Question

Structures on a bird feather act like a reflection grating having$\mathbf{8000}$lines per centimeter. What is the angle of the first-order maximum for $\mathbf{600}\mathbf{‐}\mathbf{nm}$light?

Short answer

The angle for the first-order maximum is $28.7^{°}$.

Step-by-step solution

Given data

The wavelength of the light is$\mathrm{\lambda }=600\hspace{0.33em}\mathrm{nm}\left(\frac{{10}^{‐9}\mathrm{m}}{1\mathrm{nm}}\right)=6\times {10}^{‐7}\hspace{0.33em}\mathrm{m}$

The order of maximum is 1.

Grating lines on the reflection grating are 8000 liner per centimeter, which means

that the distance between the two slits is $\mathrm{\lambda }=600\hspace{0.33em}\mathrm{nm}\left(\frac{1\mathrm{cm}}{8000}\right)\left(\frac{1\mathrm{m}}{100\mathrm{cm}}\right)=1.25\times {10}^{‐6}\hspace{0.33em}\mathrm{m}$

Evaluating the spacing between the structures

The equation of the constructive interference can be expressed as

$\mathbf{\theta sin\lambda }$ …………………(1)

Where, d,$\mathrm{\lambda }$, $\mathrm{\theta }$and m is the separation between slits, the wavelength of the incident light, and the angle made with the maximum and order of maximum, respectively.

For a given value of $\mathrm{\lambda }$we can determine the angle for the first-order maximum (m=1) using the formula using equation (1),

role="math" localid="1654061489643" $\mathrm{\theta }={\sin }^{‐1}\left(\frac{\mathrm{\lambda }}{\mathrm{d}}\right)$………………….(2)

Substituting the given data in equation (1) will give,

$$\begin{gathered} \mathrm{\theta }={\sin }^{‐1}\left(\frac{\mathrm{\lambda }}{\mathrm{d}}\right) \\ ={\sin }^{‐1}\left(\frac{6\times {10}^{‐7}\mathrm{m}}{1.25\times {10}^{‐6}\mathrm{m}}\right) \\ =28.7^{°} \end{gathered}$$

Therefore, the angle for the first-order maximum is $28.7^{°}$.