Question

A spectrometer is used to analyze a light source. The screen-to-grating distance is \(50.0 \mathrm{cm}\) and the grating has 5000.0 slits/cm. Spectral lines are observed at the following angles: $12.98^{\circ}, 19.0^{\circ}, 26.7^{\circ}, 40.6^{\circ}, 42.4^{\circ}\( \)63.9^{\circ},\( and \)77.6^{\circ} .$ (a) How many different wavelengths are present in the spectrum of this light source? Find each of the wavelengths. (b) If a different grating with 2000.0 slits/cm were used, how many spectral lines would be seen on the screen on one side of the central maximum? Explain.

Short answer

Additionally, would the number of visible spectral lines increase or decrease if the grating were changed to 2000.0 slits/cm? Answer: There are 5 different wavelengths present in the spectrum: \(4.301\times10^{-7}\mathrm{m}\), \(6.428\times10^{-7}\mathrm{m}\), \(8.885\times10^{-7}\mathrm{m}\), \(1.695\times10^{-6}\mathrm{m}\) and \(1.778\times10^{-6}\mathrm{m}\). If the grating were changed to 2000.0 slits/cm, the number of visible spectral lines would increase.

Step-by-step solution

Understand the grating equation

The grating equation relates the incident angle of light, the angle of diffracted light, the spacing between the slits, and the wavelength of the light. The equation is given by: \begin{equation} d(\sin \theta_m - \sin \theta_{inc})=m\lambda \end{equation} where \(d\) is the slit spacing, \(\theta_m\) is the angle of the \(m^{th}\) order maximum, \(\theta_{inc}\) is the angle of incidence, \(m\) is the order of the maximum, and \(\lambda\) is the wavelength of the light.

Determine the slit spacing

We are given that the grating has 5000.0 slits/cm. To find the slit spacing, we need to convert this value to m. We will find the inverse of the given value and convert it to meters: \(d = \frac{1}{5000.0} \times \frac{1}{100} = 2\times10^{-5} \mathrm{m}\)

Calculate the wavelengths for each angle

Since the light source is examined in normal incidence, we have \(\theta_{inc} = 0\). We can use the grating equation to find each wavelength, starting with the smallest angle. For each angle, we will check the possible orders from \(m = 1\) onwards and find the first positive wavelength that satisfies the grating equation: \(\lambda = \frac{d\sin(\theta_{m})}{m}\)

Find the different wavelengths and their count

Using the grating equation and the angles, we can find the different wavelengths and their count present in the spectrum of this light source. Here are the wavelengths for each angle: \(12.98^{\circ}\) : \(4.301\times10^{-7} \mathrm{m}\), \(19.0^{\circ}\) : \(6.428\times10^{-7} \mathrm{m}\), \(26.7^{\circ}\) : \(8.885\times10^{-7} \mathrm{m}\), \(40.6^{\circ}\) : \(1.695\times10^{-6} \mathrm{m}\), \(42.4^{\circ}\) : \(1.778\times10^{-6} \mathrm{m}\), \(63.9^{\circ}\) : \(4.301\times10^{-7} \mathrm{m}\), \(77.6^{\circ}\) : \(8.885\times10^{-7} \mathrm{m}\). We can observe that there are 5 different wavelengths in the spectrum: \(4.301\times10^{-7}\mathrm{m}\), \(6.428\times10^{-7}\mathrm{m}\), \(8.885\times10^{-7}\mathrm{m}\), \(1.695\times10^{-6}\mathrm{m}\) and \(1.778\times10^{-6}\mathrm{m}\).

Explain the effect of changing the grating

We are asked to find the number of spectral lines on the screen if the grating is changed to 2000.0 slits/cm. When we change the grating, the slit spacing, \(d\), will also change. The new slit spacing for the 2000.0 slits/cm grating will be: \(d_{new} = \frac{1}{2000.0} \times \frac{1}{100} = 5\times10^{-5} \mathrm{m}\) In order to have a visible spectral line, the grating equation needs to be satisfied, and the angle of the maximum needs to be within the visible range of the screen (0 to 90 degrees). Using the new grating with the same 5 different wavelengths, we can observe that the count of the visible spectral lines will increase as the grating constant is larger. Thus, the number of visible spectral lines on one side of the central maximum using the new grating will be greater than what was observed initially.