JEE MAIN - Physics (2024 - 6th April Evening Shift - No. 20)

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 $$\epsilon_0=9 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$$ )
486
729
243
972

Explanation

To find the intensity of the given electromagnetic wave, we need to use the formula for the intensity of an electromagnetic wave:

$$ I = \frac{1}{2} \epsilon_0 c E_0^2 $$

where:

  • $$I$$ is the intensity in $$\mathrm{W} / \mathrm{m}^2$$
  • $$\epsilon_0$$ is the permittivity of free space, given as $$9 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$$
  • $$c$$ is the speed of light in vacuum, approximately $$3 \times 10^8 \mathrm{~m/s}$$
  • $$E_0$$ is the peak electric field, given as $$600 \mathrm{Vm}^{-1}$$

Now, substitute these values into the formula:

$$ I = \frac{1}{2} \times 9 \times 10^{-12} \times 3 \times 10^8 \times (600)^2 $$

Simplify the expression step-by-step:

$$ I = \frac{1}{2} \times 9 \times 10^{-12} \times 3 \times 10^8 \times 360000 $$

First, calculate $$9 \times 3 \times 360000$$:

$$ I = \frac{1}{2} \times 9.72 \times 10^{-4} \times 360000 $$

Combine 9 and 3 into 27, giving you:

$$ I = 13.5 \times 10^{-4} \times 360000 $$

Then, calculate the multiplication:

$$ I = 13.5 \times 36 $$

Finally, multiply the remaining values:

$$ I = 486 \times 10^{-4} $$

The final value is:

The intensity, $$I$$, is 486 $$\mathrm{W} / \mathrm{m}^2$$. Therefore, the correct option is:

Option A: 486

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