To find the number of lines per centimeter on a diffraction grating, we can use the formula for the grating equation:
\(m \cdot \lambda = d \cdot \sin(\theta)\),
where:
\(m\) is the order of the maximum (for a second-order maximum, \(m = 2\)),
\(\lambda\) is the wavelength of light in meters,
\(d\) is the spacing between the lines on the grating in meters, and
\(\theta\) is the angle of diffraction in degrees.
Given:
\(\lambda = 480\) nm (blue light with a wavelength of 480 nm),
\(m = 2\) (second-order maximum),
\(\theta = 14.0^\circ\) (angle of diffraction).
First, convert the wavelength to meters:
\(\lambda = 480 \, \text{nm} = 480 \times 10^{-9}\) m.
Now, rearrange the grating equation to solve for \(d\):
\(d = \frac{m \cdot \lambda}{\sin(\theta)}\).
Substitute the values:
\(d = \frac{2 \times 480 \times 10^{-9}}{\sin(14.0^\circ)}\).
Now, calculate the value of \(d\):
\(d = \frac{2 \times 480 \times 10^{-9}}{\sin(14.0^\circ)} \approx 6.809 \times 10^{-6}\) m.
Finally, convert \(d\) to lines per centimeter:
\(1 \, \text{m} = 100 \, \text{cm}\).
\(d\) in cm = \(6.809 \times 10^{-6} \, \text{m} \times \frac{1 \, \text{cm}}{0.01 \, \text{m}} \approx 0.6809\) cm.
The number of lines per centimeter on the diffraction grating that gives a second-order maximum for 480-nm blue light at an angle of 14.0° is approximately 0.6809 lines per centimeter.