# Electric field of an electron in motion in a wire

How do I correctly model the electric field of an electron in motion in a wire? I could treat the electron as a point charge moving through the wire. If I use the Liénard Wichert equations, they will predict radiation if the wire turns, since the electron is being accelerated here. But we know that constant currents doesn't radiate like this.

Alternatively I could view the electron as a wave function distributed over the entire wire. Which equations would I then use to obtain the field? And would the wave function then be used as a chargedistribution?

• Real Wires do radiate in wire turns. However due to a continuous approximation it suggests there is no radiation. to find the field of a single electron you must use lienard wichert equations. Modelling a wire as a continuous current density is an approximation. Im sure someone else will answer with better answer on the limiting process. Commented Jan 14, 2022 at 22:15

The electrons in a conductor occupy quantum momentum and energy states in a band structure. I think that it is not possible to model them as classical moving charges as for Liénard Wichert equations.

For example, even without any external E-field applied, each electron has a momentum, so they are all moving. As the conductor has boundaries, at some time they must change their momentum, otherwise they would escape outside. But that accelerations don't generate EM waves of course.

It is similar to electrons in atomic orbitals. They can have angular momentum, but don't radiate.

Alternatively I could view the electron as a wave function distributed over the entire wire

It is the band structure. A simplified model to get it is the Kronig-Penney's. Its main hypothesis is the eletric field on the electrons from the atoms of the lattice, that results in a periodic potential. The effect of the electric field from each electron on its neighbours is not part of the calculation. I don't know a model that also includes it.

• Thank you Claudio. I agree with your position. Yet I don't think it answers the question. If I cannot model the charge as a point charge, and use the Liénard-Wichert equations, how do i then model the field of the charge? Is the wave function to be understood as a chargedistribution, and which equation then describes the field correctly? Thanks Commented Jan 15, 2022 at 21:15

If it's an Ohmic material $$\vec{J}=\sigma \vec{E}$$. Current density is proportional to electric field.

$$J=I/A, E=V/l$$ Where $$I$$ is current $$A$$ is cross sectional area, $$V$$ is voltage between endpoints of the wire, and $$l$$ is length of the wire.

From here we get $$\sigma =\frac{Il}{AV}$$

Dimensional analysis tells us the numerator is Coulomb Meters per second. That's charge times velocity. Charge is contributed by many electrons electrons. If we multiply numerator and denominator by length, we have volume times voltage in the denominator.

So $$\sigma =\frac{(N)evl}{(Al)V}=\frac{ne^2vl}{eV}$$

Where $$n=N/Al$$ is the charge carrier density and numerator and denominator have been multiplied by $$e$$ to get an energy term $$(eV)$$ in the denominator.

$$eV\approx \frac{1}{2}mu^2$$ where $$m$$ is mass of an individual electron and $$u$$ is the velocity obtained when an electron traverses that potential difference.

$$\sigma=\frac{ne^2}{m} (\frac{2vl}{u^2})$$. $$v$$ is the average speed of an individual electron, $$u^2/2l$$ is as an average acceleration. Average velocity over average acceleration gives you a characteristic time of the system, in this case the time is the mean free flight time.

So $$\sigma=\frac{ne^2\tau}{m}$$

For a more rigorous derivation, terms like momentum and mean free path can be used to deduce a relationship between the conductivity and motion of the electrons in a wire. This gives you the Drude Model

• Still is an approximation due to being continuous,I think op is trying to think about individual electrons with a vacuum gap between Commented Jan 14, 2022 at 22:28