Imagine we have axial-symmetric permanent magnet (say, in the form of torus or piece of tube), so that magnetic field in its center is along the axis. The magnet is resting in a vacuum at zero-gravity.
We also have electrically carged very small ball on a stick, placed far away from the magnet along its axis.
Magnetic field is non-zero near the magnet and effectively zero far away from it. Electrical field is non-zero near the charge and effectively zero away from it. Both charge and magnet contribute to the total electromagnetic energy somehow. However, as magnetic field and electric field are spatially isolated, the energy flux density (Poynting vector) is effectively zero.
Now we slowly move the electric charge into the magnet, right into its center and leave it there.
1. Assuming both magnet and charge are still at rest now, magnet still creates only magnetic field and the charge creates only electrical field. But fields are not spatially separated any more.
2. As the magnetic field at the axis is always directed along the axis because of the symmetry, the Lorentz force at the charge is zero, no matter if it is at rest or moves along the axis. Qulomb force at the charge should be zero too, as the electrical field from the magnet, even if it is spinning around the axis, should be zero at the axis.
3. Field energy should be the same. However, energy flux (Poynting vector) is non-zero. As magnetic field is directed along the axis at the magnet center, and electric field goes radially from the charge, the flux is around the axis. Such a circular energy flux contributes to angular momentum, so that the electrical field now has acquired angular momentum.
But where has this field angular momentum originated from?
Does it mean the magnet starts spinning to compensate that momentum? If yes, then where has the energy of spinning come from, and what can prevent us from withdrawing it? Besides, as no force has been acting on a charge all that time, we could just "throw" the charge through the magnet, and the maget should have turned by some angle, which should depend on the velocity of the electron. We could draw lines on a magnet, so we can measure the angle it turned. Hence, we just have measured the velocity of electron without interacting with it!
If the magnet does not start spinning when the charge goes through it, then what compensates the angular moment of the field?
Or the total angular moment of the field somehow totals to zero in this case?
Where is the flaw here?
Hi Roman, I guess the crucial point here is the radius of the ball: If it is zero, the energy of its electric field approaches infinity (one of the well known problems with electrons), and the whole setup is unrealistic. If the radius is greater than zero, your assumption 2 isn't correct because the magnetic field has a radial component everywhere except on the (infinitely thin) axis of the magnet.
While the charged ball is moving along the axis the radial component of the magnetic field causes a rotating force on the charge, and the ball gains angular momentum. My guess is that the sum of the angular momenta of the ball and of the energy flux in the fields is always zero.
Hope this helps.
Addendum: To be exact, the radial component of the magnetic field is zero not only on the axis but on the symmetry plane of the magnet perpendicular to the axis (this is the location where the radial field reverses its sign). But this is pure bean counting on my part, and doesn't contribute to the problem. :-)
Methinks . E and B only interact when there is relative motion of the sources. Buried in your puzzle is the charge moves my some undeclared mechanism . Whatever that mechanism it is creating the interaction and hence possibly a torque. The work done by that mechanism has to be included in the energy budget
Dear Roman,
The angular momentum of your device is zero. The reason is very simple. The Poynting vector only gives you density of linear momentum, and therefore if you only consider that the magnetic field is on one line, by summetry the whole integral of the torus will give you zero. No problem at all!
According to Lorenz, the force on a moving charge q with velocity v upon the influence of a magnetic induction B can be expressed by:
F= q v xB = vB sin theta, where theta is the angle between the vectors v and B.
The force will be zero for v=B = zero and theta=0.
So if the the charge is static a force will never exist.
Best wishes
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