A diffraction grating with 75 slits per millimeter means that there are 75 lines (or slits) per 1 millimeter of the grating's surface. When light is incident on the grating, it undergoes diffraction, which is the bending and spreading of light waves as they pass through the slits.
The angle of diffraction, θ, depends on the wavelength of light (λ) and the spacing between the slits (d):
n * λ = d * sin(θ)
where:
n is the order of the diffraction (0, 1, 2, 3, ...),
λ is the wavelength of light,
d is the spacing between adjacent slits, and
θ is the angle at which the diffracted light is observed.
Since the diffraction grating has 75 slits per millimeter, the spacing (d) between adjacent slits is:
d = 1 mm / 75 slits = 1/75 mm = 0.01333... mm
Now, let's assume the grating is illuminated with white light, which consists of various wavelengths. In this case, different colors (wavelengths) will be diffracted at different angles, causing a rainbow-like pattern known as a spectrum.
If you want to calculate the angles for specific wavelengths or orders of diffraction, you can use the formula above. For example, if you are interested in the first-order diffraction (n = 1) of a certain wavelength (λ), you can rearrange the equation to solve for θ:
θ = arcsin(n * λ / d)
Keep in mind that the angles will be measured with respect to the incident direction of the light (usually taken as the normal to the grating surface). Also, higher-order diffraction peaks will appear at larger angles from the incident direction.
For example, if you want to find the angle of first-order diffraction for a green light with a wavelength of approximately 550 nm, you would calculate:
θ ≈ arcsin(1 * 550 nm / 0.01333 mm) ≈ arcsin(41.26) ≈ 24.54 degrees
Remember that this is just one of the many diffraction angles that will be present in the spectrum produced by the diffraction grating. Different colors (wavelengths) will be spread out at different angles, forming a full spectrum.