Two coherent monochromatic light beams of intensities 4I and 9I are superimposed. The difference between the maximum and minimum intensities in the resulting interference pattern is $x \mathrm{I}$. The value of $x$ is___________ .

<p><strong>Calculate Maximum Intensity ($I_{\text{max}}$):</strong></p> <p>$ I_{\max} = \left(\sqrt{I_1} + \sqrt{I_2}\right)^2 $</p> <p>Substituting the values, we have:</p> <p>$ I_{\max} = \left(\sqrt{4I} + \sqrt{9I}\right)^2 $</p> <p>$ I_{\max} = \left(2\sqrt{I} + 3\sqrt{I}\right)^2 = \left(5\sqrt{I}\right)^2 = 25I $</p></p> <p><p><strong>Calculate Minimum Intensity ($I_{\text{min}}$):</strong></p> <p>$ I_{\min} = \left(\sqrt{I_1} - \sqrt{I_2}\right)^2 $</p> <p>Substituting the values, we have:</p> <p>$ I_{\min} = \left(\sqrt{4I} - \sqrt{9I}\right)^2 $</p> <p>$ I_{\min} = \left(2\sqrt{I} - 3\sqrt{I}\right)^2 = \left(-1\sqrt{I}\right)^2 = I $</p></p> <p><p><strong>Calculate the Difference Between Maximum and Minimum Intensities:</strong></p> <p>$ I_{\max} - I_{\min} = 25I - I = 24I $</p></p> <p>Thus, the value of $x$, which represents the difference between the maximum and minimum intensities, is $24$.</p>