The light shining on a diffraction grating has a wavelength of 470 nm (in vacuum). The grating produces a second-order bright fringe whose position is defined by an angle of 9.35°. How many lines per centimeter does the grating have?
To find the number of lines per centimeter on the grating, we need to use the diffraction grating equation. The diffraction grating equation is given by:
sin(θ) = mλ / d
where:
θ is the angle between the incident light and the normal to the grating,
m is the order of the bright fringe,
λ is the wavelength of the incident light,
and d is the spacing between the adjacent lines on the grating.
In this case, we know that the light has a wavelength of 470 nm (or 470 × 10^(-9) m) and the angle of the second-order bright fringe is 9.35°. We need to solve for d, which represents the spacing between two adjacent lines on the grating.
First, we need to convert the angle from degrees to radians. Since 1° = π/180 radians, we have:
θ = 9.35° * (π/180) radians.
Next, we rearrange the equation to solve for d:
d = mλ / sin(θ).
Substituting the given values into the equation:
d = (2 * 470 × 10^(-9) m) / sin(9.35° * (π/180)).
Evaluate the sine of the angle:
d = (2 * 470 × 10^(-9) m) / sin(0.163254 radians).
Finally, to find the number of lines per centimeter, we use the relationship:
Number of lines per centimeter = 1 / (d * 100).
Substituting the value of d into the equation:
Number of lines per centimeter = 1 / (d * 100).
Now, plug in the value of d calculated earlier, and evaluate the expression to find the number of lines per centimeter.