# Structures on a bird feather act like a reflection grating having 8000 lines per centimeter. What...

## Question:

Structures on a bird feather act like a reflection grating having 8000 lines per centimeter. What is the angle of the first-order maximum for 452 nm light?

## Gratings:

Gratings are special devices that are used for diffraction experiments. Compared to just a one slit or two slit system, gratings have thousands of slits. Gratings have a quantity called the diffraction ruling, which can be interpreted as how many slits the grating has per unit length of measurement.

Given:

• {eq}\displaystyle \frac{1}{d} = 8000\ cm^{-1} {/eq} is the diffraction ruling
• {eq}\displaystyle \lambda = 452\ nm = 452\ \times\ 10^{-9}\ m {/eq} is the light wavelength

Let us first note that the diffraction equation is given as:

{eq}\displaystyle d\sin\theta = n\lambda {/eq}

So for a first-order maximum, we have:

• {eq}\displaystyle n = 1 {/eq}

This leaves us with:

{eq}\displaystyle d\sin\theta = \lambda {/eq}

We isolate the angle here:

{eq}\displaystyle \sin\theta = \frac{\lambda}{d} {/eq}

{eq}\displaystyle \theta = \sin^{-1} \left(\frac{\lambda}{d} \right) {/eq}

Let us first make sure that we are substituting the same units so that the units cancel out. We convert our diffraction ruling to meters:

{eq}\displaystyle \frac{1}{d} = 8000\ \frac{1}{cm} \left(\frac{100\ cm}{1\ m} \right) {/eq}

We get:

{eq}\displaystyle \frac{1}{d} = 8000\ \frac{1}{\require{cancel}\cancel{cm}} \left(\frac{100\ \require{cancel}\cancel{cm}}{1\ m} \right) {/eq}

{eq}\displaystyle \frac{1}{d} = 800,000\ m^{-1} {/eq}

We substitute:

{eq}\displaystyle \theta = \sin^{-1} \Big[(800,000\ m^{-1})(452\ \times\ 10^{-9}\ m)\Big] {/eq}

We thus get:

{eq}\displaystyle \theta = \sin^{-1} (0.3616) {/eq}

{eq}\displaystyle \boxed{\theta = 21.2^\circ} {/eq}