A double-slit system with individual slit widths of $\mathbf{\text{0.030\hspace{0.17em}mm}}$and a slit separation of localid="1663157041233" $\mathbf{\text{0.18\hspace{0.17em}mm}}$ is illuminated with localid="1664276472239" $\mathbf{\text{500\hspace{0.17em}nm}}$light directed perpendicular to the plane of the slits. What is the total number of complete bright fringes appearing between the two first-order minima of the diffraction pattern? (Do not count the fringes that coincide with the minima of the diffraction pattern.)

The wavelength of light, $\lambda =\text{500\hspace{0.17em}nm}$

The slit width, $a=\text{0.030\hspace{0.17em}mm}$

The slit separation,$d=\text{0.18\hspace{0.17em}mm}$

In a single-slit diffraction pattern, the central maximum is larger than the maxima on either side and that the intensity decreases rapidly on either side. In contrast, light passing through the double slit produces evenly spaced lines that slowly darken on either side of the center.

Upon passing waves through a long narrow slit of width $a$, diffraction patterns are produced on a viewing screen containing a central maximum and another maximum separated by minima lying at angles $\theta$to the central axis.

$a\sin \theta =n\lambda$for $n=1,2,3,...$(minima)

Where, $a$is the slit width and $\lambda$is the wavelength of the light.

The schematic of diffraction pattern produced by a single slit is shown below.

From this you can understand that the central peak lies between two minima that is $-{\theta }_1<{\theta }_{\max }<+{\theta }_1$.

Where, ${\theta }_{\max }$is angle of central maxima (which is not zero degrees as shown in the graph above. It is just schematic diagram.). And ${\theta }_1$is the angle of first order minima.

In double slit experiment, each slit produces a diffraction pattern like the above pattern but the pattern seen on the screen is the interference of two diffraction pattern of the slits. The angle at which bright fringes are observed can be expressed as,

$d\sin {\theta }_{\max }=m\lambda$

The interference of two diffraction patterns produces number of bright and dark fringes. In this problem you are asked to determine the number of bright fringes which are enveloped in the central maxima. As we know from the above step, the central set of bright fringes span between $-{\theta }_1<{\theta }_{\max }<+{\theta }_1$.

The angle of central maxima can be written as,

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

The first minima $\left(m=1\right)$is observed at an angle,

${\theta }_1={\sin }^{-1}\left(\frac{\lambda }{a}\right)$

Therefore,

$$\begin{gathered} -{\sin }^{-1}\left(\frac{\lambda }{a}\right)<{\sin }^{-1}\left(\frac{m\lambda }{d}\right)<+{\sin }^{-1}\left(\frac{\lambda }{a}\right) \\ -\left(\frac{\lambda }{a}\right)<\left(\frac{m\lambda }{d}\right)<+\left(\frac{\lambda }{a}\right) \\ -\left(\frac{1}{a}\right)<\left(\frac{m}{d}\right)<+\left(\frac{1}{a}\right) \\ -\left(\frac{d}{a}\right)<m<+\left(\frac{d}{a}\right) \end{gathered}$$

Inserting the values of and in above expression, you get,

$$\begin{gathered} -\left(\frac{0.18{\displaystyle \text{mm}}}{0.030{\displaystyle \text{mm}}}\right)<m<+\left(\frac{0.18\text{\hspace{0.17em}mm}}{0.030\text{\hspace{0.17em}mm}}\right) \\ -6<m<+6 \end{gathered}$$

As $m$is an integer, all the possible values of $m$that satisfy above relation is,

$m=0,\pm 1,\pm 2,\pm 3,\pm 4,\pm 5$

Therefore, in total, $11$values of $m$is obtained. Hence, eleven bright fringes are observed which are enveloped between two first order minima.

A generic pattern of double slit pattern is shown below