Question

When laser light of wavelength 632.8nm passes through a diffraction grating, the first bright spots occur at $\mathbf{\pm }\mathbf{17}\mathbf{.}\mathbf{8}\mathbf{°}$ from the central maximum. (a) What is the line density (in lines/cm) of this grating? (b) How many additional bright spots are there beyond the first bright spots, and at what angles do they occur?

Short answer

1. The line density of this grating is 4831 lines/cm.
2. There are two additional bright spots beyond the first bright spots at angle $37.7°and65.5°$respectively.

Step-by-step solution

Formulas used to solve the question

angles of bright fringes is given by

$\sin \theta =\frac{m\lambda }{d}$ (1)

Density of grating is given by,

$N=\frac{1}{d}$ (2)

Determine the line density

Given: $$\begin{gathered} \lambda =632.8nm=632.8*{10}^{-9}m \\ {\theta }_1=\pm 17.8^{\circ } \end{gathered}$$

From equation (1) solve for and plug the given,

$\mathrm{d}=\frac{\mathrm{m\lambda }}{\sin \mathrm{\theta }}=\frac{632.8*{10}^{-9}}{\sin 17.8^{\circ }}=2067.4*{10}^{-9}\mathrm{m}$

Now, line density can be given as,

$\rho =\frac{{10}^{-2}}{2067.4*{10}^{-9}}=4830$lines/cm

Determine the additional bright spots beyond the first bright spots and their angles

Here, take the condition $sin{\theta }_m<1$which gives

role="math" localid="1664127629344" $$\begin{gathered} sin{\theta }_2=2sin{\theta }_1=2*0.3056=0.62\Rightarrow {\theta }_2=37.7^{\circ } \\ \Rightarrow {\theta }_2=37.7^{\circ } \end{gathered}$$

And

$$\begin{gathered} sin{\theta }_3=3sin{\theta }_1=3*0.3056=0.62\Rightarrow {\theta }_3=0.{93}^{\circ } \\ \Rightarrow {\theta }_3=66.5^{\circ } \end{gathered}$$

Thus, the line density of this grating is lines/cm. There are two additional bright spots beyond the first bright spots at angle $37.7°\mathrm{and}66.5°$respectively