A diffraction grating has 6 * 10^5 lines/m. Find the angular spread in the second order spectrum between red light of wavelength 7 * 10^-7 m and blue light of wavelength 4.5 * 10^-7 m.

To find the angular spread in the second order spectrum between red light and blue light using a diffraction grating, we can apply the diffraction grating formula, which relates the angle of diffraction to the wavelength of light and the grating spacing. The formula is given by:

Understanding the Diffraction Grating Formula

The formula is expressed as:

d sin(θ) = mλ

Where:

• d = distance between adjacent grating lines (grating spacing)
• θ = angle of diffraction
• m = order of the spectrum (in this case, m = 2 for the second order)
• λ = wavelength of light

Calculating the Grating Spacing

First, we need to determine the grating spacing, d. Given that the grating has 6 × 10⁵ lines/m, we can find d as follows:

d = 1 / (number of lines per meter)

d = 1 / (6 × 10⁵) = 1.67 × 10^-6 m

Finding the Angles for Red and Blue Light

Next, we will calculate the angles for both red light (λ = 7 × 10^-7 m) and blue light (λ = 4.5 × 10^-7 m) using the diffraction grating formula.

For Red Light

Using the formula:

sin(θ_red) = (mλ_red) / d

Substituting the values:

sin(θ_red) = (2 × 7 × 10^-7) / (1.67 × 10^-6)

sin(θ_red) = 0.839

Now, we find θ_red:

θ_red = sin^-1(0.839) ≈ 57.5°

For Blue Light

Now, applying the same process for blue light:

sin(θ_blue) = (mλ_blue) / d

Substituting the values:

sin(θ_blue) = (2 × 4.5 × 10^-7) / (1.67 × 10^-6)

sin(θ_blue) = 0.539

Now, we find θ_blue:

θ_blue = sin^-1(0.539) ≈ 32.5°

Calculating the Angular Spread

The angular spread between the red and blue light in the second order spectrum can be found by taking the difference between the two angles:

Angular Spread = θ_red - θ_blue

Angular Spread = 57.5° - 32.5° = 25°

Final Result

The angular spread in the second order spectrum between red light and blue light is approximately 25°. This shows how different wavelengths of light can be separated by a diffraction grating, allowing us to observe distinct colors in the spectrum.