Understanding 4-Momenta in Electromagnetism: Solutions and Formulas Explained

[physicsforumsfan]

Hey all,

I have a three part question:

Homework Statement

If A^(3)=(Ax,Ay,Az) is the three-vector potential, J^(3)=(Jx,Jy,Jz) is the three-vector current density, \varphi is the scalar potential and p is the charge density, then the four-current density J is given by:

The Attempt at a Solution

I have read through literature and believe the answer is:

c=speed of light

J=(cp Jx Jy Jz) -> I am confused because can this answer be written transposed in a matrix and still be correct or is this answer only valid in the form:

J=(cp,Jx,Jy,Jz)?

Part 2

Homework Statement

A photon is produced in frame S with 4-momentum P=(E/c,p,0,0) and frequency f , where hf=pc=E, p is the magnitude of the 3-momentum and E is the energy. Frame S' is traveling at speed v in the direction of the positive x-axis of frame S. What is the relationship between the 4-momenta P and P in the two frames?

The Attempt at a Solution

Using invariance of 4-momentum, Squaring P' will yield:

P'^2= (γP)^2 where γ is Lorentz factor 1/√1-v^2/c^2

Thus sqrt answer is:

P' = P/√(1-v^2/c^2)

Is this correct?

Part 3

Homework Statement

In the previous question, what is the relationship between the frequencies f and f' of the photon in the two frames?

The Attempt at a Solution

Taking inverse of f', you get

f' = 1/(γf) = f*√(1-v^2/c^2) or is it take the square and then sqrt it to give:

f'= f/√(1-v^2/c^2)

That's it,

Thanks

 

[physicsforumsfan]

Hi,

no reply?

Help?

 

[physicsforumsfan]

Hi,

Looks like no reply, must be difficult questions...

Anyways, I just like to added that for part two:

\overline{P}=γP

where P (NOT dashed) is the 4 momentum for frame S.

Now if anyone can agree with me on this, easier to read, equation.

Now for Part 3...

Anyone?

 

[vela]

Homework Statement

A photon is produced in frame S with 4-momentum P=(E/c,p,0,0) and frequency f , where hf=pc=E, p is the magnitude of the 3-momentum and E is the energy. Frame S' is traveling at speed v in the direction of the positive x-axis of frame S. What is the relationship between the 4-momenta P and P in the two frames?

The Attempt at a Solution

Using invariance of 4-momentum, Squaring P' will yield:

P'^2= (γP)^2 where γ is Lorentz factor 1/√1-v^2/c^2

Thus sqrt answer is:

P' = P/√(1-v^2/c^2)

Is this correct?

No, it isn't. You made two logical errors. First, in the very first step, you assumed $p'^\mu = \gamma p^\mu$, yet this is what you're claiming to deduce. You can't assume what you're trying to prove. Second, even if the first line were valid, the second one doesn't follow from it logically. Just because two vectors have the same norm, it doesn't mean the vectors are proportional to each other.

What you need to find out is how a four-vector transforms under a boost. This should be explained in your textbook.

 

[physicsforumsfan]

What you need to find out is how a four-vector transforms under a boost. This should be explained in your textbook.

Hi vela,

Thanks for replying. The lorentz boost (or the transformation) of any four vector is the lorentz matrix multiplied by the vector according the the article http://en.wikipedia.org/wiki/Lorentz_transformation#Transformation_of_other_physical_quantities.

Essentially, the article is saying what I am saying:

\overline{P}= γP

These questions are basically warm up questions to more 'applied' ones so the proofs aren't necessary as their comprehension is seen in the more latter questions.

Does the rule apply similarly for the frequency of a proton (next part)?

Thanks

 

[vela]

No, the article is not saying what you're saying. $\gamma$ is a scalar, not a matrix. As you noted,
$$\gamma = \frac{1}{\sqrt{1-(v/c)^2}}.$$ How are you getting that multiplying by $\gamma$ is the same as multiplying by a matrix?

 

[physicsforumsfan]

No, the article is not saying what you're saying. $\gamma$ is a scalar, not a matrix. As you noted,
$$\gamma = \frac{1}{\sqrt{1-(v/c)^2}}.$$ How are you getting that multiplying by $\gamma$ is the same as multiplying by a matrix?

Hi,

I just noticed by assumption which as you said, is incorrect. I am using the gamma factor because I know the answer involved is P multiplied by a form/shape of the gamma factor (ie. sqrt (1-(v/c)^2)).