A spectrometer is used to analyze a light source. The...

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.

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.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component featuring a regular pattern of closely spaced lines or slits that diffract light into several beams traveling in different directions. The pattern splits the light into its constituent wavelengths, making it a fundamental tool for spectroscopy and the analysis of light spectra.

Grating Equation

The grating equation, given by d sin ? = m?, is the mathematical relationship connecting the slit spacing (d), the diffraction angle (?), the order of the spectrum (m), and the wavelength (?) of light. This equation is central to determining the wavelengths present in a light source based on the measured diffraction angles.

Grating Spacing

Grating spacing refers to the distance between adjacent slits in the diffraction grating. It is typically derived from the number of slits per unit length and directly influences the diffraction pattern. A smaller spacing (or higher number of slits per unit length) results in greater angular separation between spectral orders.

Spectral Order

The spectral order (denoted by m in the grating equation) indicates the sequence of diffraction maxima produced by a grating. Each order corresponds to a set of wavelengths diffracted at specific angles, with higher orders representing larger diffraction angles. Understanding which orders are physically admissible is essential for interpreting the observed spectrum.

Angular Dispersion

Angular dispersion is the rate at which the diffraction angle changes with respect to wavelength. It determines the ability of the grating to separate closely spaced wavelengths, making it a key concept in assessing the spectral resolution. A higher angular dispersion often leads to a finer separation of spectral lines.

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