Question

The central star of a planetary nebula emits ultraviolet light with wavelength 104nm. This light passes through a diffraction grating with 5000 slits per mm. What is the first-order diffraction angle?

Answer 1

Answer: 31.33 degrees

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)

Now we have to find the value of $\theta_{1}$:

$sin\theta_{1}=\frac{\lambda}{d}$  

$\theta_{1}=arcsin(\frac{\lambda}{d})$   (3)

We know:

$\lambda=104nm=104(10)^{-9}m$

In addition we are told the diffraction grating has 5000 slits per mm, this means:

$d=\frac{1mm}{5000}=\frac{1(10)^{-3}m}{5000}$

Substituting the known values in (3):

$\theta_{1}=arcsin(\frac{104(10)^{-9}m}{\frac{1(10)^{-3}m}{5000}})$

$\theta_{1}=arcsin(0.52)$

Finally:

$\theta_{1}=31.33\º$ >>>This is the first-order diffraction angle