Interesting Electrodynamics Problem

1. Oct 24, 2015

ChrisVer

I am not sure if this fits under this title, but oh well... I guess it goes for students but also for higher in hierarchy people (they also like some problems which can have an extension)
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. Oct 25, 2015

vanhees71

3. Oct 29, 2015

muscaria

Are you looking for unsolved problems, or just generally conceptual problems over a fairly wide range of difficulty that reveal interesting physical/mathematical structure?

4. Oct 29, 2015

ChrisVer

Both can be fancy ... But to keep it down a little, I'd go with your 2nd category

5. Oct 30, 2015

muscaria

Not that it's a favourite of mine or particularly difficult, just something i noticed earlier the other day which I kind of found interesting.. The aim of the problem is to gain insight into the role played by vector potential fields in generalising the relationship $U(q_1,q_2,\cdots,q_n) = -V(q_1,q_2,\cdots,q_n)$ between a velocity independent work function and the potential energy function, to the situation of a velocity dependent work function of a charge interacting with the E-M field.

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 $U(q_1,q_2,\cdots,q_n;\dot{q}_1,\dot{q}_2,\cdots , \dot{q}_n)$.

6. Oct 31, 2015

vanhees71

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

7. Oct 31, 2015

muscaria

Sure why not! That would give you all the time in the world :p.. And would lead nicely to further head scratching:
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.

8. Oct 31, 2015

Staff: Mentor

A charge undergoing uniform linear acceleration radiates. So why doesn't a charge at rest in a gravitational field?

9. Oct 31, 2015

vanhees71

One should be fair and note that the uniformly accelerating charge (hyperbolic motion) is a singular problem, where the standard treatment of the retarded fields does not properly work without a careful analysis. The correct treatment can be found in the following paper:

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. 83, 349 (2015)
http://arxiv.org/abs/1409.1569

10. Oct 31, 2015

Staff: Mentor

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

11. Nov 4, 2015

ChrisVer

@vanhees71 I don't understand how to get the Green's function in that problem...
$G= \frac{\delta \big( c(t- t^\prime) - R \big)}{R} \Theta(t-t')$
I understand that the delta function but not the 1/R.

12. Nov 4, 2015

GiuseppeR7

Suppose you are calculating the Poynting Vector of a point charge near a magnet...you can discover that the energy flows in circles around the magnet, from where this angular momentum come from?

13. Nov 4, 2015

muscaria

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")?