Question

Perhaps to confuse a predator, some tropical grind beetles (whirling beetles) are colored by optical interference that is due to scales whose alignment forms a diffraction grating (which scatters light instead of transmitting it). When the incident light rays are perpendicular to the grating, the angle between the first order maxima (on opposite sides of the zeroth-order maximum) is about $26^\circ $ in light with a wavelength of $550nm$ what is the grating spacing of the beetle?

Answer 1

Hint: In order to solve this question we need to understand diffraction.Diffraction is a phenomenon of light which only happens if the wavelength of light is in comparison with slit width then light upon falling on slit get diffracted from its corner. This phenomena clearly depicts the wave nature of light.

Complete step by step answer:
Since the angle between two first order maxima on either side of zeroth order maxima is, Given, $2\theta = 26^\circ $
$\Rightarrow \theta = \dfrac{{26^\circ }}{2}$
$\Rightarrow \theta = 13^\circ $
Let the spacing between two slits be $d$.Since order of maxima given is, $n = 1$.Using diffraction formula for maxima we get,
$d\sin \theta = n\lambda $
Here $\lambda $ is the wavelength of monochromatic light source, $\lambda = 550\,nm$.
Putting values we get,
$d\sin (13^\circ ) = 550\,nm \\ $
$\Rightarrow d = \dfrac{{550\,nm}}{{0.224}} \\ $
$\Rightarrow d = 2455.357\,nm \\ $
$\therefore d = 2.45\,\mu m$

So the grating spacing of the beetle is $d = 2.45\,\mu m$.

Note: It should be remembered that diffraction phenomena only exhibit when the wavelength of light used is in comparison with slit width. It is only due to the wave nature of light. For electrons it is untrue that electrons behave as waves when not observed but when observed they behave as particles. Also monochromatic sources of light are used so that only one wavelength falls, otherwise so many wavelengths for different patterns of bright and dark fringe.