Question

In a typical Young's double slit experiment a point source of monochromatic light is kept as shown in the figure. If the source is given an instantaneous velocity $v=1 \,mm$ per second towards the screen, then the instantaneous velocity of central maxima is given as $\alpha \times 10^{-\beta} cm / s$ upward in scientific notation. The value of $\alpha+\beta$ is ________.

Answer 1

The central maxima
$\frac{d y}{D}=\sqrt{d^{2}+x^{2}}-x$
$=x\left[1+\frac{d^{2}}{2 x^{2}}\right]-x=\frac{d^{2}}{2 x}$

$y=\frac{D d}{2 x} \Rightarrow \frac{d y}{d t}=-\frac{D d}{2 x^{2}}\left(\frac{d x}{d t}\right)$
$=\left(\frac{1 \times 0.01}{2 \times 0.5 \times 0.5}\right) \times(0.001)=0.02\, mm / s$
$\Rightarrow y=2 \times 10^{-3} cm / s$
$\Rightarrow \alpha+\beta=5$