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(C) The path difference $\Delta x$ at a distance $x$ from the center is given by $\Delta x = \frac{xd}{D}$.Since the fringe width $\beta = \frac{\lamb

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$I_0 \cos^2 \frac{\pi x}{\beta}$

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(C) The path difference $\Delta x$ at a distance $x$ from the center is given by $\Delta x = \frac{xd}{D}$.Since the fringe width $\beta = \frac{\lambda D}{d}$,we have $\frac{d}{D} = \frac{\lambda}{\beta}$.Thus,the path difference is $\Delta x = x \cdot \frac{\lambda}{\beta}$.The phase difference $\phi$ is given by $\phi = \frac{2\pi}{\lambda} \Delta x = \frac{2\pi}{\lambda} \cdot \frac{x\lambda}{\beta} = \frac{2\pi x}{\beta}$.The intensity $I$ at any point is given by $I = I_{max} \cos^2(\frac{\phi}{2})$,where $I_{max} = I_0$.Substituting $\phi$,we get $I = I_0 \cos^2(\frac{2\pi x / \beta}{2}) = I_0 \cos^2(\frac{\pi x}{\beta})$.

In Young's double-slit experiment,$I_0$ is the intensity of the central maximum and $\beta$ is the fringe width. The intensity at a point $P$ at a distance $x$ from the center is:

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