Q23E - University Physics with Modern Physics

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?

Step-by-Step Solution

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 $37.7° and 65.5°$respectively.

Step 1: 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)

Step 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

Step 3: Determine the additional bright spots beyond the first bright spots and their angles

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

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

 

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