In this problem, we will study the interference of light passing through thin vertical slits in an opaque screen. Alternatively, you can think of the interfering circular wave patterns in a ripple tank filled with water.

Remember that the sources are along the edge X=0, the
screen is along X=**X Screen**, and measures intensity or
amplitude along constant Z, for varying Y.

Measure the distance in Y from the central maximum to the first
(You can change the screen size to around 30, and then
zoom in with the mouse. You
should be able to measure it to at least three significant
figures.) **Write the answer in your lab write-up.**

Give the formula which tells where you expect the waves from the two
sources to interfere constructively, in terms of
**d**, **X Screen**, and **Lambda**.
**Include this formula in your write-up.**
Using the small angle approximation, evaluate the distance you theoretically
expect between the central peak and the first side peak, and
**compare with the distance you measured.**

Now reset the **Screen size** to 300 m. Notice
that the peaks are not equally
spaced (unlike the prediction of the small-angle theoretical prediction).
Find the position of the seventh peak from the center, to three significant
figures or more. **Include it in your lab write-up.**
Now calculate the theoretical prediction, *with and without* making the
small angle approximation. **Compare the three in your write-up.**
Is the varying distance between the peaks due to large values of the
angle?

Vary **d** and **Lambda**. Convince yourself
that you understand, whether the distance between peaks increases or decreases
when you increase these parameters. (It's important that you have a gut
feeling for diffraction: formulas aren't enough.)
**Write down whether the distance between peaks on the screen
varies proportionately or inversely with d and with Lambda.**

Vary the screen position **X Screen** inward and outward
by a factor of ten. How much does the intensity at the peak Y=0 change?
**Write down your measurements. Give a formula with your guess
as to how the intensity varies with distance.**

We can understand this in a simple way. If energy is conserved,
and if it doesn't keep building up in a region, then the average flow into
the region must equal the average flow out. Consider the region between
radius R_{1} and radius R_{2}. The average energy per
second passing through a radius R must equal that passing through R'.
Since the intensity times the circumference is the energy per second,
the intensity must go as one over the circumference. **Compare
this theoretical prediction with your measurements.**
A similar argument works for the intensity from a light-bulb or from
the Sun, except that there the energy is spread over a sphere instead
of a circle. **With what exponent of R will the intensity decay
for the Sun?**

Finally, as an optional problem, measure the decay of intensity as a function
of Y for a fixed screen position, and compare with the decay expected
from the increasing distance of the screen from the source. What is
the formula giving the decay as a function of **theta**?

If **d**=6, and the screen is set at a distance 8m away,
how far is the point Y=3 from the upper source?
How far is it from the lower source? (Use the Pythagorean theorem
for 3-4-5 triangles.) For what largest value of **Lambda**
will the waves from these two sources interfere constructively?
What's the largest value of **Lambda** for destructive
interference at Y=3? **Include these predicted
values in your write-up.**

Set **Lambda**=4, and locate the point Y=3, X=8.
Does it have destructive or constructive interference?
(Remember, destructive interference means the amplitude of the
wave is zero, which yields grey on the wave-tank.) Vary the phase,
and make sure the color stays grey as the waves fly past.

You can get a more accurate reading using the screen.
Move the screen to **X Screen**=8 m, and shrink the screen
size to 10m. Vary **Lambda**, and test if your predictions
for constructive and destructive interference are correct!

It's also easy to figure out where the interference is destructive
near **X**=0. If **Lambda**=4, at which points
along the line between the two sources will they interfere destructively?
**Write your predictions in your write-up.** Check using
the screen and using the wave-tank, and note to how many decimals they
agree with your prediction.

We can visualize the curves of destructive interference. Use the
**configure...** button to make the wave-tank size
**L**=30, keeping **Lambda**=4 and
**d**=6. You can see that the grey line showing the
interference minimum aims to the midpoint between the two sources,
but that it curves away near X=0. Put the screen back at
**X Screen**=100, and compare
**d sin(theta)** with **Lambda/2** at the
minimum. How well do they agree?

We should note that, at large angles **theta**, the effects
of increasing distance from the screen discussed in problem H1b will
distort the positions of the maxima of the interference patterns away
from those predicted by constructive interference. (The maxima will
shift in a bit, to get closer to the source!) At large distances
the entire pattern is multiplied by **cos(theta)**:
this shifts the maxima but doesn't change the zeros. Thus
**d sin(theta) = (m+1/2) Lambda** is correct for the
zeros even at large theta, but
**d sin(theta) = m Lambda**, the formula for constructive
interference, gives the positions of the maxima perfectly only for small
**theta**.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).