1. Electromagnetic Waves

2. 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.

3. 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

4. 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

5. 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

6. .. And let there be light!

7. 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:

8. 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

9. 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

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

11. 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

12. Not necessary in bound space with surface charges and currentsz 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

13. TE vs TM modes

14. Frequency selectivityz y
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