1. What is the spacing between structures in a feather that acts as a reflection grating, given...

Question:

1. What is the spacing between structures in a feather that acts as a reflection grating, given that they produce a first-order maximum for 525-nm light at a 30.0-degree angle?

2. 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 600-nm light?

Interference

When light passes a single slit, it forms a diffraction pattern that is different from what was formed by double slits or what is called diffraction gratings.

Below is the equation for destructive interference for a single slit:

$$d sin\theta = m \lambda $$

where {eq}d {/eq} is the slit width,

{eq}\theta {/eq} is the angle relative to the original direction of the light,

{eq}m {/eq} is the order of the minimum and {eq}\lambda {/eq} is the light's wavelength.

Answer and Explanation:

Part A.

Given:

{eq}m = 1 \\ \lambda = 525 nm \\ \theta = 30.0 {/eq}

Solution:

{eq}d sin \theta = m \lambda \\ d = \frac {m \lambda }{ sin \theta } = \frac {5.25 \times 10^\text{-7}}{ sin (30.0) } =1.05 \times 10 ^\text{-6} m {/eq}

PART B.

Given:

8000 lines/cm

{eq}m = 1 \\ \lambda = 600 nm {/eq}

Solution:

For d:

{eq}d = \frac {0.01 m }{8000} = 1.25 \times 10^\text {-6} {/eq}

{eq}\theta = sin^\text{-1} \left ( \frac {m \lambda}{d} \right) = sin^\text{-1} \left ( \frac {6.00 \times 10^\text{-7}}{1.25 \times ^\text{-6}} \right) = 28.7^o {/eq}