An infinite charged line moving with velocity V and its energy current

[Verdict]

Homework Statement

Consider an infinite charged line along the z-axis, with linear charge density .
The charge moves uniformly with velocity v in the positive z direction.
1. Give an expression for the electric and magnetic field.
2. Give an expression for the energy
flux density (or energy current).
3. Why does the energy
ow in the direction it does?
4. Bonus question. Instead of a charged thin rod, one may imagine a
neutral conductor, with positive ions, and negative electrons carrying the
current. In this case there is no electric field, and thus not energy current.
Explain the difference.

Homework Equations

If I am correct, the energy flux density or the energy current is the Poynting vector, which is 1/$\mu₀$ E cross product B.

For the E field and the B field, I somehow feel like it's just Gauss's law and Ampère's law. This is a rather important point though, as I'm not sure if I can just use them here. (In class we've already treated retarded potentials and such, but this question 'feels' like it is of less advanced material.)

The Attempt at a Solution

Alright, so I used Gauss's law, using a cylinder with radius r, to compute the E field. Similarly, I used a circle with radius r to compute the B field. E has a radial direction, and B is perpendicular to that, and I indicated that direction as the phi-hat direction, can I do that?
Then, as E and B are mutually perpendicular, I could use the right hand rule to compute the direction of S. Actually doing the cross product might have gone wrong, I thought it was the length of E times the length of B times the sine of their angle, which is just E multiplied by B.
I end up with

Now, I have no idea how to explain the direction of the S vector. But most importantly, is what I have done up to this point correct? The bonus question I am not too worried about (as in, I don't really feel the urge to do it), as I don't know what to do there, but if it is not too hard, maybe someone could help me get started with that one too?

Kind regards,
Verdict

 

[mfb]

Apart from the prefactor for S (where did the second 2 and the mu go?), I don't see an error so far. You should specify the velocity definition ("in positive z-direction"?).

Concerning the bonus question, I don't see why someone would expect an energy flow there... the setup has a perfect symmetry.

 

[Verdict]

Apart from the prefactor for S (where did the second 2 and the mu go?), I don't see an error so far. You should specify the velocity definition ("in positive z-direction"?).

Concerning the bonus question, I don't see why someone would expect an energy flow there... the setup has a perfect symmetry.

Ah oops. That went wrong in word, it should have said (2pi)^2. Should I specify the velocity definition in a separate line, or do you mean that it has to be reflected in the equation itself?

And then, if it is indeed correct, explaining why the energy flow has the same direction as the flow of charge.. It just seems 'obvious' that this would be so, but that is hardly an explanation.

 

[mfb]

Should I specify the velocity definition in a separate line

I would do this, or call it v_z.

Energy flow: If we add some (hypothetical) start of the current flow, you have to accelerate particles against the electric field of the existing charges.

 

[Verdict]

Hm. Sure. And the direction of the acceleration is the direction of the energy flow?

 

[mfb]

The direction of the particles, right.
You need energy at the source, and send this along the line of moving charges.

 

[Verdict]

Hmm alright. Would you also agree with the explanation that in this case, the current flows in the Z direction, carrying positive charges and thus positive potential energy in this direction, and thus there is an energy flow in that direction?

 

[mfb]

The charge type does not matter (S is proportional to lambda squared), the argument works for both charge types.

 

[Verdict]

Great, thanks a bunch