Question

A diffraction grating is 1.50 cm wide and contains 2400 lines. When used with light of a certain wavelength, a third-order maximum is formed at an angle of 16.0°. What is the wavelength (in nm)?

Answer 1

The wavelength (in nm) is 370 nm.

Determining the wavelength of the light:

Here's how to find the wavelength of the light in nanometers:

Define the variables and given values:

Grating width (d) = 1.50 cm = 0.015 m

Number of lines (N) = 2400

Grating spacing (g) = d / N = 0.015 m / 2400 =
`6.25\:X\:10^-6 m`

Third-order maximum angle (θ3) = 16.0° = 0.2793 radians (converted to radians for calculations)

Wavelength (λ) (unknown)

Use the grating equation for the third-order maximum:

The grating equation relates the grating spacing, wavelength, and diffraction angle for a specific order:

nλ = g sin(θn)

where n is the diffraction order (3 in this case).

Solve for the wavelength:

Rearranging the equation for λ:

λ =
`(g sin(\theta n))/(n)`

λ =
`6.25\:X\:10^-6 m\:X\:(sin(0.2793))/(3)`

`\lambda \approx 3.70 X 10^-7 m`

Convert the wavelength to nanometers:

λ ≈
`3.70 \:X \:10^-7 m\: X \:10^9 nm/m`

λ ≈ 370 nm

Answer 2

The wavelength of the light is approximately 475.5 nm. We can find the wavelength of the light by using the formula: λ = (d × sin(θ))/n.

Step-by-step explanation:

The diffraction grating is 1.50 cm wide and contains 2400 lines. The angle of the third-order maximum is given as 16.0°.

To find the wavelength of the light, we need to use the formula:

λ = (d × sin(θ))/n, where d is the distance between adjacent lines on the grating, θ is the angle of the maximum, and n is the order of the maximum. In this case, n = 3.

First, we need to find the distance between adjacent lines.

Since the diffraction grating contains 2400 lines within 1.50 cm,

we can convert it to lines per centimeter by dividing 2400 by 1.50, giving us 1600 lines/cm.

Now we can calculate the wavelength using the formula: λ = (1/1600) × sin(16°)/3.

Plugging in the values, we find that the wavelength is approximately 475.5 nm.