequency selectivity z y http://www.tpub.com/ Variation of E determined by k components along y and z Wavelengths must be set by dimensions (e.g. maximum wavelength = 2 x lateral width) This imposes cut-off frequency for various modes  allows selectivity of frequency e.g. TE10: w10 = pa/L is the cut-off In general, kx = √(w2-wmn2)/c, with wmn/c = √[(mp/a) 2 + (np/b) 2], m,n: integers w w20, w02 w11 w10, w01 b