Answer to Question #246732 in Electricity and Magnetism for mrsupersuper

Question

If possible, I would like explanations on how to arrive at these answers rather than just answering them outright as I'm trying to better understand and check the work I've already done to see if I'm correct.

As part of a laser light show, a technician is adjusting a diffraction grating. A green laser with a wavelength of 532 nm. it shines on a diffraction grating with 2700 lines/cm. The screen is 35.0 m from the diffraction grating and n = 1.

a. What is the distance between the slits of the diffraction grating?

b. How far from the central maximum, is the bright fringe?

Answer 1

Hello!

The first part (a) of the problem is quite simple. It's written that there are 2700 lines (slits) in 1 cm. Therefore, the distance between two adjacent slits is simply 1 cm divided by the number of slits:

$$d=1/2700=3.7·10^{-4}\text{ cm, or 3.7}·10^{-6}\text{ m}.$$

For part (b), we know the relation

$$d\sin\theta=m\lambda,$$

where θ is the angle between the normal to the screen and the direction (line) from center of the diffraction grating to the $m_\text{th}$ bright fringe. As m increases, the fringes become dimmer and dimmer. So, find the angle for the first (bright) fringe:

$$d\sin\theta=\lambda,\\\space\\ \theta=\arcsin\frac \lambda d.$$

Then apply simple geometry to find the distance to the bright fringe:

$$x=L\tan\theta=L·\tan\bigg(\arcsin\frac\lambda d\bigg)=5.09\text{ m}.$$