Animated Solution for Physics - Electromagnetic Waves: An electromagnetic wave of intensity $50{\text{ Wm}}^{-2}$ enters in a medium of refractive index '$n$' without any loss. The ratio of the magnitudes of electric fields and the ratio of the magnitudes of magnetic fields of the wave before and after entering into the medium are respectively, given by

Imagine an electromagnetic wave traveling through the vast emptiness of space. It carries energy, oscillating with electric and magnetic fields. Suddenly, it hits a transparent medium—like a thick slab of glass. The problem tells us a crucial piece of information: the wave enters the medium without any loss. This means the energy flowing through a unit area per second—the intensity $I$—remains perfectly constant.

Before we dive into the math, let's understand what happens physically. When the wave enters a medium with a refractive index $n$, it slows down. The new velocity $v$ is given by $v=\frac{c}{n}$.

But what causes this slowdown? It's the electrical properties of the medium. For most transparent materials, the magnetic permeability $\mu$ is practically identical to the vacuum permeability ${\mu }_0$. However, the electrical permittivity $\epsilon$ changes significantly. Since $n=\sqrt{\frac{\mu \epsilon }{{\mu }_0{\epsilon }_0}}$, assuming $\mu \approx {\mu }_0$, we get a beautiful relation: $\epsilon =n^2{\epsilon }_0$. The medium is much more 'permissive' to electric fields.

The intensity of an electromagnetic wave can be expressed entirely in terms of its electric field amplitude. In a vacuum, the intensity is $I=\frac{1}{2}{\epsilon }_0{E}_0^{2}c$. Inside the medium, it becomes $I^{\prime }=\frac{1}{2}\epsilon E^{2}v$.

Because energy is conserved ($I=I^{\prime }$), we can equate these two expressions:

Now, we substitute our physical insights: $\epsilon =n^2{\epsilon }_0$ and $v=\frac{c}{n}$.

Notice how elegantly the terms cancel out. The ${\epsilon }_0$ and $c$ vanish from both sides, leaving us with:

Taking the square root, we find the ratio of the electric fields:

This tells us that the electric field amplitude actually decreases inside the medium ($E=\frac{E_0}{\sqrt{n}}$). Because the medium has a higher permittivity, it takes less electric field strength to store the same amount of energy.

We can perform a similar analysis for the magnetic field. The intensity can also be written as $I=\frac{B_0^2}{2{\mu }_0}c$ in vacuum, and $I^{\prime }=\frac{B^2}{2\mu }v$ in the medium.

Equating them and assuming $\mu \approx {\mu }_0$:

Again, the speed of light $c$ cancels out:

Taking the square root gives us the ratio of the magnetic fields:

Fascinatingly, the magnetic field amplitude increases inside the medium ($B=B_0\sqrt{n}$). As the wave slows down, the energy gets spatially compressed. Since the magnetic permeability didn't change to help store this compressed energy, the magnetic field itself must grow stronger to carry its share of the constant intensity.

Combining our results, the ratios are $\left(\sqrt{n},\frac{1}{\sqrt{n}}\right)$, which perfectly matches option (d).