# Conservation of Momentum vs Faraday's Law of Induction

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### Rich L.

Jul 27, 2014, 1:10:02 PM7/27/14
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I've been meditating on a question for a week or more and not seeing how to resolve it. It concerns conservation of momentum with electromagnetic forces.

Consider a wire carrying a current and a stationary charge a small distance from it. Let's assume the current is carried by negative charges moving is a static matrix of positive charges, as is usually the case. The stationary charge for now is positive.

If the current is increasing with time, the magnetic vector potential at the stationary charge will likewise be increasing, and pointing in the same direction as the current (i.e. opposite to the actual electron motion). Because the vector potential is increasing, there will be an electric field directed in the opposite direction. As a result the positive charge will accelerate in the same direction as the electrons.

The acceleration of the electrons will induce a "radiation resistance", which will appear as a force on the accelerating electrons that will resist the acceleration. In other words, the force required to accelerate the electrons will be slightly greater than that required to accelerate their mass. This force is a rate of change of momentum. This extra momentum has to go somewhere, but it isn't going into the electrons.

Conservation of momentum requires that this extra momentum be accounted for. The obvious answer is that the extra momentum is transferred to the stationary charge (and other charges much further away). This works, because the stationary charge will be accelerated in the same direction as the electrons are being accelerated, provided the stationary charge is positive.

But what about if the stationary charge is negative? In that case the charge will accelerate in the opposite direction, and its momentum will be in the opposite direction. The electrons in the wire will feel the same radiation resistance, however, so how is momentum conserved?

### Tom Roberts

Aug 2, 2014, 1:30:02 PM8/2/14
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On 7/27/14 7/27/14 1:07 PM, Rich L. wrote:
> [...]

As with most such cases, there is electromagnetic radiation which carries off
energy and momentum. The radiation will be different for your two situations.
Computing the effects of radiation is complicated and difficult, but we know
they occur -- that is guaranteed because energy and momentum are conserved at
each and every point in the manifold (i.e. d_i T^ij = 0, where d_i is the
partial derivative with respect to coordinate i, and the {T^ij} are the
components of the energy-momentum tensor; i,j = 0,1,2,3).

Tom Roberts

### Roland Franzius

Aug 3, 2014, 1:20:03 PM8/3/14
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As it is known since 1900 (Lorentz et al) energy-momentum conservation
of the classical Maxwell field does not work in the case of point
charges and unbounded electromagnetic local fields.

QED is the first theory that turns energy-momentum conservation of
interacting bounded quantum mechanically charge distributions together
with their free radiation fields into a working principle.

And as we all know, this only works in the frame of a bit of fermionic
matter, relativistc an overall redefinition of nearly all observable
physical quantities by a renormalization procedure.

--

Roland Franzius