A grating in a spectrometer is illuminated with red light (λ=690 nm) and blue light (λ=460 nm)

A grating in a spectrometer is illuminated with red light...

A grating in a spectrometer is illuminated with red light $(\lambda=690 \mathrm{~nm})$ and blue light $(\lambda=460 \mathrm{~nm})$ simultaneously. The grating has $10,000.0$ slits per $\mathrm{cm}$. Sketch the pattern that would be seen on a screen $2.0 \mathrm{~m}$ from the grating. Label distances from the central maximum. Label which lines are red and which are blue.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with many equally spaced slits that disperse incident light into several beams traveling in different directions. The interference among light waves emerging from these slits produces a pattern of bright and dark regions, which is essential in applications like spectroscopy.

Grating Equation

The grating equation, expressed as sin(?) = m?/d, relates the angle at which a particular wavelength is diffracted to the order number (m) and the spacing between the slits (d). This formula is fundamental to predicting the positions of the bright fringes produced in a diffraction grating setup.

Order Maxima

The bright spots or maxima in a diffraction grating pattern occur at specific angles that satisfy the grating equation for integer order numbers. Each order corresponds to a set of constructive interference conditions, with higher orders typically farther from the central maximum.

Wavelength Dependence in Diffraction

The diffractive behavior depends on the wavelength of the incident light; different wavelengths will satisfy the grating equation at different angles. This wavelength dependence leads to angular dispersion, where light of various wavelengths spreads out spatially, allowing for their separation and analysis in spectroscopic instruments.

Transcript

00:20 The location for maxima for a grating is d -sign theta equals m -lamda for a maxima.

00:56 Approximating theta to be x over d, we have x equals m lambda, upper case d over -case d.

01:40 It is better to put these values as 2m times lambda d over d now we are going to find the value of lambda d over d for both of the colors so for red let's use 690 nanometer screen is two meter away and d is given by 10 000 slits per centimeter so that comes out to be 10 to the minus 2 meter divided by 10 ,000 so 10 to the 5 so that comes out to be 690 times 10 to the negative 9 times 2 times 10 to the 7 meter which is 6 .9 times 2 times 10 to the 7 meter which is 6 .9 times 2 so 10 ,006 per centimeter, this is 6.

04:34 Here we have 690 nanometer, 2 meter is the uppercase d, and lower case d is 1 centimeter or 10 to minus 2 meter divided by 10 to the 5, 10 ,0006 per 10 to minus 2 meter.

04:53 So this comes out to be this much meter.

05:05 So over here we are trying to find 2 meter over lambda d over 2 ,000, over 2.

05:09 2d so if we try to find that then this becomes 6 .9 meter for red for blue lambda d over 2d will come out to be 4 .6 meter so on the screen the red the red pattern will look like like this where this is at 0 .6 meter so on the screen the red pattern will look like like this is at 0 this is at 6 .9 meter the second maxima is a 6 .9 times 2 so that's 13 .8 this point is 3 times 6 .9 20 .7 meter this maxima is 4 times 6 .9 27 .6 meter so for blue these numbers will be given as 4 .6 times 2 is 9 .2 times 3 .6 times 3 .2 times 13 .8, 4 .6 times 4, 18 .4.

07:34 So the blue will have a maxima over here...

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