Question

Structures on a bird feather act like a reflection grating having 450 lines per centimeter.

Answer 1

The structures on a bird feather act like a reflection grating, creating an interference pattern. The angle of the first-order maximum for 600-nm light can be calculated using the formula θ = arcsin(m * λ / d), where m is the order of the maximum, λ is the wavelength of the light, and d is the spacing between the structures.

Step-by-step explanation:

The structures on a bird feather act like a reflection grating, which is a device that creates an interference pattern when light passes through it. The spacing between the structures determines the interference pattern. In this case, the feather has 8000 lines per centimeter. To find the angle of the first-order maximum for 600-nm light, we can use the formula:

θ = arcsin(m * λ /d)

Where:

• θ is the angle of the maximum,
• m is the order of the maximum (in this case, m = 1 for first-order maximum),
• λ is the wavelength of the light (600 nm = 0.6 μm),
• and d is the spacing between the structures in the feather (in centimeters).

Plugging in the values, we have:

θ = arcsin(1 * 0.6 μm / 8000 lines per cm)

Simplifying the equation, we get:

θ ≈ 0.813°

Therefore, the angle of the first-order maximum for 600-nm light is approximately 0.813 degrees.

Answer 2

The angle of the first-order maximum for 452 nm light using the given feather structure acting as a reflection grating with 8000 lines per centimeter is approximately
`\(2.08^\circ\)`.

The formula to calculate the angle of diffraction
`(\(\theta\))` for the first-order maximum using a diffraction grating is:

`\(\sin(\theta) = \frac{{m \cdot \lambda}}{{d}}\)`

Where:

`\(m\) =` order of the maximum (first order in this case)

`\(\lambda\) =` wavelength of light
`(452 nm = \(452 * 10^(-9)\) meters)`

d = spacing between lines on the grating (in meters)

Given that the grating has 8000 lines per centimeter, first, convert this to lines per meter:

8000 lines per centimeter = 80000 lines per meter

Then,
`\(d = \frac{1}{{80000}}\)` meters.

Now, substitute the values into the formula:

`\(\sin(\theta) = \frac{{1 \cdot 452 * 10^(-9)}}{{\frac{1}{{80000}}}}\)`

`\(\sin(\theta) = 0.036\)`

To find the angle
`\(\theta\)`:

`\(\theta = \arcsin(0.036)\)`

`\(\theta \approx 2.08^\circ\)`

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?