GREPhysics.NET: GR8677 #72

GR8677 #72

Problem

GREPhysics.NET Official Solution

Wave Phenomena $\Rightarrow$ }Group Velocity

Recall the definition of index of refraction $n$ in terms of the speed of light in vacuum $c$ and the velocity of the light in medium $v$ , $n=c/v$ . Since the velocity of light in any medium is $v\leq c$ , the condition $n \geq 1$ usually holds. However, even if rock salt has $n > 1$ , the wave does not exceed the speed of light. The group velocity can travel faster than the speed of light, and apparently it is the group velocity at work in the equation $n=c/v$ .

One can also arrive at this conclusion via MOE (Method of Elimination):

(A) Relativity is a pretty general theory that supposedly applies everywhere. (Newtonian mechanics can be achieved using the proper approximation technique.) X-out this choice.

(B) An x-ray is a specific frequency in the electromagnetic spectrum. Nothing forbids an electromagnetic wave from transmitting signals, and thus this choice is out.

(C) Imaginary mass? If confused, save for last comparison.

(D) Historical precedence shouldn't change the correctness of a theory (at least not in the ideal world ETS lives in)... X-this out.

(E) One recalls that there's a difference between group and phase velocities. Could this difference allow one of them to exceed the speed of light? Probably. In either case, this is a much better choice than the other remaining candidate, choice (C). So, choose this.

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Comments

kpb
2009-12-18 12:55:18

Refractive index= C/V
group velocity is always less than C, so refractive index is less than one. But phase velocity can be more than C, hence refractive index can be greater than one. so the result E makes sence. For better explanation, please see the link below.

http://www.mathpages.com/home/kmath210/kmath210.htm

kpb
2009-12-18 13:20:28

Please ignore the above explanation.
Refractive index= C/V, group velocity is always less than C, so refractive index is greater than one. But phase velocity can be more than C, hence refractive index can be less than one. so the result E makes sence. For better explanation, please see the link below. http://www.mathpages.com/home/kmath210/kmath210.htm

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gliese876d
2008-10-17 11:37:38

I have to say I just laughed out loud at the "imaginary mass" choice :)

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nitin
2006-11-16 12:10:49

No! The group velocity $v_g$ cannot be greater than c, and it is not $v_g$ which is "at work in the equation n=c/v". Concerning the latter, if $v=v_g$ , then we have

$n_g=\frac{c}{v_g}$ ,

which is called the "group index", and which is NOT the commonly known refractive index $n$ . Instead, $n$ is defined with respect to the phase velocity $v_p=\frac{\omega}{k}$ .

prismofmoonlight
2007-10-28 13:21:58

To clear things up, the index of refraction n=c/v_phase. rnrnn<1 does not violate the theory of relativity, which says that no information-carrying signal can propagate faster than c, because it is v_group that describes the velocity of the transmitted information.

prismofmoonlight
2007-10-28 13:22:43

That's just supposed to be n<1 in the second line.

temujin
2008-10-28 15:31:02

speed of light in a material is less than c; so, the group velocity in the medium can be greater than the the speed of light (for that medium).

Jeremy P
2008-11-04 10:53:01

nitin is right.rnrn $V_{group}$ carries information and so cannot travel faster than $c$ rnrn $V_{phase}$ carries no information and is greater than $c$ rnrnuse $n=\frac{c}{V_{p}}$ to arrive at n<1

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globalphysics
2006-10-29 22:19:21

Shouldn't it be n<1 for the 4th line?

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I have to say I just laughed out loud at the "imaginary mass" choice :)

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