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ChrisVer

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What has been your favourite to talk-about problem in Electrodynamics ? I am looking for something that would make me think over it apart from just solving.

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- #1

ChrisVer

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What has been your favourite to talk-about problem in Electrodynamics ? I am looking for something that would make me think over it apart from just solving.

- #2

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R. Medina, Radiation reaction of a classical quasi-rigid extended particle, J.Phys. A

http://iopscience.iop.org/article/10.1088/0305-4470/39/14/021/meta

http://arxiv.org/abs/physics/0508031

- #3

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Are you looking for unsolved problems, or just generally conceptual problems over a fairly wide range of difficulty that reveal interesting physical/mathematical structure?What has been your favourite to talk-about problem in Electrodynamics ? I am looking for something that would make me think over it apart from just solving.

- #4

ChrisVer

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Both can be fancy ... But to keep it down a little, I'd go with your 2nd categoryAre you looking for unsolved problems, or just generally conceptual problems over a fairly wide range of difficulty that reveal interesting physical/mathematical structure?

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Consider a charge ##+e## travelling at some moment in time with non-relativistic velocity ##\textbf{v}_0## in a uniform static magnetic field orthogonal to ##\textbf{v}_0## and uniform time independent background scalar potential ##\phi##. In such a situation, the charge undergoes uniform circular motion with energy $$E=\frac{1}{2}m\textbf{v}_0\cdot\textbf{v}_0 +e\phi,$$ which remains constant during the motion and is independent of the strength of the uniform magnetic field. Thus, the effect of the rotational vector field ##\textbf{A}## is simply to rotate ##\textbf{v}_0## at a constant rate in time without changing its magnitude.

1) Show that the Hamiltonian function is a constant of the motion which can be interpreted as the sum of kinetic energy ##T## and potential energy ##V##, provided the potential energy is defined as ##V=\sum_i A_iv_i - U(\textbf{x},\textbf{v})##.

2) What form should ##V## take generally (in terms of ##U##) for such an interpretation to be valid?

3) Thus, find the relationship between ##A_i## and the work function ##U##. Compare this expression to the definition of canonical momentum and comment on the nature of the mathematical relationship between the potential and work functions and of the variable ##A_i##.

Bonus (purely analytical mechanics really):

Given the invariant differential form of the work function ##dU=\sum_i F_i dq_i##, find an expression for the generalised forces ##F_i## in terms of a velocity dependent work function ##U(q_1,q_2,\cdots,q_n;\dot{q}_1,\dot{q}_2,\cdots , \dot{q}_n)##.

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4) Generalize the problem to relativistic motion :-).

- #7

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Sure why not! That would give you all the time in the world :p.. And would lead nicely to further head scratching:4) Generalize the problem to relativistic motion :-).

5) In a similar fashion to considering the motion of the phase fluid as a continual succession of infinitesimal canonical transformations, show that the motion of the velocity vector of a charge in an EM field can be viewed as a continual sequence of infinitesimal Lorentz transformations, provided by the components of the EM stress tensor.

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- #9

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D. J. Cross, Completing the Liénard-Wiechert potentials: The origin of the delta function fields for a charged particle in hyperbolic motion, Am. J. Phys.

http://arxiv.org/abs/1409.1569

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Which is one of the many things that makes it an interesting problem.

- #11

ChrisVer

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[itex] G= \frac{\delta \big( c(t- t^\prime) - R \big)}{R} \Theta(t-t')[/itex]

I understand that the delta function but not the 1/R.

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- #13

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Depends what you mean by near as well as the shape of the magnetic and whether or not the charge is initially moving. I'm guessing by the last part of your question that you are trying to describe a situation identical to the problem I suggested, right? Are you trying to get to the idea that the canonical momentum is the quantity which represents the flow and the gauge potential curves the path associated with this flow, turning linear momentum into angular momentum (from magnet "off" to magnet "on")?

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