Transcript (pptx)

Two-Source
Constructive and Destructive
Interference Conditions
Crests
Troughs
P
S1
Crests
Troughs
S2
P
S1
l1
Crests
l2
S2
Troughs
P
S1
l1
Crests
Troughs
l2
Path length difference:
l = l2-l1
Travel time difference:
S2
t = t2- t1
t = (l2- l1)/v
P
S1
l1
Crests
l2
S2
Troughs
P
S1
l1
Crests
l2
S2
Troughs
P
l1
S1
h1 g1 f1
S2
h2
g2
f2
e1 d1
Crests
c1 b1 a1
l2
e2
d2
c2
b2
a2
Troughs
P
Q1
S1
h1 g1 f1
S2
h2
g2
f2
e1 d1
e2
d2
c1 b1 a1
Crests
Troughs
c2
b2
a2
(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 crest
or 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 ?
P
Q2
l1
S1
h1 g1 f1
e1 d1
Crests
c1 b1 a1
l2
Troughs
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
S2
h2
g2
f2
e2
d2
c2
b2
a2
(B) Assume P is positioned so that l2 and l1
are equal:
l2=l1
How 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 crests
and/or troughs, from either source, which arrive
simultaneously at P.
(D) Is there constructive or destructive
interference at P? Or neither? Explain!
P
Q3
l1
S1
h1 g1 f1
e1 d1
Crests
c1 b1 a1
l2
Troughs
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
S2
h2
g2
f2
e2
d2
c2
b2
a2
(B) Assume P is positioned so that l2 exceeds l1
by one wavelength, λ:
l2=l1 + λ
How 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 crests
and/or troughs, from either source, which arrive
simultaneously at P.
(D) Is there constructive or destructive
interference at P? Or neither? Explain!
P
Q4
l1
S1
h1 g1 f1
e1 d1
Crests
c1 b1 a1
l2
Troughs
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
S2
h2
g2
f2
e2
d2
c2
b2
a2
(B) Assume P is positioned so that l2 exceeds l1
by two wavelengths, 2λ:
l2=l1 + 2λ
How 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 crests
and/or troughs, from either source, which arrive
simultaneously at P.
(D) Is there constructive or destructive
interference at P? Or neither? Explain!
P
Q5
l1
S1
h1 g1 f1
e1 d1
Crests
c1 b1 a1
l2
Troughs
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
S2
h2
g2
f2
e2
d2
c2
b2
a2
(B) Assume P is positioned so that l2 is shorter
than l1 by two wavelengths, 2λ:
l2=l1 - 2λ
How 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 crests
and/or troughs, from either source, which arrive
simultaneously at P.
(D) Is there constructive or destructive
interference at P? Or neither? Explain!
P
Q6
l1
S1
h1 g1 f1
e1 d1
Crests
c1 b1 a1
l2
Troughs
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
S2
h2
g2
f2
e2
d2
c2
b2
a2
(B) Assume P is positioned so that l2 exceeds l1
by one half-wavelengths, λ/2:
l2=l1 + λ/2
How 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 crests
and/or troughs, from either source, which arrive
simultaneously at P.
(D) Is there constructive or destructive
interference at P? Or neither? Explain!
P
Q7
l1
S1
h1 g1 f1
e1 d1
Crests
c1 b1 a1
l2
Troughs
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
S2
h2
g2
f2
e2
d2
c2
b2
a2
(B) Assume P is positioned so that l2 is shorter
than l1 by three half-wavelengths, 3λ/2:
l2=l1 - 3λ/2
How 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 crests
and/or troughs, from either source, which arrive
simultaneously at P.
(D) Is there constructive or destructive
interference at P? Or neither? Explain!
P
Q8
l1
S1
h1 g1 f1
e1 d1
Crests
c1 b1 a1
l2
Troughs
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
S2
h2
g2
f2
e2
d2
c2
b2
a2
(B) Assume P is positioned so that l2 is shorter
then l1 by one quarter-wavelength, λ/4:
l2=l1 - λ/4
How 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 crests
and/or troughs, from either source, which arrive
simultaneously at P.
(D) Is there constructive or destructive
interference at P? Or neither? Explain!
P
Q9
l1
S1
h1 g1 f1
e1 d1
Crests
c1 b1 a1
l2
Troughs
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
S2
h2
g2
f2
e2
d2
c2
b2
a2
(B) Assume P is positioned so that l2 is exceeds l1
by two third-wavelengths, 2λ/3:
l2=l1 + 2λ/3
How 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 crests
and/or troughs, from either source, which arrive
simultaneously at P.
(D) Is there constructive or destructive
interference at P? Or neither? Explain!
P
Q10
l1
S1
h1 g1 f1
e1 d1
Crests
c1 b1 a1
Troughs
l2
Summarize your results for constructive and
destructive interference at P in terms of two
simple mathematical conditions for the
Path length difference:
S2
h2
g2
f2
e2
d2
c2
b2
a2
l = l2-l1
and, equivalently, for the
Travel time difference:
t = t2- t1
t = (l2- l1)/v
Constructive Interference
=Intensity Maximum:
Destructive Interference
=Intensity Minimum:
Path length difference:
Path length difference:
l = l2-l1= m λ
Travel time difference:
t = t2- t1 = m T
m= 0, +1, -1, +2, -2, …
l = l2-l1= (m+1/2) λ
Travel time difference:
t = t2- t1 = (m+1/2) T
(m+1/2) = +1/2, -1/2, +3/2, -3/2, …
where the period T is:
T= λ/v
Interference
Pathlength Geometry
P
Q11.1
l1
S1
l2
d
S2
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!
P
Q11.2
S1
l1
l2
d
S2
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 ?
P
Q11.3
S1
l1
y
l2
d
L
O
S2
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
P
Q11.3 (contd.)
l1
S1
y
d/2
d
l2
L
d/2
S2
(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!
O
P
Q12.1
l1
S1
y
l2
d
Θ
L
O
S2
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.
P
Q12.2
l1
S1
y
l2
d
Θ
L
S2
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?
O
Multi-Slit
Constructive Interference
Pathlength Geometry
Intensity Plots
Multi-Slit (N-Slit) Interference and Diffraction Grating (N>>1)
Notation:
Δl = lk+1 – lk
≈same for k=1, 2, …,N-1.
O
y
P
Δl = m λ
with m integer
Again, by geometry:
Δl ≅ d sin(Θ)
assuming L>> Nd; and
Δl
2 Δl
tan(Θ) = y/L
3 Δl
Δl
Δl
Maximally constructive
interference occurs when
Principal
Maxima:
sin(Θ) = m λ/d
with m integer
Secondary
Maxima
Q13
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 a
quintuple-slit (N=5) instead of a diffraction grating?