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I have read that photons always travel at the speed of light.So if I have a bunch of photons travelling in air then they will have an energy given be E=hf ;where f is the frequency.

Now if I place a tank of water in their path then they will have still be travelling at the same speed and same frequency(I have been taught that frequency depends only on the source).This would require their wavelengths in both mediums to be the same ,which is not feasible.

So where is the discrepancy ?

Is my initial assumption incorrect?

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When people say that photons always travel the speed of light, they meant in terms of reference frames. And in fact, even through water, photons do travel the speed of light. The only reason why they appear to travel more slowly is because they "run into" molecules, and the path is not straight. In a vacuum, there are no molecules to hinder the light, and it appears to travel faster.

When people say that photons always travel the speed of light, they mean that regardless the speed of another object, they will be traveling at the speed of light. From the point of view of the photon, the photon travels instantly.

---EDIT: in the case of transparent crystals, the light will still travel in a straight line. The time will be lost in absorbtion and emission.

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Light does not travel at the same speed in all mediums. In water the speed of light is less than that in vacuum.

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    $\begingroup$ This comment is factually incorrect. Photons still travel the speed of light in water. It's just that because their path is no longer straight and they run into water molecules that it seems longer. $\endgroup$ – DevilApple227 Jan 22 '16 at 15:44
  • $\begingroup$ Please specify phase velocity, group velocity, or photon velocity when referring to "speed of light". $\endgroup$ – Rick Jan 22 '16 at 16:21
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The formula E=h*f is a quantum mechanical formula for the elementary particle called a photon. The photon travels with velocity c in vacuum. If it is not a vacuum it will interact with appropriate quantum mechanical interactions, and in the interval between interactions it will be traveling with velocity c. If its interaction is elastic, as happens in transparent media, there is no energy loss, just direction so the frequency does not change.

The classical electromagnetic wave we call "light" travels with velocity c in vacuum but has a reduced velocity in a medium according to the index of refraction., v=c/n.

Light is not photons in the same way that water waves are not water molecules. Light is emergent from the confluence of zillions of photon wavefunctions, which build up the electric and magnetic fields of the electromagnetic radiation. These wavefunctions in a medium change due to interaction, in a transparent medium they rearrange themselves in such a way as to give the observed index of refraction for the collective wave.

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In one word the answer is interference.

Now more details.

Assume you are traveling through something which has refractive index different then 1, like water. It actually means that for "slightly" different frequency of light water molecules will start to absorb the photons, i.e. shake together with electric field. However if you stay away from that frequency their electron distribution will only slightly shake in the electric field of light. Because you are out of resonance there will be a lag between incident field and actual vibration of electron density. This vibration however will result in the emission of a new photon which will be slightly delayed with regards to the incident. Put together all atoms let all the emitters interfere and this would look like your light wave has changed it's speed and wavelength.

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What bugs you here is that the word "speed" isn't precise at all when we talk about a wave in a dispersive medium, such as water. What we usually refer to when we say the "speed of light" is the constant we call $c = 299792458$ m/s, which is equal to the velocity of light in the vacuum. Vacuum is non-dispersive, so the group and phase velocity are the same, and are equal to $c$.

Also, it is not very precise to say that "photons" travel at a certain "speed" inside a medium. Photons are quantum excitations of the electric field in the vacuum.

When we talk about a transparent dielectric medium, the excitations are quantised polarization waves : depending on the precise microscopic details of your medium, the dispersion relation can be different of just $\omega = k c$ ($k$ being the wavevector), which holds in vacuum. If the medium is dispersive, group and phase velocity are different. If you take a polychromatic light travelling through water, its group velocity will be less than $c$, thus the light will appear to travel slower.

To sum it all up, you can think of it like this : as long as you consider a linear propagation (which means you light is not too intense), frequency will never change, regardless of which medium you're travelling through. If you are inside the vacuum, velocity is $c$ and the wavelenght is just $\frac{c}{\lambda}$. Once you enter the medium, you get a non-trivial dispersion relation $\omega(k)$. Then, the light appears to be moving at a velocity given by $\frac{d \omega(k)}{dk}$, which is the group velocity, it is usually lower than $c$.

To awnser your original question yes, wavelenght of "photons" depend on the medium, it is just that light in a medium is a bit more complicated than just a photon.

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Only in vacuum light travels without disturbances. To be precise, really without any disturbance light would travel only in a space with constant gravitational potential (and such a potential distribution does not exist). Every source of gravitation deflects light. But this does not has an influence on different trajectories (geodesic lines) nor leads to different speeds of light of different frequencies.

A different situation we get when light goes through a medium. A medium c an be glass, metal, gas, liquid, you name it. In all of this cases the speed of light is smaller c ($300,000 km/s$). In some cases we do not see remarkable change of frequency of the light, going through a medium (air, water, glass). In other cases we have to understand, how the change of the frequency happens. Take metal. You illuminate a metal with visible light. The light that gets reflected, we do not take in account here. But some light gets absorbs by the metal. Do s this light goes through the metal. Yes, if you accept that infrared light is light too. The absorbed light excite the atoms of the metal and gets re-emitted in lower frequencies. there are a lot of phenomenons inside the metal (scattering, phonons).

Karan Singh, are this facts helpful for you and you find the way? Or which additional explanations do you need?

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