Colorful Electrons

A narrow beam of electrons passes close to the surface of a large metallic mirror with a diffraction grating of period d=2 μmd=2~\mu\text{m} inscribed on it as shown in the figure. The electrons move with v=0.9cv=0.9c (cc is the speed of light) at right angles to the lines of the grating. As a result of the etchings, the electrons may emit visible radiation whose wavelength depends on the angle of observation θ\theta. At which angle θ\theta in degrees will the observed radiation be green? Assume that the velocity of the electrons is not affected by the mirror.

Details and assumptions

Assume that green light has wavelength λg=0.55 μm.\lambda_{g}=0.55~\mu \text{m}.)

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