An electromagnetic wave of frequency v = 3.0 MHz passes from vacuum into a dielectric medium with pe

Q: Why doesn't the frequency change when the wave enters a new medium?

Frequency is the number of oscillations per second. It is determined entirely by the source that generated the wave. The medium can only affect how fast the wave travels and how squished the wave becomes, but it cannot create or destroy wave cycles.

Q: What is the difference between permittivity $\varepsilon$ and relative permittivity $\varepsilon_r$?

Permittivity $\varepsilon$ is the absolute measure of how a medium resists the formation of an electric field. Relative permittivity $\varepsilon_r$ (also called dielectric constant) is the ratio of the medium's permittivity to the permittivity of free space ($\varepsilon_0$). In many textbook problems, when they say 'permittivity is 4.0', they mean the relative permittivity $\varepsilon_r = 4.0$.

Q: Does the amplitude of the wave change?

Yes, typically when a wave enters a new medium, some of it is reflected and some is transmitted. The transmitted wave will generally have a different amplitude than the incident wave due to the conservation of energy and the change in the medium's impedance. However, this question only asks about frequency and wavelength.

Q: What happens to the speed of the wave?

The speed of the wave decreases. In a vacuum, it travels at $c$. In a medium with refractive index $\mu = 2$, it travels at $v = c/2$.

Q: Is this true for sound waves as well?

The principle that frequency remains constant across boundaries is true for all waves, including sound. However, unlike light, sound waves typically travel *faster* in denser media (like water or steel) compared to air, so their wavelength would *increase*.

Question

An electromagnetic wave of frequency $
u = 3.0 	ext{ MHz}$ passes from vacuum into a dielectric medium with permittivity $\varepsilon = 4.0$. Then,

Options:

(A) wavelength is doubled and the frequency remains unchanged

(B) wavelength is doubled and frequency becomes half

(C) wavelength is halved and frequency remains unchanged

(D) wavelength and frequency both remain unchanged

Sigma Percentile · Accepted Answer

### The Unchanging Heartbeat of a Wave

Imagine an electromagnetic wave as a runner sprinting across different terrains. When the wave moves from the empty expanse of a vacuum into a dense dielectric medium, it experiences a sudden shift in its environment. But amidst this change, one fundamental property remains absolutely untouched: its **frequency**.

Why is that? Frequency is the "heartbeat" of the wave. It is determined entirely by the source that originally created the wave, not by the medium it travels through. Whether the wave is cruising through empty space or struggling through thick glass, the number of oscillations it completes per second stays exactly the same. So, right off the bat, we know our frequency remains unchanged at $3.0 	ext{ MHz}$.

### The Speed Limit of the Medium

Now, while the frequency is stubborn, the speed of the wave is highly adaptable. When the wave enters the dielectric medium, it interacts with the atoms inside, which slows it down. We quantify this slowdown using the **refractive index** ($\mu$).

For an electromagnetic wave, the refractive index is intimately tied to the electrical and magnetic properties of the medium. Specifically, it is the square root of the product of the medium's relative permittivity ($\varepsilon_r$) and relative permeability ($\mu_r$). Since we are dealing with a standard non-magnetic dielectric, its relative permeability is just $1$. This simplifies our life immensely! The refractive index becomes just the square root of the relative permittivity:

$$ \mu = \sqrt{\varepsilon_r} $$

The problem tells us the permittivity of the medium is $4.0$. Plugging this in, we find:

$$ \mu = \sqrt{4.0} = 2.0 $$

This means the wave travels exactly twice as slow in this medium compared to a vacuum!

### The Wavelength Squeeze

So, the wave is moving slower, but its frequency (its heartbeat) is still the same. How does the wave physically adjust to this? It has to squeeze its steps!

We know the universal wave equation relates speed ($v$), frequency ($
u$), and wavelength ($\lambda$):

$$ v = 
u \lambda $$

In the vacuum, the wave travels at the speed of light ($c$), so $c = 
u \lambda$. In the new medium, its speed is halved ($v = c/2$). Since the frequency $
u$ cannot change, the only way for the math to balance is if the wavelength also halves.

$$ \lambda' = \frac{v}{
u} = \frac{c/2}{
u} = \frac{\lambda}{2} $$

The physical distance between the wave crests is compressed to exactly half of what it was in the vacuum.

### The Final Verdict

Putting it all together, the journey into the dielectric medium forces the wave to slow down and compress its wavelength by a factor of two, while its frequency remains completely unfazed. Therefore, the **wavelength is halved and the frequency remains unchanged**. This elegant interplay of wave properties leads us straight to the correct answer!

Animated Solution for Physics - Electromagnetic Waves: An electromagnetic wave of frequency $ν = 3.0 MHz$ passes from vacuum into a dielectric medium with permittivity $ε = 4.0$ . Then,

Select Answer:

Visualized Solution

$Frequency Invariance$

$Refractive Index$

$Calculating μ$

$Wavelength in Medium$

$Final Result$

The Sigma Insight: Characteristics of Electromagnetic Waves

The Unchanging Heartbeat of a Wave

The Speed Limit of the Medium

The Wavelength Squeeze

The Final Verdict

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