Intensity In Interference

When two coherent waves of equal intensity $I_0$ overlap, the resultant intensity varies with their phase difference $\varphi$ as $I=4I_0{\cos }^2\left(\frac{\varphi }{2}\right)$, and the maximum $I_{max}=4I_0$ occurs when $\varphi =0$. Here the phase difference is $\varphi =90°=\frac{\pi }{2}$, so $\frac{\varphi }{2}=45°$ and ${\cos }^2(45°)={\left(\frac{1}{\sqrt{2}}\right)}^2=\frac{1}{2}$. Therefore $I=4I_0\times \frac{1}{2}=2I_0=\frac{I_{max}}{2}$. It is worth noting that the resultant is not the simple sum $2I_0$ of the two individual intensities; interference redistributes energy, so the combined brightness depends on phase, ranging from $4I_0$ at full reinforcement down to zero at complete cancellation. The cosine-squared dependence guarantees energy conservation overall, since the excess brightness piled into the maxima is exactly balanced by the darkness of the minima across the pattern. The value $I_{max}/4$ is wrong because it uses ${\cos }^2(\varphi )$ rather than ${\cos }^2(\varphi /2)$. The value $I_{max}$ is wrong as it corresponds to zero phase difference, full constructive interference. The value $3I_{max}/4$ is wrong since it misapplies the half-angle and would need $\varphi =60°$. This NCERT intensity relation underlies the smooth cosine-squared brightness profile of fringes. A sanity check confirms a quarter-wave path difference gives exactly half the peak brightness, the natural midway point of the fringe.