Question

A diffraction grating is illuminated simultaneously with red light of wavelength$660\mathrm{nm}$and light of an unknown wavelength. The fifth-order maximum of the unknown wavelength exactly overlaps the third-order maximum of the red light. What is the unknown wavelength$?$

Short answer

The unknown wavelength is$369\mathrm{nm}$.

Step-by-step solution

Formula for diffraction angle

We know that for an incident light of wavelength$\mathrm{\lambda }$, the$\mathrm{m}$-th order diffraction angle is,

${\mathrm{\theta }}_{\mathrm{m},\mathrm{\lambda }}={\sin }^{-1}\left(\frac{\mathrm{m\lambda }}{\mathrm{d}}\right)$

$\mathrm{\theta }$is diffraction angle

$\mathrm{\lambda }$is wavelength

$\mathrm{d}$is diffraction spacing

Calculation for unknown wavelength

We might equate the overlapping situation with

${\mathrm{Y}}_{3,\text{red}}={\mathrm{Y}}_{5,\text{ukn.}}\Rightarrow {\mathrm{Ltan\theta }}_{3,\text{red}}$

${\mathrm{Ltan\theta }}_{5,\text{ukn.}}\Rightarrow {\mathrm{\theta }}_{3,\text{red}}$

${\mathrm{Ltan\theta }}_{5,\text{ukn.}}\Rightarrow {\mathrm{\theta }}_{3,\text{red}}={\mathrm{\theta }}_{5,\text{unk}}$

${\mathrm{Y}}_{3,\text{red}}={\mathrm{Y}}_{5,\text{ukn.}}\Rightarrow {\mathrm{Ltan\theta }}_{3,\text{red}}={\mathrm{Ltan\theta }}_{5,\text{ukn.}}\Rightarrow {\mathrm{\theta }}_{3,\text{red}}={\mathrm{\theta }}_{5,\text{unk}}$

So,

$\frac{3\times {\mathrm{\lambda }}_{\text{red}}}{\mathrm{d}}=\frac{5\times \mathrm{\lambda }}{\mathrm{d}}$

$\mathrm{\lambda }=\frac{3\times {\mathrm{\lambda }}_{\text{red}}}{5}=0.6\times 660$

$=396\mathrm{nm}$