What is the spacing between structures in a feather that acts as a reflection grating, given that they produce a first- order maximum for 525-nm light at a 30.030.0^\circ angle?
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Final Answer

1.05 μm1.05\textrm{ }\mu\textrm{m}

Solution video

OpenStax College Physics for AP® Courses, Chapter 27, Problem 30 (Problems & Exercises)

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Video Transcript
This is College Physics Answers with Shaun Dychko. A bird feather acts as a reflection grating and produces a first order maximum at an angle of 30.0 degrees for 525 nanometer light. So this is the formula for the maxima in a diffraction pattern and the spacing between lines and the diffraction grating multiplied by sin of the angle to the maximum equals the order of the maximum times the wavelength of the light. So we can solve for d here which will be the spacing between the features and the feather that are acting as a grating and we divide both sides by sin Θ and so d is the order times the wavelength divided by sin of the angle. So that's 1 times 525 nanometers divided by sin 30.0 and since we use nanometers here for the wavelength, our answer is in nanometers—1050— and we'll convert this into micrometers just because that's a more traditional way of writing a number like this... multiply by 1 micrometer for every 1000 nanometers and we have 1.05 micrometers.