Animated Solution for Physics - Optics: A light wave of frequency $5\times 10^{14}\text{ Hz}$ enters a medium of refractive index $1.5$. In the medium the velocity of the light wave is ...... and its wavelength is ......

Imagine a light wave traveling freely through the vast emptiness of a vacuum. In this state, it moves at the ultimate speed limit of the universe, $c=3\times 10^8\text{ m/s}$. But what happens when this wave suddenly encounters a denser medium, like a block of glass or a pool of water?

When light enters a medium with a refractive index $\mu >1$, it experiences a "slowdown." The atoms in the medium interact with the electromagnetic wave, causing its phase velocity to decrease.

The relationship between the speed of light in a vacuum ($c$), the speed of light in the medium ($v$), and the refractive index ($\mu$) is beautifully simple:

In our problem, the refractive index of the medium is given as $\mu =1.5$. Let's substitute the known values into our master equation:

Dividing $3$ by $1.5$ gives us exactly $2$. Therefore, the new velocity of the light wave inside the medium is:

Now, we need to find the new wavelength. But before we do, we must understand a profound physical truth: the frequency of a wave is its fundamental identity.

Think of frequency as the "heartbeat" of the wave, determined entirely by the source that created it. Whether the wave is traveling through empty space, water, or diamond, its frequency ($f$) remains absolutely constant. In this case, $f=5\times 10^{14}\text{ Hz}$.

Since the velocity has decreased but the frequency remains constant, something else must give. That "something" is the wavelength ($\lambda$). The wave equation connects these three properties:

Rearranging this to solve for wavelength, we get:

Let's substitute the velocity we just found and the constant frequency:

Let's break down the math. First, divide the coefficients: $2/5=0.4$. Next, divide the powers of ten: $10^8/10^{14}=10^{-6}$.

To write this in standard scientific notation, we shift the decimal point one place to the right, which decreases the exponent by one:

And there we have it! By understanding how a medium affects velocity and wavelength while leaving frequency untouched, we've successfully decoded the behavior of the light wave.