In Young's double slit experiment, the fringe width is $\beta = \frac{\lambda D}{d}$ where $d$ is the slit separation.

Here, the slit separation varies with time as $d(t)=d_0+a_0\sin\omega t$ Since $\sin\omega t$ varies between $-1$ and $+1$, the slit separation varies between: $d_{\min}=d_0-a_0$ $d_{\max}=d_0+a_0$

Since fringe width $\beta$ is inversely proportional to $d$, we get:

Therefore, the required difference is \beta_{\max}-\beta_{\min}=rac{\lambda D}{d_0-a_0}-\frac{\lambda D}{d_0+a_0} Taking LCM: $=\lambda D\left(\frac{(d_0+a_0)-(d_0-a_0)}{(d_0-a_0)(d_0+a_0)}\right)$ $=\lambda D\left(\frac{2a_0}{d_0^2-a_0^2}\right)$ $=\frac{2\lambda D a_0}{d_0^2-a_0^2}$

Matching with the options, this is Option B.

Comparison with stored correct answer: Stored correct answer = B, which matches our derived result.