# Time varying currents and Newton's Third Law?

My book says

It turns out that when we have time-dependent currents and/or charges in motion, Newton's third law may not hold for forces between charges and/or conductors. An essential consequence of Newton's third law in mechanics is conservation of momentum of an isolated system. This, however, holds even for the case of time-dependent situations with electromagnetic fields, provided the momentum carried by fields is also taken into account.

This unit in my book first described Ampere's experiment with long parallel wires and their magnetic attraction and repulsion. Then it points out that for steady currents in the wires, the magnetic force on the first wire due to the second is equal and opposite to the force on the second due to the first.

After this, my textbook gives a footnote saying that while Newton's Third Law may not be valid for time varying currents, the conservation of momentum still holds. What I understand about this is that the difference in momentum is carried by the field, but here is my question: which field, electric, magnetic or both? Also, what exactly is happening, why aren't there equal and opposite forces in case of a time varying current?

I'm in highschool, so a simple explanation would be appreciated. Edit: this question is not the same as angular momentum of charges and Newton's third law since I also ask which field stores the deficit in momentum.

• $\uparrow$ Which book? Sep 17, 2017 at 13:49
• NCERT Physics Volume 2, Grade 12. Sep 18, 2017 at 14:29
• Sep 18, 2017 at 17:15
• @sammygerbil That's very closely related, but not quite a dupe, I think. The core question here ("which field?") isn't addressed there. Sep 18, 2017 at 18:35
• @EmilioPisanty The formulae for $g$ and $j$ are given in your other answer. Both depend on $E$ and $B$. Doesn't that answer the question "which field?" Sep 18, 2017 at 18:44

You need both. More specifically, the fields carry a global momentum $$\mathbf P_\mathrm{fields} = \int\mathbf g\:\mathrm dV = \int \epsilon_0 \mathbf E\times\mathbf B \:\mathrm dV,$$ where $\mathbf g= \epsilon_0 \mathbf E\times\mathbf B = \mathbf S/c^2$ can be interpreted as the momentum density of the fields, and $\mathbf S$ is the Poynting vector. If either field vanishes, the whole thing defaults to zero, so you need regions that have simultaneously nonzero electric and magnetic fields for the fields to carry nonzero momentum.

The rest of the conceptual parts of your question are addressed in Apparent violation of Newton's 3rd law and the conservation of angular momentum for a pair of charged particles interacting magnetically.

Before the field concept came to be established, the interaction of two objects was thought to be the result of their action at a distance. This meant that two objects could influence rach other directly, irrespective of the distance or the nature of the medium between them. Inherent in this conceptual model is the instantaneous nature of interaction and it means that at every instant the action of A on B is equal and opposite to that of B on A. Let us consider two objects A and B whose interaction varies with the distance between them. On the action at a distance view, if an any change is made in the position of A this change will be immediately felt by B, without delay, through a change of the force it experiences due to A. On this view, then, if the earth suddenly changes its orbital course at time t, in a given reference frame then the moon will be found to start changing its course at this same time t in the same frame. If this action at a distance is included in our theoretical structure we get results contradicted by experiments.

On this action at a distance view, no temporal sequence can be assigned to phenomena associated with two interacting particles. For if A and B are two interacting particles and A is moved at time t then because B also starts moving at this same time t, we cannot say that first A was moved and then be moved that is we cannot have any casual relationship between events associated with interacting particles.

Let A and B be two charged balls kept at rest in an inertial frame S by the exertion of forces just sufficient to balance their coulomb repulsion. If agency holding A at rest stars changing the position of A, at time t, by changing the force it was initially exerting on A, and brings it over to a new position P in a very small time dt then the agency holding B at rest sould experience an added force on B right at time t on the action at a distance point of view. If careful experiments are performed along these lines, they definitely show that the agency holding B at rest, and hence B, starts experiencing and added force only at a time t+dt where dt though small is not zero. One can then conclude that interactions between objects are not transmitted instantaneously and that they are transmitted as wave disturbances. For instance in the example of the charged balls, we can say that the influence of A travels to B with a speed of D/dt. We observe that during the time dt, the force of action and reaction are not equal because during all this time B continued to experience the old force, that corresponding to the original position of A at time t, while A experienced forces greater than this.

Investigating along these lines physicists have been led to the concept of classical field. According to the field view, the influence of a particle exist at every point in the space around it and hence this space itself can be regarded as having become endowed with special attributes. The space endowed with special attributes is called the field of the particle A. Particle A interacts with another partile B not directly, but through its field. It's the field of A that acts on B and exerts a force on B. So long as the particles are static, the field view has no attractive features. But when the mutual configuration of the particles is changing, field view becomes essential owing to the possible inequality of the action reaction forces. The delay in experiencing the action-reaction forces can be explained by regarding thefields as wave disturbances emanating from their sources particles and propagating at a certain finite velocity. Both Static and variable particle configuration have there interaction explained on this model. Consider the case of two charged particles Q1 and Q2 moving relative to an inertial frame velocity of Q1 and Q2 are at right angles these moving charges can be regarded as current elements. The application of the Biot Savart law then shows that there is a magnetic field at the position of Q1 due to Q2 but the magnetic field at the position of Q2 due to Q1 is zero hence at this instant there is a force on Q1 due to Q2 but no force on Q2 due to Q1 thus the action and reaction may not be equal when we consider magnetic forces between charged particles in motion in this circumstances, the momentum of the system consisting of the two particles cannot remain conserved even if there are no forces on this system from outside this may appear perturbing because normally if one believes in the conservation of momentum of an isolated system. The difficulty lies in the action at a distance point of view. Conservation of momentum is restored if one introduces the concept of a field. The system of two charged particles is not nearly a system of two charged particles but 2 particles and a field. Not only the particles but the field also carries energy and momentum, and only when we take the momentum of the field into account, the conservation of momentum is restored. Certainly the momentum is associated with electromagnetic field.

These limitations of 3rd law is needed to be considered in electromagnetic phenomena involving fast moving particles, time dependent electric currents and in the interactions of particles separated by large distances.

• Okay, now I understand it, if one charge is at the origin and say the other is moving along the Y-axis and both their velocities are perpendicular, then there is no magnetic field on the charge at the origin. But a part of my question still remains unanswered, this difference in momentum is stored in the magnetic field or the electric field or they share it? Because the momentum imbalance is magnetic in nature. Sep 18, 2017 at 14:41
• I have already mentioned that momentum is associated with electromagnetic field. In fact in case of time dependent electric current you can not think electric and magnetic field separately. A time dependent electric current produces a time dependent magnetic field which produces an electric field which varies with space and time .The time rate of change of the sum of particle momentum and electromagnetic field momentum is equal to the total force which'd be exerted by the Maxwell stress on the region considered. Sep 18, 2017 at 15:26