GREPhysics.NET
GR | # Login | Register
   
  GR8677 #59
Problem
GREPhysics.NET Official Solution    Alternate Solutions
Verbatim question for GR8677 #59
Wave Phenomena}Group Velocity


Group velocity is v_g=d\omega/dk. So, take the derivative of the quantity \omega = \sqrt{c^2 k^2 + m^2} to get


Use the above equation to test the 5 choices:

(A) As k \rightarrow 0 \Rightarrow \omega = 0, not infinity. The first condition doesn't work, no need to test the second (don't have to remember L'Hopital's rule).

(B) Wrong for the same reason as (A).

(C) Wrong because v_g approaches 0, not c, as k \rightarrow 0.

(D) As k \rightarrow \infty, v_g \approx \frac{kc^2}{\sqrt{c^2 k^2}}=c, since m \ll (ck)^2. So, v_g doesn't tend towards \infty. This choice is wrong.

(E) This is it. The conditions work (and it's the only choice left).



See below for user comments and alternate solutions! See below for user comments and alternate solutions!
Alternate Solutions
blackwind
2011-10-16 14:47:35
Simply use limits to evaluate v_g :

\lim _{k\rightarrow 0} v_p \rightarrow w = m
\lim _{k\rightarrow \infty} v_p \rightarrow w = ck

Derivative by k then gives you the limits
Alternate Solution - Unverified
Comments
varsha
2016-02-19 15:14:17
In option A,why have they taken w.r.t w.As k tends to 0,we should see Vg right??\r\nVg should be infinity,not w..NEC
blackwind
2011-10-16 14:47:35
Simply use limits to evaluate v_g :

\lim _{k\rightarrow 0} v_p \rightarrow w = m
\lim _{k\rightarrow \infty} v_p \rightarrow w = ck

Derivative by k then gives you the limits
Alternate Solution - Unverified
anmuhich
2009-03-21 09:45:15
I used a similar approach as chrisfizzix. I didn't even remember much at all about group velocities but I did remember it's like the velocity of a particle. I didn't know what a propogation vector was per se but I relized that it must be directly related to the group velocity just by the name of it. This idea combined with the idea that it shouldn't move faster than the speed of light led me to E. Of course it's much better to know the formula for the group velocity and solve as shown above, but if all else fails use a little common sense and you'll come out ahead of the 1/4 point off for each wrong answer.NEC
chrisfizzix
2008-10-03 13:16:24
I solved this by some quick common sense. The group velocity is just the velocity of the particle associated with the wave motion. For any physical system, the group velocity can never be larger than c. Thus, any answer that has v \rightarrow \infty is out - goodbye A, B, D. The propagation vector k is absolutely related to group velocity, and if k \rightarrow 0 then the wave better not be going anywhere. Thus, C is out, leaving E.
neon37
2008-11-05 00:07:59
Hi I read this article in wiki http://en.wikipedia.org/wiki/Group_velocity that seems to suggest group velocity can be greater than speed of light. It says quote:
Since the 1980s, various experiments have verified that it is possible for the group velocity of laser light pulses sent through specially prepared materials to significantly exceed the speed of light in vacuum. However, superluminal communication is not possible in this case, since the signal velocity remains less than the speed of light.

well i can sorta visualize it but if someone is expert in this subject could you please explain it so that I can confirm.

Thanks
neo55378008
2012-09-06 10:43:35
neon37, if you read the next line in the Wiki entry, it says that while the group velocity is greater than c, communication is still less than c. I don't claim to be an expert on this, but from my understanding the vocabulary we use is changing. Physicists have stopped using the term matter/energy when talking about relativity, and lately have been saying information. Information cannot travel faster than the speed of light. It seems like a subtle difference, but I believe this saves relativity and causality.
NEC

Post A Comment!
Username:
Password:
Click here to register.
This comment is best classified as a: (mouseover)
 
Mouseover the respective type above for an explanation of each type.

Bare Basic LaTeX Rosetta Stone

LaTeX syntax supported through dollar sign wrappers $, ex., $\alpha^2_0$ produces .
type this... to get...
$\int_0^\infty$
$\partial$
$\Rightarrow$
$\ddot{x},\dot{x}$
$\sqrt{z}$
$\langle my \rangle$
$\left( abacadabra \right)_{me}$
$\vec{E}$
$\frac{a}{b}$
 
The Sidebar Chatbox...
Scroll to see it, or resize your browser to ignore it...