Maximum Intensity Between Dark Fringes, Diffraction

[JackFlash]

Homework Statement

Laser light of wavelength 632.8 nm falls normally on a slit that is 0.0210 mm wide. The transmitted light is viewed on a distant screen where the intensity at the center of the central bright fringe is 8.50 W/m².

a) Find the maximum number of totally dark fringes on the screen, assuming the screen is large enough to show them all.

b) At what angle does the dark fringe that is most distant from the center occur?

c) What is the maximum intensity of the bright fringe that occurs immediately before the dark fringe in part (b)? Approximate the angle at which this fringe occurs by assuming it is midway between the angles to the dark fringes on either side of it.

Homework Equations

sinθ = mλ/a
Intensity = I₀($\frac{sin(πa*sin(θ)/λ)}{πa*sin(θ)/λ}$)²
Intensity = $\frac{Io}{((m+.5)^2 + π^2)}$

where
m = a number based on the fringe
λ = wavelength
a = distance between the slits
π = a delicious dessert of varying flavors, and 3.14159

The Attempt at a Solution

I found the first two solutions (66 and 83.9° respectively) with some ease. The last one is what is grinding my gears.

I've calculated the angle between the two dark fringes at the end, m=33 and m=32, and got 83.9335° and 74.63699° respectively. I got the angle between them, 79.29°, by taking the average. I plug it into the first equation for intensity and I get a wrong answer. I try using m=32.5, which would be the bright fringe the question refers to, in the second equation for intensity and still no luck. It might not seem like I've tried at all on this question, but I have run through several different combinations (using different values for m to get an angle, then plugging into the second equation or just using m in the third). The solution is 7.21*10^-4, which is about .6*10^-4 less than all the solutions I've gotten.

I've a suspicion I may be using the wrong m (32.5) or I'm doing something else odd. Any help is much appreciated.

 

[ehild]

When you get π*(a/λ) *sin(θ) it is in radians, not degrees. Check if you use "RAD" when calculating sin(π*(a/λ) *sin(θ)) in the numerator of intensity. .

ehild

 

[JackFlash]

Ah yes. Radians. That was the issue.
That always seems the case. Radians is a reoccuring enemy of mine. Thanks.

 

[ehild]

You are welcome and take care. Do not let you beat by the radians.

ehild

 

[asteriatic]

Dear student,

Thank you for your question. Based on the information provided, it seems like you have correctly solved for parts (a) and (b). For part (c), you are on the right track. However, there are a few things to keep in mind.

First, for the maximum intensity at the bright fringe, you need to use the equation for intensity in terms of m, rather than the one in terms of θ. This is because the question is asking for the intensity at a specific fringe (m = 32.5), not at a specific angle (θ = 79.29°). So you need to use the equation:

Intensity = (I0/((m+0.5)^2 + π^2))

Second, when using this equation, you need to take into account the fact that this is the intensity at a specific fringe (m = 32.5), but the equation is for the intensity at any fringe. So you need to use the value of m that corresponds to the fringe just before the one you are interested in (m = 32 in this case). This is because the intensity at a given fringe is affected by the fringes on either side of it.

So the correct equation to use would be:

Intensity = (I0/((m+0.5)^2 + π^2)) where m = 32

Now, using this equation, you can solve for the maximum intensity at the bright fringe immediately before the dark fringe at m = 32.5. This should give you the correct answer of 7.21*10^-4.

I hope this helps. Let me know if you have any further questions. Good luck with your studies!

Best,