 and  (and therefore v) are properties of the material in which the wave is traveling [how much the material affects E and B]. In vacuum, the speed of EM waves is v = (  0  0 ) -1/2  c = x 10 8 m/s  3 x 10 8 m/s (and air speed is very close to vacuum speed!) Radio waves, microwaves, infrared waves, visible light, ultraviolet rays, x-rays, and gamma rays are all electromagnetic waves of different frequencies (f) and wavelengths (in vacuum/air): = c/f. v E y = E y0 sin [2π(x  vt)/ ], B z = B z0 sin [2π(x  vt)/ ] B zo =  E y0 /v Electromagnetic Waves is the wavelength of the wave and its frequency is f = v/. The intensity (energy/time/area) of the wave  E y0 2. v = (  ) -1/2

c = x 10 8 m/s  3 x 10 8 m/s earth D If there was a tunnel through the earth, it would take an electromagnetic wave  t = D/c = 1.3 x 10 4 m / (3 x 10 8 m/s) = 4.3 x s to pass through the earth earth moon R em Earth to moon:  t = R em /c = 3.8 x 10 8 m / (3 x 10 8 m/s) = 1.3 s sun earth R se Sun to earth:  t = R se /c = 1.5 x m / (3 x 10 8 m/s) = 500 s  8.3 minutes

Since the speed of electromagnetic waves is so large (seems instantaneous in everyday applications), it cannot be measured using conventional techniques (e.g. meter stick and stop watch). Fizeau (1849) had first successful terrestrial measurement: Time for light to travel between wheel and mirror and back  t = 2d/c. If the wheel has N spokes, the light will be blocked by the next spoke (and not reach the detector) if it has turned  =2  /2N in time  t, i.e. if  = 2  F =  /  t = (  /N)/ (2d/c)  c = 4dNF F light eye (detector) e.g. Fizeau: N = 720, d = 9500 m (~ 6 miles)  F = 11/sec

The boundaries between different types of waves are “fuzzy” – the different names mostly refer to how the waves are generated. Different colors of light correspond to different wavelengths/frequencies. White light is a combination of visible light of all frequencies. Visible light (what we can see) only occupies less than one octave (factor of two) in frequency and wavelength, but this is an octave where there is a lot of sunlight. Unless otherwise stated, wavelength means wavelength in vacuum: = c/f Electromagnetic Spectrum

Consider two electromagnetic waves from the same source with the same wavelength that travel different distances (  x) before coming back together: E 1 = E 0 sin[2π(x -  x/2  vt)/ ] and E 2 = E 0 sin[2  (x +  x/2  vt)/ ] E 1 = E 0 sin(  -  /2) and E 2 = E 0 sin (  +  /2) where  = 2   x /. After they come together, the total electric field is E = E 1 + E 2 = E 0 [sin(  -  /2) + sin(  +  /2)] E = [2E 0 cos(  /2)]sin(  ) = [2E 0 cos(  /2)] sin(2  (x  vt)/ ] Intensity I  4E 0 2 cos 2 (  /2) = 4 I 0 cos 2 (  /2) = 4 I 0 cos 2 (  x/ ) constructive interference destructive interference I 0 = intensity of each wave alone

An interesting instrument which uses interference: Michelson Interferometer (pp )  x = 2(L 1 – L 2 ) If |  x| = N  constructive interference If |  x| = (N+1/2)  destructive int. (or detector)

Rings (alternating constructive and destructive interference) appear because rays of light not going exactly to center travel different distances. If use white light, different colors (different wavelengths) have constructive interference at different places.

E = E 0 sin [2π(x  vt)/ ] Since E and B only depend on x and t, they are the same for all y and z  surfaces of constant phase (wave fronts) are planes: this is called a “plane wave”. Rays The electromagnetic wave is traveling in the direction perpendicular to the wavefronts along imaginary lines called rays.

A point source creates a spherical wave, where the wave fronts are spherical surfaces and the rays are radii. If you block most of a spherical or cylindrical wave, the part that gets through is approximately a plane wave. A line source (e.g. antenna) creates a cylindrical wave, where the wave fronts are cylindrical surfaces and the rays are radii.

Consider a hole (or obstacle) or size d. If >> d or even  d, a wave will spread out when passing through the hole (or around an obstacle). This effect is called diffraction and the study of the optics when one must be concerned with large is called physical optics. It will be studied in PHY232.

If << d, the rays will continue traveling in straight lines, until they bounce (reflect) off a surface or change direction (refract) when passing through a second material. Therefore, in this small limit (geometric optics), the ray approximation is good and light passing through holes or around obstacles, will cast sharp spots or shadows.

1905: Einstein pointed out that electromagnetic waves are not “continuous” but that their energy travels in bundles (now called photons); each photon has energy E = hf where f is the frequency of the wave and Planck’s constant h = 6.63 x J·s. However, if the power emitted or absorbed is much larger than the energy of one photon/period, P >> (hf)/(1/f) = hf 2, photons will overlap and the wave can usually be treated as a continuous, classical wave -- “quantum” effects can be ignored. For visible light, with f ~ 5 x Hz, this classical regime corresponds to P >> 0.2 mW. Quantum effects will be studied in PHY 361.

Electromagnetic waves slow down in materials, i.e. have speed v < c. One can view this in the following way. When a wave of frequency f hits molecule A, it is absorbed, causing the charges in A to vibrate at frequency f. A then emits an EM wave at the same frequency, which is then absorbed by B, and the pattern continues. Although the EM wave travels at speed c between A and B, the delay while A is absorbing and re-emitting the wave causes the average speed of the wave while passing through the material to be less than c. Note that the frequency of the wave doesn’t change when passing through the material, but its wavelength does change: material = v/f < c/f = vacuum : material / = v/c < 1. [Note: v/c will depend on the density of the material, the types of atoms, and the frequency.] Unless otherwise stated, wavelength means wavelength in vacuum:  vacuum.