Scientists shine a laser beam on a -wide slit and produce a diffraction pattern on a screen behind the slit. Careful measurements show that the intensity first falls to of maximum at a distance of from the center of the diffraction pattern. What is the wavelength of the laser light?
Hint: Use the trial-and-error technique demonstrated in Example to solve the transcendental equation.
Wavelength of the laser light is .
Intensity is ,
The maximum intensity is ,
Width of the slit is ,
The wavelength is
And the distance between the slit and the screen is .
Rearrange the solution above.
The intensity decreases to of the high capacity.
Use the trial-and-error method as outlined in the textbook's example.
The is in radians.
The first minimum
This amount is lower than the solution. Because the intensity has tapered off more in this case, we should take our first forecast higher than the one given in the example.
For second trial, ,
For third trial,
Average of the value is,
Find an expression for the positions of the first-order fringes of a diffraction grating if the line spacing is large enough for the small-angle approximation to be valid. Your expression should be in terms of and .
. Use your expression from part a to find an expression for the separation on the screen of two fringes that differ in wavelength by . Rather than a viewing screen, modern spectrometers use detectors-similar to the one in your digital camera-that are divided into pixels. Consider a spectrometer with a grating and a detector with located behind the grating. The resolution of a spectrometer is the smallest wavelength separation that can be measured reliably. What is the resolution of this spectrometer for wavelengths near , in the center of the visible spectrum? You can assume that the fringe due to one specific wavelength is narrow enough to illuminate only one column of pixels.
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