Calculating the angle of a double-slit experiment

[MadMartigan]

Homework Statement

So, my physics professor has been behind all semester long and basically taught the entire light wave and optics chapters in a single day and explained absolutely nothing, hence massive confusion on the following problem:[In a double-slit experiment, the slit separation is 1.75 mm, and two coherent wavelengths of light, 425 nm and 510 nm, illuminate the slits. At what angle from the centerline on either side of the central maximum will a bright fringe from one pattern first coincide with a bright fringe from the other pattern?]

d (split separation) = 1.75 mm = 1.75E-3m
λ1 = 425 nm = 425E-9m
λ2 = 510 nm = 510E-9m
θ = ?

Homework Equations

ΔX = dsinθ
dsinθ = mλ

The Attempt at a Solution

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Now, I think it's a constructive interference since the question references the bright fringes, not that I think that necessarily matters.

I've tried working this a few different ways and I honestly don't know if I'm on the right track since my professor spent of all 5 minutes on this and just spat out the equations he gave us on the board.

My first thought was to just use ΔX = dsinθ and plug and chug using path length difference.

510nm-425nm = 1.75E-3m*sinθ

Which would result in a tiny angle of 2.78E-3° which seems super small and has me questioning if I've screwed up.

And if I use the 2nd equation and add the wavelengths and have m=1, I get an angle of 0.031°. Small, but larger.

Am I on the right track on either of these methods or am I missing a concept entirely because my professor didn't explain jack?

 

[Charles Link]

Homework helpers on the Physics Forums aren't supposed to just give out the answer, but I will try to steer you to the answer. $\\$ Bright spots occur in a two slit pattern when $m \lambda=d sin(\theta)$ where $m$ is an integer. Two bright spots will coincide if $m_1 \lambda_1=m_2 \lambda_2$ for $m_1$ and $m_2$ being integers. The idea is to find the smallest $m_1$ and $m_2$ that works besides $m_1=m_2=0$ which is the central maximum for both colors/wavelengths. Once you find that, then compute $\theta$ from the equation $m_1 \lambda=d sin(\theta)=m_2 \lambda_2$. Notice that your solution for $m_1$ and $m_2$ can be both negative or positive=the pattern is symmetric so that $\theta$ can be positive or negative. $\\$ Editing: Additional item: You don't need to have $m_1 \lambda_1=m_2 \lambda_2$ to 3 decimal places. If you can find $m_1$ and $m_2$ that satisfy this approximately, that should be a correct answer. Editing some more... In fact, now that I have computed the answer, this one is exact.

 

[Biker]

You can always find your topics on the internet, Optics is a really interesting topic once you get the hang of it. Any member will be happy to help you.
I think you should look at this link first, So you can understand where the equation came from and how you can use it:
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/slits.html

 

[Charles Link]

@MadMartigan I'm interested in your feedback=were you able to solve it? Please let us know what you came up with.

 

[MadMartigan]

Hahah. Haven't gotten to it. But I also checked with physics major friend and I think I understand the process now. We'll see when I go about answering.

I'll update with what I came up with after dinner.

 

[MadMartigan]

Okay, I'm not sure if my math skills/reasoning are garbage right now, but I got m1λ1=m2λ2 to equal 425 nm using m1=1 and m2=0.8333333333

This leads me back to getting that angle of .0139°.

unless I've gone stupid, that is the actual angle right?

 

[Charles Link]

$m_1$ and $m_2$ must be integers. .8333333 is not an integer. Try again=it's really kind of simple once you figure out the answer.

 

[Charles Link]

A hint that almost makes it too easy is what fraction is the number .833333333?

 

[MadMartigan]

Brain fart.

m1 should equal 6 when λ1= 425nm. That results in 2550. As a result when λ2= 510nm, m2 should equal 5, resulting in 2550.

 

[Charles Link]

Brain fart.

m1 should equal 6 when λ1= 425nm. That results in 2550. As a result when λ2= 510nm, m2 should equal 5, resulting in 2550.

Good. Now all you need to do is finish it up by calculating $\theta$, remembering that these integers can also both be negative as well. The pattern of bright spots for the two-slit interference pattern is symmetric about m=0 ==>> i.e. symmetric about $\theta=0$.

 

[MadMartigan]

angle is approx .088 right?

 

[Charles Link]

angle is approx .088 right?

I assume the units for your angle is degrees (and not radians). I get a slightly different answer. Scratch that=I remebered the slit spacing incorrectly. With d=1.75 mm, I think your answer is correct=in degrees.

 

[MadMartigan]

yes, that was in degrees.

 

[Charles Link]

yes, that was in degrees.

I get .084, but I'm doing it without a calculator. Please check your arithmetic=I think I calculated it correctly.

 

[MadMartigan]

Might have had a typo. You're closer. It should come out to 0.0834