Two-Source Constructive and Destructive Interference Conditions

Crests

Troughs

S1

P

S2

Crests

Troughs

S1

P

l1

l2

S2

Crests

Troughs

S1

P

l1

l2

S2

l = l2-l1

Path length difference:

t = t2- t1

Travel time difference:

t = (l2- l1)/v

Crests

Troughs

S1

P

l1

l2

S2

Crests

Troughs

S1

P

l1

S2

l2

Crests

Troughs

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

S1

P

S2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

(A) How long ago, before this snapshot was taken, did

a1, b1, c1, d1, e1, f1, g1, h1 leave source S1 ?How long ago did

a2, b2, c2, d2, e2, f2, g2, h2 leave source S2 ?Express all your results, here and in the following in terms of the period of oscillation, T !Tabulate the results!

Reminder: It takes 1 period for a crestor trough to travel 1 wavelength

(B) Tabulate all pairs of crests and/or troughs which left their resp. sources simultaneously.

(C) Do the results in (A) depend on l1 or l2 ?

Q1

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

(A) How long after a1,willb1, c1, d1, e1, f1, g1, h1

arrive at the detector, P? Tabulate!

(B) Assume P is positioned so that l2 and l1 are equal: l2=l1

How long after a1 willa2, b2, c2, d2, e2, f2, g2, h2

arrive at the detector, P? Tabulate!

(C) Using (A) and (B), tabulate all pairs of crestsand/or troughs, from either source, which arrive simultaneously at P.

(D) Is there constructive or destructive interference at P? Or neither? Explain!

Q2

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

(A) How long after a1,willb1, c1, d1, e1, f1, g1, h1

arrive at the detector, P? Tabulate!

(B) Assume P is positioned so that l2 exceeds l1 by one wavelength, λ: l2=l1 + λHow long after a1 will

a2, b2, c2, d2, e2, f2, g2, h2

arrive at the detector, P? Tabulate!

(C) Using (A) and (B), tabulate all pairs of crestsand/or troughs, from either source, which arrive simultaneously at P.

(D) Is there constructive or destructive interference at P? Or neither? Explain!

Q3

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

(A) How long after a1,willb1, c1, d1, e1, f1, g1, h1

arrive at the detector, P? Tabulate!

(B) Assume P is positioned so that l2 exceeds l1 by two wavelengths, 2λ: l2=l1 + 2λHow long after a1 will

a2, b2, c2, d2, e2, f2, g2, h2

arrive at the detector, P? Tabulate!

(C) Using (A) and (B), tabulate all pairs of crestsand/or troughs, from either source, which arrive simultaneously at P.

(D) Is there constructive or destructive interference at P? Or neither? Explain!

Q4

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

(A) How long after a1,willb1, c1, d1, e1, f1, g1, h1

arrive at the detector, P? Tabulate!

(B) Assume P is positioned so that l2 is shorter than l1 by two wavelengths, 2λ: l2=l1 - 2λHow long after (+) or before (-) a1 will

a2, b2, c2, d2, e2, f2, g2, h2

arrive at the detector, P? Tabulate!

(C) Using (A) and (B), tabulate all pairs of crestsand/or troughs, from either source, which arrive simultaneously at P.

(D) Is there constructive or destructive interference at P? Or neither? Explain!

Q5

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

(A) How long after a1,willb1, c1, d1, e1, f1, g1, h1

arrive at the detector, P? Tabulate!

(B) Assume P is positioned so that l2 exceeds l1 by one half-wavelengths, λ/2: l2=l1 + λ/2How long after (+) or before (-) a1 will

a2, b2, c2, d2, e2, f2, g2, h2

arrive at the detector, P? Tabulate!

(C) Using (A) and (B), tabulate all pairs of crestsand/or troughs, from either source, which arrive simultaneously at P.

(D) Is there constructive or destructive interference at P? Or neither? Explain!

Q6

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

(A) How long after a1,willb1, c1, d1, e1, f1, g1, h1

arrive at the detector, P? Tabulate!

(B) Assume P is positioned so that l2 is shorter than l1 by three half-wavelengths, 3λ/2: l2=l1 - 3λ/2How long after (+) or before (-) a1 will

a2, b2, c2, d2, e2, f2, g2, h2

arrive at the detector, P? Tabulate!

(C) Using (A) and (B), tabulate all pairs of crestsand/or troughs, from either source, which arrive simultaneously at P.

(D) Is there constructive or destructive interference at P? Or neither? Explain!

Q7

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

(A) How long after a1,willb1, c1, d1, e1, f1, g1, h1

arrive at the detector, P? Tabulate!

(B) Assume P is positioned so that l2 is shorterthen l1 by one quarter-wavelength, λ/4: l2=l1 - λ/4How long after (+) or before (-) a1 will

a2, b2, c2, d2, e2, f2, g2, h2

arrive at the detector, P? Tabulate!

(C) Using (A) and (B), tabulate all pairs of crestsand/or troughs, from either source, which arrive simultaneously at P.

(D) Is there constructive or destructive interference at P? Or neither? Explain!

Q8

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

a2b2c2

d2e2f2 g2

h2

(A) How long after a1,willb1, c1, d1, e1, f1, g1, h1

arrive at the detector, P? Tabulate!

