How to determin the momentum eA of electromagnetic field?

[Dale12]

I got a question when I was reading a paper, it describe a periodic structure composed of many infinite metal wire, and then try to calculate the momentum per unit length of the wire like these when there is current through wire:


electrons in a magnetic field have an additional contribution to theri momentum of eA,where e is charge value and A is vector potential of magnetic field B, and therefore the momentum per unit length of the wire is.
$$e\pir^2nA(r)=\frac{\mu_0 e^2(\pi r^2n)^2}{2\pi}ln(a/r)$$

because
$$H(R)=\frac{I}{2\pi R}=\frac{\pi r^2nve}{2\pi R}$$
where I is the current flowing in the wire, R is the distance from the wire and v is the mean electron velocity. we can express this field as
$$\vec{H(R)}={\mu_o}^{-1}\nabla\times\vec{A}$$
where A(R)=\frac{\mu_0\pi r^2nve}{2\pi}ln(a/R)[/tex]
and a is the lattice constant as listed in the picture.

I know that from classical mechanics we could write momentum of a particle in EM field as P=mv+eA, and here when talking about the momentum of wire, we ignore "mv" because it is from electrons in the wire.but we should calculate moment of the EM field.

but when calculate A in classical electro-magnetic theory, we use
$$A_z=\frac{\mu I}{4\pi}\int_{-\infty}^{+\infty}\frac{dz}{\sqrt{z^2+a^2}} and if we want to get the result like A(R)=\frac{\mu_0\pi r^2nve}{2\pi}ln(a/R)$$
we should choose a point as zero vector potential, here we should choose it as R.
but since A is arbitrary if we another point, so my question is that here why we choose R to get the momentum of wire.

thanks for your patience.

 

[Evalesco]

It makes sense that you would be confused about why we choose R as a reference point for the vector potential. The reason for this is because the momentum of the wire is dependent on the electric current running through it, which is affected by the distance from the wire (R). By choosing this point as our reference, we can calculate the momentum of the wire more accurately. Additionally, it is important to note that the vector potential is arbitrary, so you can choose any reference point that you feel is most appropriate.

 

[astrokreng]

Hello,

Thank you for your question. Determining the momentum of an electromagnetic field is a complex task that requires a thorough understanding of classical electromagnetism and quantum mechanics.

To begin, it is important to note that the momentum of an electromagnetic field is not a well-defined concept in classical electromagnetism. In classical mechanics, momentum is defined as the product of mass and velocity, but electromagnetic fields do not have mass. Therefore, the concept of momentum for an electromagnetic field is more accurately described as the momentum density, which is the momentum per unit volume of the field.

In terms of the specific scenario described in the paper you were reading, the momentum per unit length of the wire can be calculated by considering the momentum contribution from the electrons in the wire and the momentum contribution from the electromagnetic field. The momentum contribution from the electrons is given by eA, where e is the charge value and A is the vector potential of the magnetic field B. This is a well-known result in classical electromagnetism.

The momentum contribution from the electromagnetic field is more complicated to calculate. It involves considering the Lorentz force on the electrons in the wire, as well as the energy and momentum of the electromagnetic field itself. The equation provided in the paper, which is derived from classical electromagnetism, takes into account the current flowing through the wire, the distance from the wire, and the mean electron velocity.

In terms of your question about the choice of R as the reference point for the vector potential, this is simply a mathematical convenience. The vector potential is an arbitrary quantity and can be chosen at any point. In this case, choosing R as the reference point simplifies the calculation and allows for a clear interpretation of the result in terms of the momentum density of the electromagnetic field.

I hope this helps to clarify the concept of momentum for an electromagnetic field. It is a complex topic and requires a deep understanding of both classical and quantum mechanics. I encourage you to continue exploring this topic and seeking out additional resources for a more comprehensive understanding.