Animated Solution for Physics - Electromagnetic Waves: An electromagnetic wave of frequency $5\text{ GHz}$, is travelling in a medium whose relative electric permittivity and relative magnetic permeability both are 2. Its velocity in this medium is ...... $\times 10^7\text{ m/s}$.

Imagine an electromagnetic wave cruising through the vast emptiness of a vacuum. It travels at the ultimate cosmic speed limit, the speed of light, denoted by $c$, which is exactly $3\times 10^8\text{ m/s}$. But what happens when this wave encounters a physical medium?

When an electromagnetic wave enters a medium, it interacts with the atoms and molecules within it. The electric and magnetic fields of the wave induce tiny oscillations in the medium's charges, which in turn produce their own fields. This complex interaction effectively slows down the propagation of the wave. The new speed, $v$, is always less than $c$.

To quantify how much a medium slows down an electromagnetic wave, we use a dimensionless number called the refractive index, denoted by $n$. The relationship is beautifully simple:

But where does $n$ come from? James Clerk Maxwell, in his elegant formulation of electromagnetism, showed that the speed of an EM wave is fundamentally tied to two intrinsic properties of the space it travels through: its ability to store electrical energy (permittivity, $ϵ$) and its ability to store magnetic energy (permeability, $\mu$).

For any medium, the refractive index is given by the square root of the product of its relative permittivity (${ϵ}_r$) and relative permeability (${\mu }_r$):

In our specific problem, we are given a hypothetical medium where both the relative electric permittivity and the relative magnetic permeability are equal to 2.

Let's substitute these values into Maxwell's relation:

This tells us that the refractive index of this medium is exactly 2. Consequently, the wave will travel at half its vacuum speed. Let's calculate this new speed $v$:

We have the speed, but JEE questions often test your attention to detail regarding units and formats. The question asks for the velocity in the specific format of $x\times 10^7\text{ m/s}$.

We need to mathematically massage our answer to match this format. By shifting the decimal point one place to the right, we must decrease the exponent of 10 by one:

Comparing this to the requested format $x\times 10^7\text{ m/s}$, it is crystal clear that:

And there we have it! A beautiful intersection of Maxwell's electromagnetic theory and careful algebraic formatting.