Predicting Changes in Interference Patterns using Laser Interference Equations

[fatcats]

Homework Statement

Imagine that you are conducting an activity with a laser to create an interference pattern. Use the appropriate equations to predict two ways (other than the way described in the following example) to change the interference pattern in order to have closer fringes. Explain your predictions.
Their example : Δx = Lλ/d
the distance between the fringes (Δx) is proportional to the wavelength (since they are both numerators). This means that increasing one will increase the other. Therefore, decreasing the wavelength will decrease the distance between the fringes. (Hint: You can use the "Thomas Young's Double-slit Experiment" simulation to verify predictions).

Homework Equations

Δx = Lλ/d
(n-1/2)λ=dx/L
(n-1/2)λ=d sintheta n

The Attempt at a Solution

x = mLλ/d
x is directly proportional to the distance from the slits to the screen. Increasing distance between the slits and the projection area will make the bands closer.

Where I am struggling is a second way to show how to alter d, without using their example. This is what I tried:

(n-1/2λ)=d sintheta n
Please see my attached work document for my work, it is very legible.
Basically I rearranged (n-1/2)λ=dx/L for D, and substituted for d (n-1/2)λ=d sintheta n, then rearranged for x. I got:

L(sintheta) = x
How does that make sense? If the angle is 90 then the distance of the fringes will be equal to the distance from the screen to the fringes? What am I doing wrong?

 

[fatcats]

Updating my second equation because I realized it was out of order and didn't make sense anyway
I still got the same answer though.

(n-1/2λ)=d sintheta n
Basically I rearranged (n-1/2)λ=dx/L for D, and substituted for d (n-1/2)λ=d sintheta n, then rearranged for x. I got:
L(sintheta) = x

 

[Alloymouse]

Perhaps you can consider distance between both slits as a variable; I think equation 3 is the one describing that relationship. the "d" in equation 3 is not distance to screen, but rather distance between both slits if I am interpreting this correctly.

 

[fatcats]

oh my goodness I am insane and I have no idea how I missed the solution right in front of my nose. thank you for your reply. I had compared d already.

so I had the right answer earlier and for some reason late at night chose to delete it and redo it... I deleted my statement about the L variable, the distance from the slits to the screen! sorry for posting this and thanks for your time.