A grating 1.600 cm wide has exactly 12,000 slits. The...

A grating $1.600 \mathrm{~cm}$ wide has exactly 12,000 slits. The grating is used to resolve two nearly equal wavelengths in a light source: $\lambda_{\mathrm{a}}=440.000 \mathrm{~nm}$ and $\lambda_{\mathrm{b}}=440.936 \mathrm{~nm}$. (a) How many orders of the lines can be seen with the grating? (b) What is the angular separation $\theta_{b}-\theta_{2}$ between the lines in each order? (c) Which order best resolves the two lines? Explain.

A grating $1.600 \mathrm{~cm}$ wide has exactly 12,000 slits. The grating is used to resolve two nearly equal wavelengths in a light source: $\lambda_{\mathrm{a}}=440.000 \mathrm{~nm}$ and $\lambda_{\mathrm{b}}=440.936 \mathrm{~nm}$.
(a) How many orders of the lines can be seen with the grating? (b) What is the angular separation $\theta_{b}-\theta_{2}$ between the lines in each order? (c) Which order best resolves the two lines? Explain.

Key Concepts

Resolving Power

Resolving power is a measure of a grating’s ability to distinguish between two closely spaced wavelengths. It depends on factors such as the number of slits in the grating and the diffraction order used. Higher orders generally enhance the resolving power, allowing for a finer distinction between wavelengths, although practical limitations can restrict the usable diffraction orders.

Angular Separation

Angular separation in the context of diffracted light refers to the small angular difference between diffraction peaks corresponding to slightly different wavelengths. This separation is crucial when resolving sources with nearly identical wavelengths, as it determines whether two close spectral lines can be distinguished from one another. It is calculated by examining the differences in the diffraction angles obtained from the grating equation for the different wavelengths.

Grating Equation

The grating equation, given by d sin(?) = m?, relates the slit spacing (d), the angle of diffraction (?), the order of the spectrum (m), and the wavelength of light (?). This equation is fundamental in determining at what angles light of a specific wavelength will constructively interfere, and it is used to calculate properties like the maximum possible diffraction order for a given grating.

Diffraction Order

Diffraction orders refer to the series of angles at which constructive interference occurs, corresponding to different integer multiples of the wavelength relative to the slit spacing. The zeroth order is the directly transmitted light, and higher orders occur at increasing angles. Not all orders are visible or allowed, as physical constraints such as the width of the grating and the condition for constructive interference limit the number of observable orders.

Diffraction Grating

A diffraction grating consists of a large number of parallel slits which diffract incoming light into several directions. The interference of the light waves emerging from these slits leads to the formation of a pattern of bright and dark regions. The pattern depends on the wavelength of the light and the spacing between the slits, making diffraction gratings powerful tools in analyzing the spectral components of light.

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