A diffraction grating is calibrated by using the 546.1 nm line of mercury vapor. The first-order maximum is found at an angle of 17.62 degrees.
Calculate the number of lines per centimeter on this grating.
To calculate the number of lines per centimeter on the diffraction grating, we need to use the formula:
\( d = \frac{\lambda}{\sin \theta} \)
Where:
- \( d \) is the distance between adjacent slits on the grating in meters
- \( \lambda \) is the wavelength of light in meters
- \( \theta \) is the angle at which the first-order maximum is observed
First, let's convert the wavelength from nanometers to meters:
\( \lambda = 546.1 \, \text{nm} \times 10^{-9} \, \text{m/nm} = 5.461 \times 10^{-7} \, \text{m} \)
Next, we can substitute the values into the formula:
\( d = \frac{5.461 \times 10^{-7} \, \text{m}}{\sin(17.62^\circ)} \)
Using a scientific calculator or the sine function on a calculator, we can evaluate \( \sin(17.62^\circ) \), which gives approximately 0.305.
Now we can calculate the value of \( d \):
\( d = \frac{5.461 \times 10^{-7} \, \text{m}}{0.305} \)
This gives us the distance between adjacent slits on the grating.
Finally, to calculate the number of lines per centimeter, we need to convert the distance \( d \) from meters to centimeters and take the reciprocal:
\( \text{Number of lines per centimeter} = \frac{1}{d \times 100} \)