Animated Solution for Physics - Electromagnetic Waves: The relative permittivity of distilled water is $81$. The velocity of light in it will be (Take, ${\mu }_r=1$)

Have you ever wondered why a straw looks bent in a glass of water? Or why light, the fastest entity in the universe, suddenly decides to take a leisurely stroll when it plunges into a pool? The answer lies in the invisible electromagnetic fabric of the medium itself. In this problem, we are going to uncover the mathematical machinery that dictates exactly how much light slows down when it enters distilled water.

In the absolute emptiness of a vacuum, light travels at its maximum possible speed, denoted by the famous constant $c$. This speed is approximately $3\times 10^8\text{ m/s}$. It is the ultimate cosmic speed limit.

But what happens when light is no longer in a vacuum? What happens when it enters a tangible medium like glass, diamond, or, in our case, distilled water?

Light is an electromagnetic wave. It consists of oscillating electric and magnetic fields that perpetually regenerate each other. When this wave enters a medium, these fields interact with the atoms and molecules of that medium. The electrons in the water molecules start to jiggle in response to the incoming electric field. This jiggling creates secondary electromagnetic waves that interfere with the original wave. The net macroscopic effect of this complex microscopic dance is that the wave's phase velocity decreases. Light slows down.

Permittivity and Permeability

To quantify how much a medium slows down light, we rely on two fundamental properties of the material: its permittivity and its permeability.

Permittivity ($\epsilon$) is a measure of how much a material resists the formation of an electric field within it. Think of it as the "electrical sluggishness" of the medium. The higher the permittivity, the more the medium dampens the electric field of the light wave. In our problem, we are given the relative permittivity (${\epsilon }_r$) of distilled water, which is $81$. This means water is $81$ times more resistant to electric fields than a perfect vacuum.

Permeability ($\mu$), on the other hand, measures how much a material responds to a magnetic field. It is the "magnetic sluggishness." Most transparent dielectric materials, including water, are non-magnetic. Their response to magnetic fields is practically identical to that of a vacuum. Therefore, the relative permeability (${\mu }_r$) of water is taken to be exactly $1$.

Maxwell's Revelation

James Clerk Maxwell, the brilliant physicist who unified electricity and magnetism, gave us the master equation that connects the speed of light to these two properties. He discovered that the speed of light in any medium, $v$, is inversely proportional to the square root of the product of its permittivity and permeability.

When we express this in terms of relative permittivity and relative permeability, the formula beautifully simplifies to:

This equation is profound. It tells us that the optical properties of a material are entirely dictated by its electrical and magnetic properties. Optics is just electromagnetism in disguise!

Now, let's bring our specific values into this master equation. We are tasked with finding the velocity of light in distilled water.

We know the speed of light in a vacuum is $c=3\times 10^8\text{ m/s}$.

We are given the relative permittivity of water, ${\epsilon }_r=81$.

We know the relative permeability of water is ${\mu }_r=1$.

Let's substitute these values into our formula:

The math here is incredibly friendly. The denominator is simply the square root of $81$.

Now, we just need to simplify the fraction $\frac{3}{9}$.

We know that $\frac{1}{3}$ is the repeating decimal $\mathrm{0.333...}$

To express this in standard scientific notation, we shift the decimal point one place to the right, which decreases the exponent of $10$ by one.

And there we have it! The speed of light in distilled water is $3.33\times 10^7\text{ m/s}$.

Notice how significantly the light has slowed down. It is traveling at exactly one-third of its speed in a vacuum. This massive reduction is primarily due to the unusually high relative permittivity of water (${\epsilon }_r=81$), which is a consequence of the highly polar nature of water molecules.

This result also gives us a direct way to calculate the refractive index ($n$) of water. The refractive index is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium ($n=\frac{c}{v}$).

Looking at our master equation, we can see that $n=\sqrt{{\epsilon }_r{\mu }_r}$.

For water, $n=\sqrt{81\times 1}=9$.

Wait, is the refractive index of water $9$? No, the static relative permittivity of water is $81$ (at low frequencies or DC). At optical frequencies (the frequency of visible light), the relative permittivity of water is much lower, around $1.77$, which gives the familiar refractive index of $n\approx 1.33$.

This problem uses the static relative permittivity value ($81$) to test your understanding of the formula, even though physically, light waves oscillate too fast for the water dipoles to fully align, meaning the effective permittivity at optical frequencies is much lower. It's a classic theoretical physics problem designed to test your algebraic and conceptual grasp of Maxwell's relations!

Always remember to trust the math, but keep a critical eye on the physical reality it represents.