Question

A diffraction grating with 750 slits per mm is illuminated by light which gives a first-order diffraction angle of 34.0°. What is the wavelength of the light?

Answer 1

Answer: 745.59 nm

Explanation:

The diffraction angles $\theta_{n}$ when we have a slit divided into $n$ parts are obtained by the following equation:

$dsin\theta_{n}=n\lambda$ (1)

Where:

$d$ is the width of the slit

$\lambda$ is the wavelength of the light  

$n$ is an integer different from zero

Now, the first-order diffraction angle is given when $n=1$, hence equation (1) becomes:

$dsin\theta_{1}=\lambda$ (2)

We know:

$\theta_{1}=34\°$

In addition we are told the diffraction grating has 750 slits per mm, this means:

$d=\frac{1mm}{750}$

Solving (2) with the known values we will find $\lambda$:

$\lambda=(\frac{1mm}{750})sin(34\°)$ (3)

$\lambda=0.00074559mm$ (4)

Knowing $1mm=10^{6}nm$:

$\lambda=745.59nm$  >>>This is the wavelength of the light, wich corresponds to red.