TBR: Wave Frequency v. Speed

[justadream]

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TBR Physics Book II page 283 #30

“When white light passes from air to a prism, what is true?”

Answer: Light with the highest frequency is refracted by the greatest amount [I agree]



Incorrect Answer: Light with the greatest speed is refracted by the greatest amount



TBR says that “light traveling through a medium will exhibit the same wave speed regardless of frequency”



I’m pretty sure that TPR says that when light passes through a material medium, waves of different frequencies have different speeds (TPR says this is the one exception to the general idea that frequency is independent of speed).



More specifically, TPR says that higher frequency waves (violet) would be slower. This wouldn’t change the answer to the TBR question at hand but it does go against the logic in the answer explanation.

 

[Cawolf]

What is your question?

Clearly that choice is wrong because all of the light will be traveling at the same speed.

 

[justadream]

@Cawolf

My question is that TBR and TPR don't agree with regard to speed of a wave in a medium.

TBR says the speed will be the same.

TPR says that speed and frequency change when you are in a material medium (and this is the only exception to the idea that frequency and wave speed are independent)

Basically, if I have violet light and red light traveling through a prism, which is going faster and what about the frequencies?

 

[Cawolf]

The speed will decrease - that is the definition of an index of refraction.

But the speed will decrease uniformly.

The wavelength decreases as you divide it by the index of refraction. So the frequency will increase by a factor of n.

 

[justadream]

@Cawolf

Okay actually I confused myself with the frequency changing (does it actually)?

Can you clarify these 2 scenarios:

SCENARIO 1
If I have violet light at a frequency of X and a wavelength of Y that hits a prism, then the violet light that comes out of the prism will have:

Speed: Decreased (and decreases more as index of refraction of prism increases)
Wavelength: Decreased (less than Y) (and decreases more as index of refraction of prism increases)
Frequency: X (stays the same)

SCENARIO 2
I have white light entering a prism. The light gets "broken up" into different colors. Thus they must have different wavelengths and frequencies and speeds.
I think violet light (higher freq) will "bend more" than red light (lower freq).
Also, violet light will have lower speed (according to TPR).

Do you agree with those conclusions? I'm just a bit confused when comparing these things cause everything (speed, wavelength, and freq) all seem to change.

 

[Cawolf]

SCENARIO 1:

You mention the light coming out, but once it leaves the prism it will travel at c (assuming it enters a vacuum). So we will talk about the light just before it leaves the prism.

Lets call the refractive index, n, of the prism to be n = 1.5

Speed: v = c/n = c/1.5 = 0.667c
So yes, I agree the speed is decreased.

Wavelength: λ = λ(0)/n = λ(o)/1.5 = Y/1.5 = 0.667Y
So yes, I agree the wavelength is decreased.

Frequency: f = v/ λ = 0.667c/0.667Y = c/Y = X
Correct frequency is unchanged because the wave is both slowed down and lengthened by the same factor.


SCENARIO 2:

I agree with you here.

Since v = λf, and we know the frequency is constant through the medium, the longer wavelengths (red) will slow the least, and the shorter wavelengths (violet) will slow the most.

 

[justadream]

@Cawolf

With regard to scenario 2, just to be complete, would you agree that wavelength must be "somewhat drastically decreased" for violet light (since speed decreases AND it has higher frequency (after all, it's violet in color)?

Also conceptually, can you explain how white light suddenly gets "broken up" into different other colors? Like how does one component of white light that hits a prism "decide" whether to become violet or red after it leaves?

 

[Cawolf]

The wavelength is decreased by a factor of n, the refractive index of the medium. So 400 nm violet light passing through a n = 1.5 medium will have a λ = 266.7 nm. 700 nm red light through the same medium would have a λ = 466.7 nm. The change in wavelength is the same factor for all light.

White light contains all visible wavelengths. When the light is incident at an angle (such as one side of a triangular prism) each wavelength that is already in the light (and traveling at the same speed) now travels at a different speed. The light that is refracted the least will continue nearly straight, while light that is refracted the most will have a greater angle of refraction. The light then hits another angled surface and is further separated by wavelength. This is how the light is "split".

It is good to note that if the light was passing through two parallel surfaces, the angles would cancel and we would see no separation of the white light.

A picture explains it best.

 

[justadream]

@Cawolf

Can you explain this TBR question (TBR Physics Book II page 269 #14)

"What is true of different colors of light in a medium"?

Correct Answer: As frequency increases, the wavelength decreases
Incorrect Answer: As speed decreases, wavelength decreases


TBR's explanation talks about light being the same speed in a vacuum. But this is not a vacuum...

All I know is that in a material medium, light with greater frequency has lower speed.

But I'm not sure how to make any conclusions about wavelength based on that statement (since the wavelength is affected by both the freq and the speed - and both are changing).

For example, let's look at the incorrect answer:

As speed decreases, wavelength decreases

Speed = wavelength * frequency
As I said above, higher frequency light (e.g., violet) is slower than lower frequency light (e.g., red).
So if speed decreases, then I am going from red light to violet light.

If speed is decreasing (going from red to violet), but frequency is increasing (again, going from red to violet), then the equation holds that wavelength must go down.

This would seem to imply that this "incorrect" statement is true?

 

[techfan]

You're trying to memorize too many things. The best way (I think) to think about it is that the speed of light is a constant (c=299792458 m/s) and c=lambda*f. Then when light passes through a prism the constant (c) gets divided by another constant (in this case n: the index of refraction). The speed stays constant and only the wavelength and frequency are variable.

For your question, the speed DOESN'T decrease (it stays constant {c/n}) as you go from red to violet in the same medium. The wavelength of the light (lambda) decreases and the frequency (f) goes up. YOU HAVE TO HOLD THE SPEED CONSTANT otherwise you can't make any sort of meaningful relationships.

 

[justadream]

@techfan

But I read in TPR that in dispersion, higher frequency light travels slower in a material medium.

If you read from the fourth line starting with "Although one of the rules....."

 

[justadream]

I read a bunch of threads about it and I think I've found a way to reconcile everything:

Changing from one medium to another:
Frequency is Constant. Speed and Wavelength are directly proportional.

In the same medium:
Speed is approximately constant. Frequency and wavelength are inversely related.

Since this question is about being in the same medium, TBR is correct.

 

[The Brown Knight]

I read a bunch of threads about it and I think I've found a way to reconcile everything:

Changing from one medium to another:
Frequency is Constant. Speed and Wavelength are directly proportional.

In the same medium:
Speed is approximately constant. Frequency and wavelength are inversely related.

Since this question is about being in the same medium, TBR is correct.

Yup those are good rules of thumbs. For a given medium, speed does not depend on wavelength or frequency; for mechanical waves v = sqrt(tension/rope density) and for electromagnetic waves v = c/n. When crossing different media frequency remains the same while speed and wavelength change.

 

[The Brown Knight]

Also my understanding of chromatic dispersion is as follows:
dispersion occurs to keep the RATE OF ENERGY PROPAGATION the same for all the light that passes through the prism. since violet light has higher frequency and hence higher energy it needs to travel slower; if it traveled at same speed as red light, it would propagate more energy through the prism in the same amount of time. this slowing down is expressed as a higher index of refraction (since v = c/n), and a higher index of refraction according to Snell's law means greater bending. At the least, this helps me remember the whole thing.