Chapter 38, Problem 38.1QQ

Suppose the slit width in Figure 37.4 is made half as wide. Does the central bright fringe (a) become wider, (b) remain the same, or (c) become narrower?

Figure 37.4 (a) Geometry for analyzing the Fraunhofer diffraction pattern of a single slit. (Drawing not to scale.) (b) Simulation of a single-slit Fraunhofer diffraction pattern.

To determine

The change in the width of the central bright fringe.

Answer to Problem 38.1QQ

Option (a) the central bright fringe becomes wider.

Explanation of Solution

Consider the figure given below.

Figure (1)

The condition for the central diffraction maximum is,

$a\sin \theta =m\lambda$ (1)

Here;

$\theta$ is the position of fringe.

$\lambda$ is the wavelength of light.

$a$ is the slit width.

$m$ is the order of fringe.

From the figure (1),

$\tan \theta =\frac{y}{L}$

Here

$y$ is the vertical position of fringe measured from the centre of the screen.

$L$ is the distance between the slit and screen.

From the trigonometry property, for very small angle,

$\tan \theta \approx \sin \theta$

Substitute $\tan \theta$ for $\sin \theta$ in equation (1).

$a\tan \theta =m\lambda$

Substitute $\frac{y}{L}$ for $\tan \theta$ in the above equation.

$a\left(\frac{y}{L}\right)=m\lambda$

For the case of central bright fringe, the order of the fringe is $1$.

Substitute $1$ for $m$ in the above equation.

$\begin{array}{l}a\left(\frac{y}{L}\right)=\left(1\right)\lambda \\ \end{array}$

For $m=1$, $y$ is the width of the central bright fringe.

Rearrange the above equation for $y$,

$y=\frac{\lambda L}{a}$

Thus from above equation the central bright fringe width is inversely proportional to the slit width. Thus, if the slit width decreases or half of the initial value the width of central bight fringe increases.

Conclusion:

The width of the central bright fringe is inversely proportional to the slit width so, if the slit width decreases the width of central bright fringe increases. Thus option (a) is correct.

The slit width is half of the initial value and there is inverse dependence of width of central maxima and slit width so decrease in slit width widens the central bright fringe. Thus option (b) is incorrect.

The width of the central bright fringe is inversely proportional; so decrease in slit width will increase width. Thus option (c) is incorrect.