Animated Solution for Physics - Optics: A monochromatic beam of light of wavelength $6000\text{ A˚}$ in vacuum enters a medium of refractive index $1.5$. In the medium its wavelength is ......, and its frequency is ......

Imagine a light wave traveling through the emptiness of a vacuum and suddenly hitting a denser medium, like a block of glass. What happens to its properties? This is a classic scenario in optics that tests our understanding of the fundamental nature of waves.

When light enters a denser medium, it slows down. This is a direct consequence of the medium's optical density, quantified by its refractive index, $\mu$. However, the frequency of the light—the number of wave cycles passing a point per second—is a fundamental property of the source emitting the light. It acts like the heartbeat of the wave and does not change when crossing boundaries.

Because the speed decreases but the frequency remains constant, the waves get "bunched up" or compressed. The new wavelength in the medium is simply the original wavelength in a vacuum divided by the refractive index:

Let's plug in the numbers from our problem. The original wavelength is $6000\text{ A˚}$, and the refractive index is $1.5$:

So, the wavelength in the medium is compressed to $4000\text{ A˚}$.

Now, what about the frequency? As we discussed, frequency remains constant across boundaries. We can calculate it using the speed of light in a vacuum ($c$) and the original wavelength in a vacuum (${\lambda }_{\text{vacuum}}$), using the universal wave equation $c=f\lambda$:

The speed of light is $3.0\times 10^8\text{ m/s}$. The wavelength is $6000\text{ A˚}$, which we must convert to meters. Since $1\text{ A˚}=10^{-10}\text{ m}$, the wavelength is $6000\times 10^{-10}\text{ m}$, or $6.0\times 10^{-7}\text{ m}$.

Dividing the numbers gives us:

Always remember this golden rule of wave mechanics: Frequency is the heartbeat of the wave, set by the source. It never changes when crossing boundaries. However, the speed and wavelength are properties dictated by the medium, and they will change proportionally to keep the frequency constant.