1. John Cockburn (j.cockburn@... Room E15)
PHY 102: Waves & Quanta
Topic 6
Interference
John Cockburn Room E15)

2. Electromagnetic Waves
Interference of Sound Waves
Young’s double slit experiment
Intensity distribution for Young’s experiment

3. Electromagnetic Radiation
Visible light is an example of ELECTROMAGNETIC RADIATION:

4. Electromagnetic Waves
Existence predicted by James Clerk Maxwell (1865)
Consist of “crossed” time-varying electric and magnetic fields
Transverse wave, both electric and magnetic fields oscillate in a direction perpendicular to propagation direction
No medium is necessary: Electromagnetic waves can propagate through a vacuum
Constant speed of propagation through a vacuum: c ≈ 3 x 108 ms-1

5. Electromagnetic Waves

6. Electromagnetic Waves
It can be shown from MAXWELL’S EQUATIONS of Electromagnetism (See second year course) that the electric and magnetic fields obey the wave equations:
“standard” linear wave equation

7. Electromagnetic Waves
Where E0 and B0 are related by: E0 = cB0
INTENSITY of an EM wave  E02
NB. we will see later that EM radiation sometimes behaves like a stream of particles (Photons) rather than a wave………………

8. Speed of light in a material
Constant speed of propagation through a vacuum: c ≈ 3 x 108 ms-1
But, when travelling through a material, light “slows down”
n is the “refractive index” of the material.
Frequency of the radiation is constant, so from v = fλ, wavelength must decrease by a factor of 1/n.
(NB refractive index depends on the wavelength of the light)

10. Interference
First, consider case for sound waves, emitted by 2 loudspeakers:
Path difference =nλ
Constructive Interference
Path difference =(n+1/2)λ
Destructive Interference
(n = any integer, m = odd integer)

11. Interference

12. Interference For interference effects to be observed,
sources must emit at a single frequency
Sources must have the same phase OR have a FIXED phase difference between them. This is known as COHERENCE
Conditions apply to interference effects for both light and sound

13. Example calculation
For what frequencies does constructive/destructive interference occur at P?

14. Young’s Double Slit Experiment
Demonstrates wave nature of light
Each slit S1 and S2 acts as a separate source of coherent light (like the loudspeakers for sound waves)

15. Young’s Double Slit Experiment
Consider intensity distribution on screen as a function of  (angle measured from central axis of apparatus)……………………….
If light behaves as a conventional wave, then we expect high intensity (bright line) at a position on the screen for which r2-r1 = nλ
Expect zero intensity (dark line) at a position on the screen for which r2-r1=(n+1/2)λ

16. Young’s Double Slit Experiment
Assuming (justifiably) that R>>d, then lines r2 and r1 are approximately parallel, and path difference for the light from the 2 slits given by:

17. Young’s Double Slit Experiment
Constructive interference:
Destructive interference:

18. Young’s Double Slit Experiment
Y-position of bright fringe on screen: ym = Rtanm
Small , ie r1, r2 ≈ R, so tan ≈ sin
So, get bright fringe when:
(small  only)

19. Young’s Double Slit Experiment: Intensity Distribution
For some general point P, the 2 arriving waves will have a path difference which is some fraction of a wavelength.
This corresponds to a difference  in the phases of the electric field oscillations arriving at P:

20. Young’s Double Slit Experiment: Intensity Distribution
Total Electric field at point P:
Trig. Identity:
With  = (t + ), = t, get:

21. So, ETOT has an “oscillating” amplitude:
Since intensity is proportional to amplitude squared:
Or, since I0E02, and proportionality constant the same in both cases:

22. For the case where y<<R, sin ≈ y/R:

23. Young’s Double Slit Experiment: Intensity Distribution