Physics - Wave Optics Question with Solution | TestHub

The situation is shown in the figureThe path difference between the waves reaching the point $P$ is $\begin{array}{rl}\Delta p& =BP-AP\\ & ={\left(d^2+b^2\right)}^{1/2}-d=d{\left(1+\frac{b^2}{d^2}\right)}^{1/2}-d\\ & \approx d\left[1+\frac{1}{2}\frac{b^2}{d^2}\right]-d=\frac{b^2}{2d}\end{array}$ For a dark band $\Delta p=\frac{b^2}{2d}=(2n-1)\frac{\lambda }{2}$ where, $n=1,2,3$ or $\lambda =\frac{1}{(2n-1)}\cdot \frac{b^2}{d}$ Hence, the missing wavelengths are $\frac{b^2}{d},\frac{b^2}{3d},\frac{b^2}{5d},....$