1. 1 Electromagnetic waves Hecht, Chapter 2 Monday October 21, 2002

2. 2 Electromagnetic waves Consider propagation in a homogeneous medium (no absorption) characterized by a dielectric constant  o = permittivity of free space

3. 3 Electromagnetic waves Maxwell’s equations are, in a region of no free charges, Gauss’ law – electric field from a charge distribution No magnetic monopoles Electromagnetic induction (time varying magnetic field producing an electric field) Magnetic fields being induced By currents and a time-varying electric fields µ o = permeability of free space (medium is diamagnetic)

4. 4 Electromagnetic waves or, For the electric field E, i.e. wave equation with v 2 = 1/µ o 

5. 5 Electromagnetic waves Similarly for the magnetic field i.e. wave equation with v 2 = 1/µ o  In free space,  =   o =  o (  = 1) c = 3.0 X 10 8 m/s

6. 6 Electromagnetic waves In a dielectric medium,  = n 2 and  =   o = n 2  o

7. 7 Electromagnetic waves: Phase relations The solutions to the wave equations, can be plane waves,

8. 8 Electromagnetic waves: Phase relations Using plane wave solutions one finds that, Consequently i.e. B is perpendicular to the Plane formed by k and E !!

9. 9 Electromagnetic waves: Phase relations Since also k  E and k  B ( transverse wave) Thus, k, E and B are mutually perpendicular vectors Moreover,

10. 10 Electromagnetic waves: Phase relations Thus E and B are in phase since, requires that E B k

11. 11 Irradiance (energy per unit volume) Energy density stored in an electric field Energy density stored in a magnetic field

12. 12 Energy density Now if E = E o sin(ωt+φ) and ω is very large We will see only a time average of E

13. 13 Intensity or Irradiance In free space, wave propagates with speed c c Δt A In time Δt, all energy in this volume passes through A. Thus, the total energy passing through A is,

14. 14 Intensity or Irradiance Power passing through A is, Define: Intensity or Irradiance as the power per unit area

15. 15 Intensity in a dielectric medium In a dielectric medium, Consequently, the irradiance or intensity is,

16. 16 Poynting vector Define

17. 17 Poynting vector For an isotropic media energy flows in the direction of propagation, so both the magnitude and direction of this flow is given by, The corresponding intensity or irradiance is then,

18. 18 Example: Lasers Spot diameterIntensity (W/m 2 )Electric field magnitude (V/m) 2 mm1.6 X 10 3 1.1 X 10 3 20 µm (e.g. focus by lens of eye) 1.6 X 10 7 1.1 X 10 5 2 µm1.6 X 10 9 1.1 X 10 6 Laser Power = 5mW nb. Colossal dielectric constant material CaCu 3 Ti 4 O 12,  = 10,000 at 300K Subramanian et al. J. Solid State Chem. 151, 323 (2000) Near breakdown voltage in water Same as sunlight at earth  o = 8.854 X 10 -12 CV -1 m -1 (SI units)