Question

Question:Visible light passes through a diffraction grating that has 900 slits / cm, and the interference pattern is observed on a screen that is2.50 mfrom the grating. (a) Is the angular position of the first-order spectrum small enough for$\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }\mathbf{\approx }\mathit{\theta }$to be a good approximation? (b) In the first-order spectrum, the maxima for two different wavelengths are separated on the screen by 3.00 mm. What is the difference in these wavelengths?

Short answer

1. Yes, the angular position of the first-order spectrum small enough.
2. The difference in these wavelengths is 13.3 mm

Step-by-step solution

Determine if the angular position of the first-order spectrum is small enough

The angular position is given by

The first-order spectrum is for m = 1

So, the first-order angular position in radians is

Therefore, the angular position of the first-order spectrum is small enough for $\theta \approx \theta$to be a good approximation.

Determine the difference between the two wavelengths

The position of the bright fringes on the screen

So, the position of the first-order spectrum of the first and second wavelengths are

Now, we know that both the bright fringes for m = 1, thus subtract the positions of fringes

Substitute the values

Thus, the difference between the two wavelengths is 13.3 mm.