Let's Break the Law--Newton's that is.

In classical mechanics, Newton's Three Laws are considered inviolate. Well, OK, his first law is really just a special case of his second. No matter, the topic today is the third law, known and loved by all as:

For every action there is an equal and opposite reaction.

In introductory physics, this is made real and quantitative when constructing free body diagrams. If I want to use Newton’s Second Law, F = ma, applied, say, to a person, you make a diagram that includes all the forces on the person. If the problem states that the person is pushing on a wall to the right with a force of 10 units, that force does not appear on the free body diagram because you want forces on the person and not by the person. So you reason that from Newton’s Third Law the wall exerts a force 10 units to the left, that is, equal and opposite. That force, from the wall, is the one that gets put on your diagram.

Ask any number of physics students, and many physics professors: is Newton's Third Law always valid? The answer they will likely give is yes.

But it is not. Consider the picture shown, of two charged particles travelling along perpendicular axes. Take both charges as positive. The Coulomb force between the charges, which acts along the line joining them, will be equal and opposite. But what about the magnetic forces?

The detailed magnetic fields are complicated, but the directions are easy—it just requires application of various forms of the right hand rule. And we will only need directions—because to show a violation of Newton’s third we need only show the direction of the magnetic force from q₁ on q₂ is not opposite to the direction of the magnetic force from q₂ on q₁. In fact, it is equal but not opposite.

The charge q₁, moving along the x axis as shown, is effectively a current flowing along the x axis. Such a current produces loops of magnetic field, and the direction of the magnetic field comes from (a) placing the thumb of your right hand in the direction of the current (the positive x axis) and letting your fingers curl—they will curl as the magnetic field curls. Thus B anywhere on the xy plane with y > 0 will be coming up and out of the xy plane in the positive z direction, as shown by B (of q₁) in the diagram.

Now the moving q₂ is effectively a current along the positive y direction. From basic physics there will an IL×B force on q₂, where L is a vector in the direction of the current I, i.e., the plus y direction. Here we use the other form of the right hand rule: align your fingers with L (+y direction) curl them in the direction of B (+z) and your thumb will point in the direction of the force, in this case, +x, as indicated by F₂₁.

That is, the magnetic force from q₁ on q₂ is in the +x direction.

The same analysis, to get the force of q₂ on q₁ yields not the opposite direction, but the +y direction as indicated by F₁₂.

F₁₂ and F₂₁, as you see, are not opposite.

What is happening here?

Well, Newton's Third Law is actually a "usually" correct statement of a truly inviolate law (under the conditions we are discussing) which is conservation of momentum. That says that the change in q₁'s momentum is equal and opposite to the change in q₂'s. That is:

dp₁ = - dp₂ (where dp₁ means the change in p₁)

If we divide by dt, the time of the interaction, we get dp₁/dt = -dp₂/dt. Students will recognize dp/dt as the force. (Newton’s Second Law is not really F = ma, but F = dp/dt, but dp/dt is the same as ma for the usual case where the mass is constant.) Thus we have, if you will, derived Newton's Third Law from conservation of momentum.

Thus Newton's Third law is really a statement of conservation of momentum—but we have made an assumption that all the momentum is found in the two particles. If something else can absorb some momentum, then conservation of momentum really reads that dp₁ + dp₂ + dp_{something-else} is zero, and when we divide by dt we do not get that the magnetic forces are necessarily equal and opposite, because the dp_{something-else} is ruining the equation.

What is this "something else?" It is the electromagnetic field. The field can carry momentum. Careful analysis, taking the momentum of the field into account, shows that the momentum is indeed conserved.

In most classical physics problems there is no field, and the particles themselves have to bear the burden of conserving momentum—and that in turns leads to slavish obedience of Newton’s third law.

NOTE: The picture was taken from here, which has a similar discussion.