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To my understanding,in multi slit interference, 2 equations are correct. $n*\lambda=d*sin(\theta)$ and $S=(\lambda*D)/d$ In the problem below, however, the 2 equations gave different solutions. I hope to understand why that is.

The problem is below: enter image description here

Using the first equation $n*\lambda=d*sin(\theta)$, the answer I got was $1.51$. Specifically:

  • When we solve the equation for $\theta$ with $700$ nm, we get $22$ degrees. When we solve with $400$ nm, we get $12$ degrees. We then take the tangent of the 2 and multiply both by $8.13$, then subtract them. This is how we get $1.51$ m.

Using the second equation $S=(\lambda*D)/d$, the answer I got was $1.31$. Specifically:

  • I subbed the given values and $400$ nm into the equation, then did the same for $700$ nm. Then I multiplied the 2 results by $2$ (to get the distance from center to second principle maxima), then subtracted both.

So why would the 2 equations give different solutions?

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I suspect that your second formula has $S$ as the fringe separation on a screen that is $D$ away from slits which are $d$ apart.
That being so the second formula can be derived from the first formula on the assumption that the angle $\theta$ is small such that $\tan \theta \approx \sin \theta \approx\theta$.
In your problem the approximation is not a very good one and that results in a difference between your two answers.

So you have something like $n \lambda = d \sin \theta_{\rm n} \Rightarrow n \lambda \approx d \,\dfrac {s_{\rm n}}{D}$ and $(n+1) \lambda = d \sin \theta_{\rm n+1} \Rightarrow (n+1) \lambda \approx d \,\dfrac {s_{\rm n+1}}{D}$.

Subtraction and rearrangement gives $s_{\rm n+1} - s_{\rm n} = S \approx \dfrac{\lambda D}{d}$

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