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(A) The intensity $I$ of a wave is proportional to the square of its amplitude $a$,i.e.,$I \propto a^{2}$.This implies that the amplitude $a$ is propo

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$(\sqrt{I_{1}}+\sqrt{I_{2}})^{2}$

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(A) The intensity $I$ of a wave is proportional to the square of its amplitude $a$,i.e.,$I \propto a^{2}$.This implies that the amplitude $a$ is proportional to the square root of the intensity,i.e.,$a \propto \sqrt{I}$.Let the amplitudes of the two coherent sources be $a_{1}$ and $a_{2}$,where $a_{1} \propto \sqrt{I_{1}}$ and $a_{2} \propto \sqrt{I_{2}}$.In an interference pattern,the maximum intensity $(I_{\max})$ occurs at points of constructive interference,where the phase difference is an even multiple of $\pi$ $(2n\pi)$.At these points,the resultant amplitude is the sum of the individual amplitudes: $A_{\max} = a_{1} + a_{2}$.Since $I_{\max} \propto A_{\max}^{2}$,we have $I_{\max} \propto (a_{1} + a_{2})^{2}$.Substituting the expressions for $a_{1}$ and $a_{2}$,we get $I_{\max} = (\sqrt{I_{1}} + \sqrt{I_{2}})^{2}$.

Two coherent sources of intensities $I_{1}$ and $I_{2}$ produce an interference pattern on a screen. The maximum intensity in the interference pattern is

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