Violation of action-reaction law?

Two equally positively charged points are moving on a plane at a constant speed. One of them (q1) is heading towards south, the other one (q2) towards west, and they're approaching to each other. Then, magnetic fields are induced from these moving charges. And Lorentz forces on q1 and q2 according to the other ones act towards east and north respectively. And Coulomb forces on those charges act in the direction of away from each other. Then, the net forces on each charges act in the direction of somewhere between north and east.

Here is the question. It seems that action-reaction law is violated in this case. They're not equal forces in oppsite directions. Rather, their forces are heading weird directions. Is action-reaction law really violated in this case? Is there some other example that action-reaction law vilated?

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jtbell
Mentor
In a purely mechanical system, Newton's Third Law implies conservation of momentum and vice versa. When electromagnetic forces are involved, mechanical momentum and energy (i.e. kinetic energy plus potential energy of particles) are not conserved, in general. We account for this by associating energy and momentum with the electromagnetic field.

http://farside.ph.utexas.edu/teaching/em/lectures/node90.html

So Newton's Third Law fails in these situations because it is not as general as conservation of momentum, which does still hold true. I suppose one could define a "force exerted on the electromagnetic field" by way of $\vec F = d {\vec p}_{em} / dt$, but I don't remember ever seeing this done in practice.

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Please, check me if I'm right. Let the directions of the two forces on q1 and q2 be in north-east. Then the total momentum of the two charges are in the same direction. And the total momentum of the two fields are in the opposite direction, south-west(I'm not sure). So, the total momentum of the system cancels out and the total momentum is conserved. And so, we can think that action-reaction law can be still applied between the charges and the fields. If we consider the two charges only not the fields, then the action-reaction law fails. So, Newton's third law is right once you take everything into account. Am I right?

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you are absolute right (there is even an explicit statement in my classical mechanics book (by Taylor)), Newton's third law fails for electromagnet forces when you consider the classical definitions of momentum and/or angular momentum (or more generally, forces in relativity).

if you look at the magnetic field generated by those two moving charges, the forces caused by them are very very small (in orders of something over c), thus in the classical limit, one can neglect those terms. (just look at $\mu_0$, the force constant for magnetism, which is incredibly small comparing to $\frac{1}{4\pi\epsilon_0}$.)

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This is cool. I thought magnetic fields screwed up the third law a little, but I didn't know the details.

Just to be sure I've got it right: the "momentum of the fields" we talk about is just the momentum associated with the electromagnetic wave that propagates in the direction that q1 and q2 were flying in before they started interacting magnetically. Right?

Is this the same momentum that special relativity says waves have? If so, does that mean you can't always use pure Newtonian mechanics when dealing with magnetic interactions unless you at least borrow that concept from SR?

the third law is still there, in this case, whatever is making the fields feels the opposite force.

the third law is still there, in this case, whatever is making the fields feels the opposite force.
The electrons are making the field, and if you look carefully at the forces, you'll see that they don't feel exactly the opposite force. That's the whole point.

Dosen't the forgotten, axial, magnetic force that Amper discovered and Graneau preserve the third law for you?

Does relativity play any part here in the modification of Newtonian law for electrodynamics? Perhaps in such a case there is a reference frame where action-reaction holds.

Meir Achuz
Homework Helper
Gold Member
The forces between two moving charges requires relativity, even for slowly moving charges.
Total momentum is conserved if the electromagnetic field momentum is included.
This is difficult to show for this explicit example, but it can be shown in general.

I must ask is there some experimental work about it, or all members are linear convinced and selfexplained...only theoretically....

For example

where exp should be with care for edge effect,but principle is here.
Will magnets fill force toward red arrow (in picture this is way of current)?

Furthermore, insted of one halfcircle wire , there cannot be coil (act-react will be nulled) but U shaped copper piece powered far away on left,and question is if there is radial force, or only magnetic axial....?

NS must be lose horizontal and fixed vertical, to determine is there force on them...

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