Q11E - University Physics with Modern Physics

Question

Two thin parallel slits that are $\mathbf{0}\mathbf{.}\mathbf{0116}$ mm apart are illuminated by a laser beam of wavelength $\mathbf{585}$ mm. (a) On a very large distant screen, what is the total number of bright fringes (those indicating complete constructive interferences), including the central fringe and those on both sides of it? Solve this problem without calculating all the angles! (Hint: What is the largest that $\mathbf{sin\theta }$ can be? What does this tell you is the largest value of $\mathit{m}$?) (b) At what angle, relative to the original direction of the beam, will the fringe that is most distant from the central bright fringe occur?

Step-by-Step Solution

Answer

(a) The total number of bright fringes are 39.

(b) A $73.4^{°}$, the fringe is most distant from the central bright fringe.

Step 1: Given Data

Distance between slits:

$$\begin{gathered} \mathrm{d}=0.0116\mathrm{mm} \\ =0.0116\times {10}^{-3 }\mathrm{m} \end{gathered}$$

 

Wavelength: 

$\begin{array}{l}\mathrm{\lambda }=585 \mathrm{nm}\\ =585\times {10}^{-9 }\mathrm{m}\end{array}$

Step 2: Maximum angle for the farthest bright from the central bright fringe

The maximum angle for the farthest bright from the central bright fringe is ${\mathbf{90}}^{\mathbf{°}}$.

Step 3: Calculation of the total number of bright fringes

(a) First, find the maximum angle of this fringe.

The maximum angle for the farthest bright fringe  from the central bright fringe is .

Hence, 

$\mathrm{d} \sin {90}^{\circ }={\mathrm{m}}_{{\max }^{\mathrm{\lambda }}}$

Solve for m,

$m_{\max }=\frac{d}{\lambda }$

Plug the given,

${\mathrm{m}}_{\max }=\frac{0.0116\times {10}^{-3}}{585\times {10}^{-9 }\mathrm{m}}=19.8$

Now, $m$ must be the integer number. This means that the farthest bright fringe range is at

${\mathrm{m}}_{\max }=19$

Hence, there are $19$ bright fringes above the central bright fringe, plus $19$ fringes below the central bright fringe and the central bright fringe itself.

So, the total number of bright fringes is

$$\begin{gathered} \mathrm{N}=19+19+1 \\ =39 \end{gathered}$$

Step 4: Determine the angle at which the fringe is most distant from the central bright fringe.

(b) Solve (1) for $\theta$ to find its angle.

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

Plug the given,

$$\begin{gathered} \theta ={\sin }^{-1}\left(\frac{19\times 585\times {10}^{-9}}{0.0116\times {10}^{-3}}\right) \\ =73.4 \end{gathered}$$ 

Hence, the total number of bright fringes are $39$ and  $73.4^{°}$ is the angular separation of the farthest fringe from central fringe.