Question

The wings of some beetles have closely spaced parallel lines of melanin, causing the wing to act as a reflection grating. Suppose sunlight shines straight onto a beetle wing. If the melanin lines on the wing are spaced $2.0\mu m$apart, what is the first-order diffraction angle for green light $\left(\lambda =550nm\right)$?

Short answer

For green light, the first-order diffraction angle is${\theta }_m={16}^{\circ }.$

Step-by-step solution

Step: 1 Diffraction angle:

The smaller the value of d for a given order and wavelength, the higher the diffraction angle. To put it another way, the greater the number of slits per metre, the greater the diffraction angle.

Step: 2 Finding angle:

By grating equation,

$d\sin \left({\theta }_m\right)=m\lambda$

The angle of $m^{th}$fringe is there

$$\begin{gathered} \sin \left({\theta }_m\right)=\frac{m\lambda }{d} \\ {\theta }_m={\sin }^{-1}\left(\frac{m\lambda }{d}\right) \end{gathered}$$

Step: 3 Diffraction angle for green light:

The angle of light by

$$\begin{gathered} {\theta }_m={\sin }^{-1}\left(\frac{1\times 5.50\times {10}^{-7}}{2\times {10}^{-6}}\right) \\ {\theta }_m={\sin }^{-1}\left(0.275\right) \\ {\theta }_m={16}^{\circ }. \end{gathered}$$