Animated Solution for Physics - Electromagnetic Waves: Intensity of sunlight is observed as $0.092{\text{ Wm}}^{-2}$ at a point in free space. What will be the peak value of magnetic field at that point? (${\epsilon }_0=8.85\times 10^{-12}{\text{ C}}^2{\text{N}}^{-1}{\text{m}}^{-2}$)

Finding the Magnetic Field from Sunlight's Intensity

Imagine sunlight streaming through the vast emptiness of space. It's not just light; it's a dynamic, oscillating electromagnetic wave carrying energy across the cosmos. We are given the intensity of this sunlight, which is exactly $0.092{\text{ W/m}}^2$. This means that every second, $0.092$ Joules of energy crosses a one-square-meter area perpendicular to the wave. Our mission is to uncover the peak magnetic field hidden within this wave.

How is the intensity of an electromagnetic wave related to its fields? The intensity $I$ is given by the formula:

Here, ${\epsilon }_0$ is the permittivity of free space, $c$ is the speed of light, and $E_0$ is the peak electric field. But wait, we need the magnetic field! Remember that the electric and magnetic fields in an EM wave are intimately connected. The peak electric field is simply the speed of light times the peak magnetic field:

Let's substitute this relationship directly into our intensity equation. This gives us the intensity entirely in terms of the magnetic field:

Rearranging this, we get a beautiful master equation for the peak magnetic field:

Now, let's carefully plug in the numbers. We have $2\times 0.092$ in the numerator. In the denominator, we have ${\epsilon }_0$, which is $8.85\times 10^{-12}$, multiplied by the cube of the speed of light, $(3\times 10^8{)}^3$, which is $27\times 10^{24}$.

Let's simplify the denominator first. Multiplying $8.85$ by $27$ gives $238.95$. Combining the powers of ten ($-12$ and $24$) gives $10^{12}$. Dividing $0.184$ by this massive number yields $0.00077\times 10^{-12}$.

We can rewrite this elegantly to make taking the square root straightforward:

The square root of $10^{-16}$ is $10^{-8}$, and the square root of $7.7$ is approximately $2.77$. So, our peak magnetic field is:

Did you notice how incredibly small the magnetic field is compared to the electric field? It's smaller by a factor of the speed of light! Yet, fascinatingly, both fields carry exactly the same amount of energy in an electromagnetic wave. The energy density of the electric field ($\frac{1}{2}{\epsilon }_0{E}^2$) is perfectly equal to the energy density of the magnetic field ($\frac{B^2}{2{\mu }_0}$). They share the wave's energy perfectly 50-50, a beautiful symmetry in the laws of physics.