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(A) The intensity in an interference pattern is given by $I = 4I_0 \cos^2(\frac{\phi}{2})$,where $\phi$ is the phase difference.At the central bright

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$2$

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(A) The intensity in an interference pattern is given by $I = 4I_0 \cos^2(\frac{\phi}{2})$,where $\phi$ is the phase difference.At the central bright fringe,the phase difference $\phi = 0$,so $I_1 = 4I_0 \cos^2(0) = 4I_0$.For a point at a distance $x = \frac{\beta}{4}$ from the centre,where $\beta$ is the fringe width,the phase difference $\phi$ is given by $\phi = \frac{2\pi}{\beta} \cdot x$.Substituting $x = \frac{\beta}{4}$,we get $\phi = \frac{2\pi}{\beta} \cdot \frac{\beta}{4} = \frac{\pi}{2}$.The intensity at this point is $I_2 = 4I_0 \cos^2(\frac{\pi/2}{2}) = 4I_0 \cos^2(\frac{\pi}{4}) = 4I_0 (\frac{1}{\sqrt{2}})^2 = 4I_0 \cdot \frac{1}{2} = 2I_0$.The ratio of the intensities is $\frac{I_1}{I_2} = \frac{4I_0}{2I_0} = 2$.

The ratio of the intensity at the centre of a bright fringe to the intensity at a point one-quarter of the fringe width from the centre is

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