How is energy conserved when a moving charge has false ideas about positions of other charges

Question

An electron is shot towards a target that is negatively charged. While the electron is traveling, the target makes an abrupt move towards the electron. While the information that the target moved is traveling from the target to the electron, the electron behaves like an electron that is moving towards a target that is in the original position.

How can energy be conserved when an electron that is moving towards a nearby charge behaves like it was moving towards a far away charge? Seems we end up with electron being at 2 meters distance from the target, while the electron had enough energy to travel to at most 4 meters distance from the target.

It also seems to me that "moving the target requires energy" is not a solution to this problem.

Answer 1

Moving the target requires energy.

Suppose the target is an infinite plane with constant charge density $\sigma$. It will not radiate when you move it because the electric field is constant everywhere. Suppose the test charge $q$ is small enough that its radiation is negligible.

The electric field of the plane is $2\pi\sigma$ in the direction perpendicular to the plane.

The charge begins a distance $d$ from the plane. The potential energy in the system is $-2\pi\sigma q d$ (define to be zero when $d=0$). The force on the charge is $2\pi\sigma q$ and by Newton's third law there is an equal and opposite force on the plane. (We are assuming there is no radiation, so momentum of the charge carriers is conserved, and Newton's third law holds.)

The charge also has some kinetic energy $T$.

We move the plane towards the charge a distance $\Delta d$. This takes energy because the force of the charge on the plane does negative work. The energy required is the force multiplied by the distance, or $2\pi\sigma q \Delta d$.

The new distance of the charge from the plane is $d - \Delta d$, so the new potential energy is $-2\pi\sigma q (d-\Delta d)$. The potential energy has increased by $2\pi\sigma q \Delta d$, exactly the amount of work that had to be put into the system to move the plane. The charge still has kinetic energy $T$, so energy is conserved in that the change in energy of the system is equal to the energy that was used to move the plane.

Answer 2

The heart of your concern is the non-locality, right? The way to relieve your mind is to think of everything locally instead of globally.

We first learn about energy conservation as a global law: The total energy in some (large) region remains constant as a function of time. But there's another, local formulation of energy conservation: The rate of change of the energy density at any given point equals the rate of energy flow into that point. Formally, $$ {\partial\rho\over\partial t}=-\nabla\cdot{\bf J}, $$ where $\rho$ is energy density and ${\bf J}$ is energy current.

(Technical note: Actually, $\rho$ is part of a 4-vector and ${\bf J}$ is part of a rank-2 tensor, and they're usually called other things rather than $\rho,{\bf J}$.)

Roughly speaking, the second version says that the energy conservation accounting works out correctly point by point in space, which is a stronger statement than global energy conservation. In fact, global energy conservation follows from local energy conservation, but the reverse isn't true. A world in which energy spontaneously disappeared in this room and appeared in Antarctica, with nothing flowing from here to there, would satisfy global but not local energy conservation.

The theory of electricity and magnetism satisfies local energy conservation, not just global energy conservation. That is, when you take into account the energy stored in the electromagnetic fields, you can keep track of the energy density and energy current at each point, and the books balance everywhere. Moreover, the energy density and energy current are well-defined functions of the electric and magnetic fields, so they change smoothly and causally with time.

Once you adopt that view of energy conservation, it becomes obvious that energy is conserved, and there are no non-locality or causality issues, even in situations such as the ones you describe. The energy flows around smoothly and causally, in such a way that the total is conserved.

Answer 3

As I mention in the comments, gravitational effects take care of conservation of energy when gravitating masses are accelerated non-locally, as described in the comment.

Surprisingly, when charged objects are accelerated non-locally, gravitational effects take care of conservation of energy. Energy gravitates after all, and energy is needed to accelerate charges.

Electro-magnetics effects have some role in here too: they ensure that a lot of energy is required to very rapidly accelerate a charge to speed 0.99999999999 c.

I bet this is how it works.