Animated Solution for Physics - Electromagnetic Waves: An electromagnetic wave of frequency $3\text{ GHz}$ enters a dielectric medium of relative electric permittivity $2.25$ from vacuum. The wavelength of this wave in that medium will be ......... $\times 10^{-2}\text{ cm}$.

From Vacuum to Dielectric

Imagine you are running on a smooth, paved road, and suddenly you hit a stretch of thick, muddy water. What happens? Your speed drops instantly, and your strides become shorter, even though you are trying to maintain the same rhythm.

This is exactly what happens to an electromagnetic wave when it travels from the vacuum of space into a denser dielectric medium. Let's break down the physics behind this fascinating transition and solve the problem step-by-step.

Before the wave hits the medium, it is traveling through a vacuum. In a vacuum, all electromagnetic waves travel at the ultimate speed limit of the universe: the speed of light, denoted by $c$.

We are given that the frequency of the wave is $3\text{ GHz}$. Frequency is the "rhythm" of the wave—how many cycles pass a point per second.

The relationship between speed, frequency, and wavelength is one of the most fundamental equations in wave mechanics:

Let's substitute our known values to find the wavelength in the vacuum:

So, in the vacuum, the wave has a wavelength of $0.1\text{ meters}$.

The Medium's Resistance

Now, the wave enters the dielectric medium. A dielectric is an electrical insulator that can be polarized by an applied electric field. This polarization interacts with the electromagnetic wave, effectively slowing it down.

The factor by which the wave slows down is called the refractive index ($n$). The refractive index is determined by the electrical and magnetic properties of the medium, specifically its relative permittivity (${\epsilon }_r$) and relative permeability (${\mu }_r$).

The problem states that the medium is a dielectric, which implies it is non-magnetic. For non-magnetic materials, the relative permeability is essentially that of free space, so ${\mu }_r=1$. We are given the relative permittivity ${\epsilon }_r=2.25$.

Let's calculate the refractive index:

This means the wave travels $1.5$ times slower in this medium than it does in a vacuum.

Here is a crucial conceptual checkpoint: When a wave changes mediums, its frequency never changes. The source dictates the frequency, and the medium cannot alter it.

Since the speed ($v$) decreases by a factor of $n$, and the frequency ($f$) remains constant, the wavelength (${\lambda }_m$) must also decrease by the exact same factor to keep the equation $v=f{\lambda }_m$ balanced.

Let's find the new, "shrunk" wavelength:

Evaluating this fraction gives us:

We have the correct physical answer, but JEE questions often require the answer in a very specific format. We need to express our answer in the form of $.........\times 10^{-2}\text{ cm}$.

First, let's convert meters to centimeters by multiplying by $100$:

Now, we need to manipulate this into the $10^{-2}$ format. We can do this by multiplying and dividing by $100$:

Thus, the integer value that fills the blank is $667$.

This problem beautifully illustrates the interplay between the macroscopic properties of a wave (speed, wavelength) and the microscopic electromagnetic properties of the medium it travels through. Always remember: speed and wavelength adapt to the environment, but frequency remains loyal to the source!