An electromagnetic wave is given as E = 200 sin(1.5x – 4.5 × 10⁸t), here E is electric field in N/C. If energy density in electromagnetic field is given as N × 10^–8 J/m³. Then N is (ɛ₀ = 9 × 10^–12 SI units.)

The correct option is 18.

Step 1: Understand the given wave equation
The given electric field of the electromagnetic wave is:

$E=200\sin (1.5x-4.5\times {10}^{8}t)$

From this equation, we can identify the amplitude (maximum value) of the electric field, $E_0$:

$E_0=200\text{ N/C}$

Step 2: Formula for the average energy density of an electromagnetic wave
The average energy density ($u$) in the electromagnetic field is given by the formula:

$u=\frac{1}{2}{\epsilon }_0{E}_0^2$

where:
${\epsilon }_0$ is the permittivity of free space, given as $9\times {10}^{-12}\text{ SI units}$.
$E_0$ is the amplitude of the electric field, which is $200\text{ N/C}$.

Step 3: Calculate the average energy density
Substitute the given values into the average energy density formula:

$u=\frac{1}{2}\times (9\times {10}^{-12})\times {\left(200\right)}^2$

Simplify the square term:

${\left(200\right)}^2=40000=4\times {10}^4$

Now, plug this back into the equation for $u$:

$u=\frac{1}{2}\times 9\times {10}^{-12}\times 4\times {10}^4$

$u=2\times 9\times {10}^{-12+4}$

$u=18\times {10}^{-8}{\text{ J/m}}^3$

Step 4: Determine the value of N
We are given that the energy density is equal to $N\times {10}^{-8}{\text{ J/m}}^3$.
Comparing this with our calculated value:

$N\times {10}^{-8}=18\times {10}^{-8}$

This gives:

$N=18$