3.Blue light (λ = 450 nm) shines on a diffraction grating that has 4000 lines/inch.
A.Calculate the angle of the fourth-order image
To calculate the angle of the fourth-order image formed by the diffraction grating, we can use the formula for diffraction:
sinθ = mλ/d
where:
- θ is the angle of the diffraction.
- m is the order of the image.
- λ is the wavelength of the light.
- d is the separation between adjacent lines on the grating.
First, we need to convert the separation between lines from inches to meters:
1 inch = 0.0254 meters
4000 lines/inch = 4000 lines * (1 inch / 0.0254 meters) = 157480 lines/meter
Now we have all the required values, let's plug them into the formula:
sinθ = mλ/d
sinθ = 4 * (450 nm) / 157480 lines/meter
Note: Make sure to convert the wavelength from nanometers (nm) to meters (m) by dividing by 10^9.
sinθ = (4 * 450 * 10^(-9) m) / 157480 lines/meter
Calculating the value of sinθ gives us:
sinθ = 0.0000114118
Now, we need to find the angle θ. To do this, we take the inverse sine (sin^(-1)) of the value we obtained:
θ = sin^(-1)(0.0000114118)
Finally, calculate the angle θ (in degrees) using a calculator, which yields:
θ ≈ 0.000653 degrees
Therefore, the angle of the fourth-order image formed by the diffraction grating is approximately 0.000653 degrees.