Consider the following interaction between a proton and an electron:
The electric forces between the proton and electron exhibit “reciprocity.” That is, the electric force that the proton exerts on the electron is equal in magnitude and opposite in direction to the electric force that the electron exerts on the proton. The reason of course is that the electric force is proportional to q1q2 (which equals q2q1) and is also proportional to the vector that points from one charge toward the other. (Similarly, gravitational forces also exhibit reciprocity, because the gravitational force is proportional to m1m2 (which equals m2m1) and is also proportional to the vector that points from one mass toward the other.)
The magnetic force is quite different. The moving proton contributes zero magnetic field at the location of the electron and so no magnetic force acts on the electron, whereas the electron does contribute a nonzero magnetic field at the location of the proton, into the page, so there is a magnetic force on the proton, in the +y direction. The magnetic force does not have the property of reciprocity. To put it another way, the forces do not obey Newton’s third law, which means that this “law” is not fundamental but rather a relationship that applies to electric and gravitational interactions but not to all types of interaction.
This raises an interesting issue. Choose the proton plus electron as the system of interest. The net force on this seemingly isolated system is nonzero, which means that the total momentum of the system will change, which implies that momentum is not a conserved quantity in the presence of magnetic forces. In a 1942 paper by J. M. Keller, “Newton’s Third Law and Electrodynamics” (American Journal of Physics 10, 302-307, https://doi.org/10.1119/1.1990405), the author discusses this situation and shows that if one adds the field momentum to the particle momentum, this total momentum indeed does not change.
In magnetic interactions between closed current loops, reciprocity of the forces (Newton’s third law) is valid. It is interesting to see in detail how this comes about. In the Web VPython program shown below, the two current-carrying loops of wire are divided into many short segments. The Biot-Savart law is used to calculate the magnetic field that a loop contributes at the location of each segment of the other loop, and the magnetic force exerted on that segment. Also calculated is the torque of this force around the center of the scene (at the left edge of the right loop, which is the origin of the coordinate system, where x is to the right, y is up, and z is out of the page). The cyan arrows represent the magnetic field at a location around a loop, due to the other loop, and red arrows represent the magnetic force on that segment, due to the magnetic field of the other loop. Ignoring tiny components due to numerical roundoff, the calculation shows that the net forces on the two loops have the same magnitude and opposite directions. The result is not obvious from looking at the very different distribution of forces over the two loops.
Right button drag to rotate “camera” to view scene.
To zoom, drag with middle button or use scroll wheel.
On a two-button mouse, middle is left + right.
Touch screen: pinch/extend to zoom, swipe or two-finger rotate.
Since the net force on the left loop is in the –z direction and the net force on the right loop is in the +z direction, one might expect the torques about the center of the scene would both be in the –y direction. The forces on the segments of the left loop are all in the –z direction, so the torque must necessarily be in the –y direction. However, on the right loop the force is in the +z direction only on about a third of the loop, and the distance to the center of the screen is small (recall that the center of the screen is at the left edge of the right loop). There are smaller forces on the rest of the loop, but they are all in the –z directions and the distances from the center of the screen are large. The net effect is that the net torque on the right loop is in fact in the +y direction, with the same magnitude as the torque on the left loop. The net torque on the combined system is zero.
If the loops are free to move and start from rest, in the first short time interval the left loop will acquire some momentum in the –z direction and the right loop will acquire the same magnitude of momentum, in the +z direction. The left loop will acquire some angular momentum in the –y direction and the right loop will acquire the same magnitude of angular momentum, in the +y direction.