A Diffraction Grating GOAL Calculate different-order principal maxima for a diffraction grating. PROBLEM Monochromatic light from a helium–neon laser (λ = 632.8 nm) is incident normally on a diffraction grating containing 6.00 × 10³ lines/cm. Find the angles at which one would observe the

Question

A Diffraction Grating

GOAL Calculate different-order principal maxima for a diffraction grating.

PROBLEM Monochromatic light from a helium–neon laser (λ = 632.8 nm) is incident normally on a diffraction grating containing 6.00 × 10³ lines/cm. Find the angles at which one would observe the first-order maximum, the second-order maximum, and so forth.

STRATEGY Find the slit separation by inverting the number of lines per centimeter, then substitute values into Equation 24.12.

[latex]d\sin heta_{\mathrm{bright}}=m\lambda\qquad m=0,\pm,1,\pm,2,\ldots[/latex] [24.12]

Accepted Answer

Invert the number of lines per centimeter to obtain the slit separation:

[latex]d={\frac{1}{6.00~ imes~10^{3}\,{\mathrm{cm}}^{-1}}}=1.67 imes10^{-4}\,{\mathrm{cm}}\,=\,1.67 imes10^{3}\,{\mathrm{nm}}[/latex]

Substitute m = 1 into Equation 24.12 to find the sine of the angle corresponding to the first-order maximum:

[latex]\sin heta_{1}={\frac{\lambda}{d}}={\frac{632.8~\mathrm{nm}}{1.67~ imes~10^{3}\,\mathrm{nm}}}=0.379[/latex]

Take the inverse sine of the preceding result to find [latex] heta_{1}[/latex]:

[latex] heta_{1}=\sin^{-1}0.379={22.3^{\circ}}[/latex]

Repeat the calculation for m = 2:

[latex]\sin heta_{2}={\frac{2\lambda}{d}}={\frac{2(632.8~\mathrm{nm})}{1.67~ imes~10^{3}\mathrm{nm}}}=0.758[/latex]

[latex] heta_{2}=\ 49.3^{\circ}[/latex]

Repeat the calculation for m = 3:

[latex]\sin heta_{3}={\frac{3\lambda}{d}}={\frac{3(632.8~\mathrm{nm})}{1.67~ imes~10^{3}\mathrm{nm}}}=1.14[/latex]

Because sin θ can’t exceed 1, there is no solution for [latex] heta_{3}[/latex].

REMARKS The foregoing calculation shows that there can only be a finite number of principal maxima. In this case only zeroth-, first-, and second-order maxima would be observed.

Question 24.7: A Diffraction Grating GOAL Calculate different-order princip...

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