In the given electromagnetic wave $\mathrm{E}_{\mathrm{y}}=600 \sin (\omega t-\mathrm{kx}) \mathrm{Vm}^{-1}$, intensity of the associated light beam is (in $\mathrm{W} / \mathrm{m}^2$ : (Given $\epsilon_0=9 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$ )

<p>To find the intensity of the given electromagnetic wave, we need to use the formula for the intensity of an electromagnetic wave:</p> <p>$I = \frac{1}{2} \epsilon_0 c E_0^2$</p> <p>where:</p> <ul> <li>$I$ is the intensity in $\mathrm{W} / \mathrm{m}^2$</li> <li>$\epsilon_0$ is the permittivity of free space, given as $9 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$</li> <li>$c$ is the speed of light in vacuum, approximately $3 \times 10^8 \mathrm{~m/s}$</li> <li>$E_0$ is the peak electric field, given as $600 \mathrm{Vm}^{-1}$</li> </ul> <p>Now, substitute these values into the formula:</p> <p>$I = \frac{1}{2} \times 9 \times 10^{-12} \times 3 \times 10^8 \times (600)^2$</p> <p>Simplify the expression step-by-step:</p> <p>$I = \frac{1}{2} \times 9 \times 10^{-12} \times 3 \times 10^8 \times 360000$</p> <p>First, calculate $9 \times 3 \times 360000$:</p> <p>$I = \frac{1}{2} \times 9.72 \times 10^{-4} \times 360000$</p> <p>Combine 9 and 3 into 27, giving you:</p> <p>$I = 13.5 \times 10^{-4} \times 360000$</p> <p>Then, calculate the multiplication:</p> <p>$I = 13.5 \times 36$</p> <p>Finally, multiply the remaining values:</p> <p>$I = 486 \times 10^{-4}$</p> <p>The final value is:</p> <p>The intensity, $I$, is 486 $\mathrm{W} / \mathrm{m}^2$. Therefore, the correct option is:</p> <p>Option A: 486</p>