Light Diffraction physics problem

[Tkdmaster]

Homework Statement

You have project in your physics class to build a diffraction grating. The key component is that you need to disperse visible light over a 30-degress spread at the first order. Visible light has a range of 400-700nm. How many lines per mm do you need for your diffraction grating?

Relevant Equations

mλ =dsin(Θ)

Been trying to figure this problem out for a couple hours now, if i use 400 nm into the equation it comes out to 800: 400=sin(30). If i do 700 nm it comes out to 1400: 700=sin(30).

I feel like i’m doing somethingwrong with the problem but i can’t figure out what.

 

[jtbell]

you need to disperse visible light over a 30-degress spread at the first order

This means that the difference between the diffraction angles for 400 nm and 700 nm needs to be 30°, not that either of those angles needs to be 30°. For example (just making up numbers here!) you might end up with something like θ = 22° for λ = 400 nm and θ = 52° for λ = 700 nm, for a spread of 52° - 22° = 30°.

 

[Tkdmaster]

The thing is though, i don't know how to determine what those angles would be.

 

[haruspex]

The thing is though, i don't know how to determine what those angles would be.

Suppose you choose a spacing d. Through what angles will the extremes of visible light be diffracted (to the first order)?

 

[jtbell]

Call the extreme wavelengths $\lambda_1$ and $\lambda_2$. They diffract through angles $\theta_1$ and $\theta_2$. Can you write an equation that gives the spread in angles, $\theta_2 - \theta_1$, in terms of $\lambda_1$, $\lambda_2$, and $d$? That is, $\theta_2 - \theta_1 = \cdots$

 

[Tkdmaster]

Like 700-400=dsin(theta2-theta1)?

 

[jtbell]

No, you can't do it by simply substituting $\theta_2 - \theta_1$ for $\theta$, and $\lambda_2 - \lambda_1$ for $\lambda$.

Start with your diffraction equation $\lambda = d \sin \theta$. Solve it (rearrange it) to get a new equation $\theta = \cdots$.

Now, using your numbers, you actually have two of these equations: $\theta_1 = \cdots$ and $\theta_2 = \cdots$. Subtract one from the other to get $\theta_2 - \theta_1 = \cdots$ with $d$ somewhere on the right-hand side.