Chapter 36 - Diffraction - Problems - Page 1111: 35

$5$

The angular locations of the diffraction minima are given by $a\sin\theta=m\lambda$ For first diffraction minima or, $\sin\theta=\frac{\lambda}{a}\;.........(1)$ The angular locations of the bright fringes of the double-slit interference pattern are given by $d\sin\theta=m\lambda$ Substituting eq. 1, we obtain $d\frac{\lambda}{a}=m\lambda$ or, $m=\frac{d}{a}\;$ According to the question, the central diffraction envelope of a double-slit diffraction pattern contains 11 bright fringes which includes the central bright fringe as well as the bright fringes on the both side of the central fringe. Thus the value of m in eq. 1 is 5 So, $\frac{d}{a}=5$ For second diffraction minima or, $\sin\theta=\frac{2\lambda}{a}\;.........(2)$ The angular locations of the bright fringes of the double-slit interference pattern are given by $d\sin\theta=m\lambda$ Substituting eq. 2, we obtain $d\frac{2\lambda}{a}=m\lambda$ or, $m=\frac{2d}{a}$ or, $m=2\times5=10$ Thus, the fringes having m=6,7,8,9,10 lie between the first and second minima of the diffraction envelope. Therefore, there are 5 bright fringes between the first and second minima of the diffraction envelop.