Does frequency change when light travels from one medium to another

Here is the bookwork answer.

Consider a boundary between two media to be the plane $y=0$. Draw a rectangular loop of side $\delta x$ and $\delta y$. Have an E-field either side of the boundary that is parallel to the boundary in the $x$ direction. The E-field is $E_1$ in medium 1 and $E_2$ in medium 2.

Now use the integral form of Faraday's law. $$ \oint {\bf E} \cdot d{\bf l} = - \int \frac{\partial {\bf B}}{\partial t} \cdot d{\bf S}$$ $$ E_1 \delta x - E_2 \delta x = -\frac{\partial {\bf B}}{\partial t} \delta x \delta y.$$ But now you can let $\delta y$ shrink to zero and you find that $E_2 = E_1$. i.e. the component of E-field that is parallel to the interface must be the same immediately either side of the boundary.

Now have the boundary be defined by the plane $y=0$, the point of incidence be ${\bf r}=0$ and have an incident wave approach it of the form $E = E_i \exp[i({\omega_i t - \bf k_i}\cdot {\bf r})] \hat{\bf k}\times \hat{\bf r}$, where $\hat{\bf k}$ is a unit vector in the direction of the wave-vector ${\bf k_i}$, and $\omega_i$ is the angular frequency.

The incident wave impacts at ${\bf r}=0$ and some of the light is transmitted and some reflected. The incident, reflected and transmitted rays are all in the same plane and because, as shown above, the parallel components must be the same either side of the boundary we can write. $$E_i \exp(i\omega_i t) \cos \theta_i + E_r \exp(i\omega_r t)\cos \theta_r = E_t \exp(i\omega_t t)\cos\theta_t,$$ where $\theta_i$ etc are the angles of incidence, reflection, transmission; and $\omega_r$ and $\omega_t$ are the frequencies of the reflected and transmitted waves.

But this relationship has to be true for all values of $t$. The only way this can be arranged is if $\omega_i = \omega_r = \omega_t$. So the frequency of the light is unchanged as it passes into the medium.

I have taken a shortcut here to get to the required result. Usually, when doing this proof you define a geometry so that the wave hits at various points along the interface and then this means that the arguments of the exponentials look like $(\omega_i t -k_i x\sin\theta_i)$, $(\omega_r t -k_rx\sin\theta_r)$ and $(\omega_t t -k_tx\sin\theta_t)$, where $x$ is a coordinate along the boundary. Demanding that these arguments are equal for all $x,t$ also gives you the law of reflection ($\theta_i = \theta_r$) and Snell's refraction law; $\sin \theta_t/\sin\theta_i = k_i/k_t$, and if $\omega_t = \omega_i$ and $\omega/k = c/n$, then $\sin \theta_t/\sin\theta_i = n_i/n_t$.

The frequency of light is associated with the number of light waves travelling from a fixed point in a unit second. So light of any wavelength possesses a certain frequency that corresponds to it.

Light has the ability to change its physical properties while it traveling through different mediums, and undergoes certain process which causes change in their properties. Frequency is also a property of the light wave. Does frequency of light change with medium? This is one of the familiar questions we are going to discuss in this post.

While talking about change in medium, light processes such as refraction and diffraction are more convenient to converse about change with the medium of propagation. In both cases, the light propagates from one medium to another medium of different indices.

Does frequency of light change with medium?

We knew that frequency is inversely related to wavelength. When white light is made to strike on the glass or on a prism, there is a change in the medium since light is propagating from air medium to prism whose indices is different from one another.

The prism has the ability to refract the incident white light wave, which can disperse into prominent colours of a different wavelength. So we can observe the change in the wavelength of the incident white light when it hits the other medium, and the velocity of the incident light also changes, but the frequency does not change. The frequency of the incident light remains as before.

When the light has to travel from one medium to another medium, it has to deviate either due to refraction or due to diffraction or interference. Consider the example of the refraction of white light on the prism. The refractive index of the prism is slightly greater than the air; thus, bending of the light takes place, causing the refraction. The refractive image we obtain is the band of colors of different wavelengths called dispersion.

The position of the dispersed colors on the spectrum is due to a change in velocity and wavelength. The color in the band whose velocity is less will accumulate the least position, consequently the wavelength is also less so looks dimmer. Thus the velocity also changed with the medium. The band of colors together gives the visual spectrum; the frequency of the whole spectrum remains the same because it is an entity decided by the source collectively.

Thus even though velocity and wavelength change, the frequency does not change with the medium; it remains constant.

In other words, the wavelength decreases as the velocity decreases proportionately when it travels between two different mediums. If we take the ratio of wavelength and velocity, it will be constant. The ratio of wavelength and velocity of light is nothing but frequency. It is evident that the frequency does not change with the medium.

Why does frequency of light remain constant?

The light has the ability to deviate when it passes through a medium of different refractive indices. According to their refractive indices, as denser or rarer, the light speed either slows down or becomes fast.

The light corresponds to a packet of energy as photons; thus, every property of light is also associated with energy. The energy of the light is constant in every medium because the magnitude of the force that causes the oscillation of the photon remains constant. This oscillation is related to the amplitude of the light, which is one of the frequency entities. Hence the frequency of the light also remains constant.

The constant frequency of the light can be mathematically described using the expression,

E = hν

Where E is the energy of the light, h is Plank’s constant and ν is the frequency of the light.

Since the energy of the light is termed as constant and there is another constant factor; thus the frequency is also constant.

The light is quantized; this means that it cannot lose or gain the energy that gives the constant frequency in every medium.

Frequently Asked Questions

Does the frequency of the sound change with the medium?

Frequency is the term that never changes in any medium as it corresponds to energy.

When the sound wave travels from a denser medium to a rarer medium, we can observe a decrease in the wavelength; thus, audibility may change, but frequency depends only on the source of the sound, not on the medium; thus it remains constant.

Is frequency is independent of the propagation medium?

Yes, of course, frequency is always independent of the medium of propagation; this means that frequency is unaffected by the properties of the medium.

The frequency is entirely a source dependent entity. If the source itself changes, the frequency will be changed. It always remains independent of the medium of propagation.

The relation between the velocity and wavelength can be expressed by considering the speed of light in the vacuum.

The above statement means that if the wavelength of light is more, its frequency will be less. Mathematically the equation for frequency and wavelength dependency can be written as,

nu=c/λ ; where ν is the frequency, λ is the wavelength, and c is the velocity of light.

What factors of light are affected by the change in the medium?

Two factors affect the change in the medium while propagating; they are

• Wavelength – the wavelength of the incident light may increase or decrease depending on the indices of the medium.
• Velocity – the speed of light slowdowns while propagating from denser to rarer medium.

How does velocity and wavelength of light change with the medium?

The velocity is inversely related to the refractive index of the medium, and wavelength is proportional to the velocity.

If the refractive index of the medium increases, the velocity of light decreases; this results in the decrease of the wavelength of the light. The increase in the refractive index refers to a denser medium; thus in the denser medium, the velocity is less and proportionally the wavelength. However, the velocity of the light is more in rarer medium and thus greater the wavelength.

What is meant by light is quantized?

In quantum mechanics, quanta refer to energy packets or photons.

Quantized light means the light carries the packets of energy depending on the frequency of the source, and the four sets of quantum numbers characterize it. It exists in the discrete state also, thus remaining constant everywhere.