(B) Assume P is positioned so that l2 is exceeds l1 by two third-wavelengths, 2λ/3: l2=l1 + 2λ/3How long after (+) or before (-) a1 will

a2, b2, c2, d2, e2, f2, g2, h2

arrive at the detector, P? Tabulate!

(C) Using (A) and (B), tabulate all pairs of crestsand/or troughs, from either source, which arrive simultaneously at P.

(D) Is there constructive or destructive interference at P? Or neither? Explain!

Q9

S1

P

l1

S2

l2

Crests

Troughs

a1b1c1d1e1f1 g1h1

l = l2-l1

Path length difference:

t = t2- t1

Travel time difference:

t = (l2- l1)/v

a2b2c2

d2e2f2 g2

h2

Q10

Summarize your results for constructive and destructive interference at P in terms of two simple mathematical conditions for the

and, equivalently, for the

l = l2-l1= m λ

Constructive Interference=Intensity Maximum:

Path length difference:

t = t2- t1 = m TTravel time difference:

l = l2-l1= (m+1/2) λ

Destructive Interference=Intensity Minimum:

Path length difference:

t = t2- t1 = (m+1/2) TTravel time difference:

m= 0, +1, -1, +2, -2, … (m+1/2) = +1/2, -1/2, +3/2, -3/2, …

where the period T is: T = λ / v

Interference Pathlength Geometry

S1

P

l1

S2

l2

d

d = source-to-source spacing, l1 = distance from S1 to P, l2 = distance from S2 to P.

Suppose S1 and S2 are two small loudspeakers, placed 6.8m apart and you can move P to any location.

What is the largest possible absolute value of the path length difference, Δl =l2 – l1 .

Explain your reasoning!

Q11.1

S1

P

l1

S2

l2

d

Suppose the two small loudspeakers, S1 and S2, spaced 6.8m apart, oscillate in phase, sending out sound

waves of wavelength λ=2.2m. Constructive interference occurs at any location of P where Δl = m λ. Here

m can be any integer: 0, +1, -1, +2, -2, … ; and |m| is called the order of the interference maximum.

What is the largest possible order of interference, |m|, that can be observed, for any location of P ?

Q11.2

S1

P

l1

S2

l2

dL

O

y

Lengths and coordinates needed to describe the positioning of sources, S1 and S2, and detector, P:

d = source-to-source spacing, L = distance from observation screen to line of sources.

y = y-coordinate of P, with y-axis along the observation screen and origin O on midline between, S1 and S2

Q11.3

S1

P

l1

S2

l2

Q11.3 (contd.)

d

d/2

d/2L

O

y

(A) Derive exact equations for l1and l2, each expressed in terms of d, L, and the y-coordinate of P.

Hint: Pythagoras!

(B) From this, obtain an exact equation for the pathlength difference, Δl, in terms of d, L and y

(C) At home: Solve the equation from (B) for y, to express y in terms of of d, L, and Δl. Very difficult!

S1

P

l1

S2

l2

dL

O

y

The result in Q11.3 (B) is greatly simplified if d << L, by the so-called Fraunhofer approximation: Δl ≅ d sin Θ where tan Θ = y/L.Test this approximation against Q11.3 (B), for fixed d=5cm, fixed Θ=65deg, increasing values of L and y: Tabulate! Hint: Keep enough signif. digits! You’re subtracting 2 large numbers with a very small difference.

Q12.1

Θ

S1

P

l1

S2

l2

dL

O

y

S1 and S2, the two loudspeakers, spaced 6.8m apart, oscillate in phase, sending out sound waves of

wavelength λ=2.2m. The detector P is moved along the y-axis from y=-∞ to y=+∞, at L = 150m.

(A) Find the angles Θ and y-locations of all intensity maxima on the y-axis. How many are there?

(B) Find the angles Θ and y-locations of all intensity minima on the y-axis. How many are there?

Q12.2

Θ

Multi-Slit Constructive Interference

Pathlength GeometryIntensity Plots

Notation:

Δl = lk+1 – lk≈same for k=1, 2, …,N-1.

Maximally constructive interference occurs when

Δl = m λwith m integer

Again, by geometry:

Δl ≅ d sin(Θ)assuming L>> Nd; and

tan(Θ) = y/L

Multi-Slit (N-Slit) Interference and Diffraction Grating (N>>1)

P

O

y

2 Δl

Δl

Δl

Δl

3 Δl

2 Δl

3 Δl

Δl

Principal Maxima:sin(Θ) = m λ/dwith m integer

Secondary Maxima

A diffraction grating placed parallel to an observation screen, 40cm from the screen,Is illuminated at normal incidence by coherent, monochromatic light (a laser beam). Assume Fraunhofer conditions (L>> Nd) are satisfied.

(a) If the 1st order principal maximum is observed on the screen 30cm above the central maximum, how many principal maxima altogether, incl. central maximum, are observable?

(b) If the 2nd order principal maximum is observed on the screen 30cm above the central maximum, how many principal maxima altogether, incl. central maximum, are observable? Find the angles, Θ, and y-coordinates of all principal maxima on the screen: Tabulate!

(c) How would your answers change if the device had been a double-slit (N=2) or aquintuple-slit (N=5) instead of a diffraction grating?

Q13