An 11-mm wide diffraction grating has rulings of 550 lines per mm. Light is incident normally
on the grating. The longest wavelength that forms an intensity maximum in the fifth order is
closest to?
The correct answer is 488nm but I cant seem to get that answer.
To determine the longest wavelength that forms an intensity maximum in the fifth order, we can use the equation for the grating spacing:
Grating spacing (d) = 1 / Number of lines per mm
From the given information, the grating has rulings of 550 lines per mm, so the grating spacing is:
d = 1 / 550 mm
Now, we need to find the angle at which the maximum intensity occurs. For a diffraction grating, the angle at which the maximum intensity occurs can be calculated using the formula:
sin(θ) = m * λ / d
Where:
- θ is the angle between the incident light ray and the normal to the grating surface.
- m is the order of the maximum intensity.
- λ is the wavelength of light.
In this case, we are interested in the fifth order maximum, so m = 5. We want to find the angle for the longest wavelength, which means we need to find the maximum value of sin(θ). The maximum value of sin(θ) is 1, which occurs when θ = 90 degrees. So, we can use the formula to find the angle for the longest wavelength:
1 = 5 * λ / d
Now, we can rearrange the equation to solve for λ:
λ = d / 5
Substituting the value of d from earlier:
λ = (1 / 550 mm) / 5
λ = 1 / (550 * 5) mm
Now we need to convert this to the desired unit, which is nanometers (nm). To convert from millimeters (mm) to nanometers (nm), multiply by 10^6:
λ = (1 / (550 * 5)) * 10^6 nm
Calculating the value:
λ = 1.818 nm
Therefore, the correct answer is approximately 1.818 nm for the longest wavelength that forms an intensity maximum in the fifth order.