[Q] Lorentz force in momentum conservation view

[a1titude]

Somebody, please explain to me about the Lorentz force between two parallel current-carrying wires in terms of the momentum conservation as the electromagnetic fields has momentum density. Especially, (1), If a short rectangular magnetic pulse due to an electric pulse in one wire will still affect the Lorentz force in another wire, and (2) If the two wires carry direct current, there does not even exist electromagnetic waves to carry momentum.

 

[Dale]

This is a bit of a difficult question mathematically. I think that it is best addressed with covariant math. For a straight wire on the $z$ axis carrying a steady current $j$ the EM field tensor is given by $$F^{\mu\nu}=
\left(
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{j x}{x^2+y^2} \\
0 & 0 & 0 & \frac{j y}{x^2+y^2} \\
0 & -\frac{j x}{x^2+y^2} & -\frac{j
y}{x^2+y^2} & 0 \\
\end{array}
\right)$$

If you work it out, the EM stress energy tensor is given by $$T^{\mu\nu}=
\left(
\begin{array}{cccc}
-\frac{j^2}{2 \left(x^2+y^2\right)} & 0 &
0 & 0 \\
0 & \frac{j^2 \left(y^2-x^2\right)}{2
\left(x^2+y^2\right)^2} & -\frac{j^2 x
y}{\left(x^2+y^2\right)^2} & 0 \\
0 & -\frac{j^2 x
y}{\left(x^2+y^2\right)^2} & \frac{j^2
(x-y) (x+y)}{2 \left(x^2+y^2\right)^2}
& 0 \\
0 & 0 & 0 & -\frac{j^2}{2
\left(x^2+y^2\right)} \\
\end{array}
\right)$$

Note that even though this is for a steady current, there are still momentum and energy fluxes in the stress energy tensor. In other words, it is not just EM waves that move energy and momentum, but EM fields in general. Indeed, EM waves carry their energy and momentum simply by virtue of being EM fields, not specifically because they are waves.

Assuming that I didn't make any mistakes in the math the conservation of energy and momentum in the absence of external charges/currents/fields can be shown simply by the fact that $$\partial_{\nu}T^{\mu\nu}=0$$ So even though momentum is being transported in this field, it is still conserved.

 

[a1titude]

Somebody, please explain to me about the Lorentz force between two parallel current-carrying wires in terms of the momentum conservation as the electromagnetic fields has momentum density. Especially, (1), If a short rectangular magnetic pulse due to an electric pulse in one wire will still affect the Lorentz force in another wire, and (2) If the two wires carry direct current, there does not even exist electromagnetic waves to carry momentum.

I'd like to make it more clearer. The power and the momentum of electromagnetic fields can be calculated from the Poynting vector which includes both the electric field E and the magnetic flux density B. In the situation (1), a magnetic pulse can be thought to have the Poyning vector since it has rising and falling moments, no matter how short they are, resulting in a symmetric radiation of power and momentum all around. But, at the position of another wire, only a single direction of momentum appears. Moreover, the situation (2) has no E at all. Where am I misleading myself? I want to understand this in undergrad level knowledges.

 

[renormalize]

Moreover, the situation (2) has no E at all. Where am I misleading myself? I want to understand this in undergrad level knowledges.

In situation (2) there is only a $B$ field in the space between two parallel wires, so (as you say) the Poynting vector is zero and there is no linear- or angular-momentum carried by the EM field (but there is an EM energy-density proportional to $B^2$). At the instant before the DC currents flow, the two wires are stationary in their center-of-mass (COM) frame and thus have no momentum. If the wires are free to move, when the current flows they begin to accelerate and each develops an individual momentum vector, but those vectors are always equal and opposite in the COM frame and add to zero. So there is zero total momentum in the COM frame for the EM field and the wires, both before and after the current flows.

 

[Dale]

Moreover, the situation (2) has no E at all. Where am I misleading myself?

That is why I calculated the full EM stress energy tensor. Although the Poynting vector is zero, the full stress energy tensor is nonzero.

 

[Ibix]

If the two wires carry direct current, there does not even exist electromagnetic waves to carry momentum.

If there is only a parallel DC current then this implies that the wires are constrained not to move, because otherwise the current flows would have components perpendicular to the length of the wire. Thus there is no momentum being transferred.

If your wires are free to move then you have a changing magnetic field, and hence an electric field and (I would expect) a non-zero Poynting vector as momentum flows through the field.

(I think this is basically a different take on what @renormalize said.)

 

[Vanadium 50]

Poynting vector

Is a vector, right?

As mentioned, energy density is a scalar. Two different things.

 

[Dale]

please explain to me about the Lorentz force between two parallel current-carrying wires in terms of the momentum conservation as the electromagnetic fields has momentum density

the power and the momentum of electromagnetic fields can be calculated from the Poynting vector

To be a little more clear than what I was saying before. In order to understand the Lorentz force in terms of momentum density, the Poynting vector is insufficient. You need the stress energy tensor. It is possible to understand the Lorentz force in terms of momentum conservation, but not if you only use the Poynting vector.

The Lorentz force can be written in terms of the fields as $f_\mu = F_{\mu \nu} J^{\nu}$ or in terms of energy and momentum conservation as $f^\mu = -\partial_\nu T^{\mu \nu}$ where $f$ is the Lorentz four-force, $F$ is the electromagnetic field tensor, $J$ is the current four-density, and $T$ is the electromagnetic stress energy tensor. The Poynting vector $S$ is just a small part of $T$.

 

[a1titude]

Two wires A and B are in parallel. A short pulse current applied to wire A, resulting in a cylindrical shaped magnetic field around the wire are increasing its radius. The momentum of the field are heading to every radial directions. During the magnetic field passing wire B where DC current is flowing, wire B feels the Lorentz force in a direction according to the direction of the DC current, and obtains momentum in the same direction as the force. In this situation, it doesn't seem that momentum is conserved.

 

[Dale]

it doesn't seem that momentum is conserved

Are you going to actually do the calculation or are you just going to say “it doesn’t seem”?

Since conservation of momentum can be proven from first principles, it seems that “it doesn’t seem” is insufficient.