I Does a Railgun's Current Violate Conservation of Momentum?

[Ibix]

But this conclusion violates the law of conservation of momentum.

Note that if the wire is accelerating upwards the current density in the wire is no longer parallel to the wire's length. So the simple analysis of a steady current along a wire is a guideline at best.

 

[alan123hk]

Note that if the wire is accelerating upwards the current density in the wire is no longer parallel to the wire's length. So the simple analysis of a steady current along a wire is a guideline at best.

Sorry, maybe my English is not good, I don't quite understand what "the current density in the wire is no longer parallel to the length of the wire" means. But the name "current density" you mentioned is relevant to what I was thinking.

My idea is that all entities in this device are actually three-dimensional, with the wire having a diameter and the two capacitor parallel plates having a thickness. When the capacitor begins to discharge, the current flowing through the circuit and the magnetic field it creates begins to change with time, so the current density distribution in all conductors also changes with time due to the influence of the magnetic fields produced by other conductors. Note that changes in the current density distribution within a wire or capacitor plate cause an equivalent current to flow in a specific direction. Could the forces resulting from the interaction between these equivalent currents and magnetic fields provide clues to answering this question?

https://www.mathworks.com/matlabcen...s/submissions/47368/versions/1/screenshot.jpg

 

[vanhees71]

Note that if the wire is accelerating upwards the current density in the wire is no longer parallel to the wire's length. So the simple analysis of a steady current along a wire is a guideline at best.

The total momentum of the system consisting of matter and the electromagnetic field is conserved. It's the way, how momentum conservation is consistent with the relativistic space-time structure and causality, given that causal effects "propagate" with the finite speed of light and interactions are thus not instantaneous as in Newtonian physics.

 

[greypilgrim]

The total momentum of the system consisting of matter and the electromagnetic field is conserved.

Is there a "Newtonian 3rd law" formulation of this, i.e. can momentum transfer to or from the field described by a force?

 

[vanhees71]

The field concept is the only hitherto known way to formulate interactions, i.e., the interaction between far-distant electric charges is due to the electromagnetic field around them. The Lorentz force on a charged body is described as $\vec{F}=q (\vec{E}+\vec{v} \times \vec{B})$ with the arguments of the fields at the position of this charged body ("locality"). The electromagnetic field itself is also an entity in its own right participating in the dynamics. Thus there is an exchange of energy, momentum, and angular momentum between matter and the electromagnetic field. It's a bit difficult to make sense of a Newtonian 3rd Law in this context.

 

[greypilgrim]

Let's do a back-of-the-envelope calculation of the maximum radiation-reaction of an antenna. To proceed, we need to establish a scale for the problem. Can you provide the following?

• The area $A$ in $\text{m}^2$ of one of the pads in the capacitive region at the rear of your device.
• The separation distance $d$ in $\text{m}$ between the pads.
• The initial voltage difference $V$ imposed on the pads.
• The total mass $M$ in $\text{kg}$ of the device.

I have no intuition about a reasonable scale. Maybe for startes let's take something lab-scale that could in principle be tested, like $A=1\ \text{m}^2, d=1\ \text{m}, V=10000\ \text{V}, M=1\ \text{kg}$?

 

[greypilgrim]

Thus there is an exchange of energy, momentum, and angular momentum between matter and the electromagnetic field. It's a bit difficult to make sense of a Newtonian 3rd Law in this context.

This is why I'm confused that some people are insisting on the 3rd law in this thread and even accuse me of denying it, even though I said multiple times that I don't question the (more general) conservation of momentum, but the field needs to be included as carrier of momentum.

 

[vanhees71]

Physics made some progress since Newton, indeed. The 3rd law in Newton's form is at odds with the relativistic space-time structure, because there's no "instantaneous action at a distance" possible in relativistic physics.

In Newtonian physics the 3rd Law immediately leads to momentum conservation. Since Noether we know that momentum conservation is a necessary condition on the dynamical laws to be consistent with spatial homogeneity, i.e., translation invariance implies momentum conservation in the sense of Noether's theorem. So there's no need to postulate the 3rd Law to get momentum conservation.

The problem with the incompatibility of actions at a distance with relativistic causality has been solved already before relativity has been discovered in the early 20th century by Faraday's ingenious discovery of the field concept and the paradigm of locality, which until today is at the heart of the most successful theories, i.e., relativistic quantum field theory and the Standard Model, describing all (known types of) matter and all interactions except the gravitational interaction and general relativity, which describes the gravitational interaction in the sense of a classical field theory. It's not fully consistent (quantum concerning matter and the electroweak and strong interactions but classical concerning the gravitational interaction), but it's all strictly based on the locality paradigm.

 

[renormalize]

I have no intuition about a reasonable scale. Maybe for startes let's take something lab-scale that could in principle be tested, like $A=1\ \text{m}^2, d=1\ \text{m}, V=10000\ \text{V}, M=1\ \text{kg}$?

OK, let's consider an idealization of your device, consisting of:

• An antenna engineered to direct electromagnetic radiation in a single direction.
• A parallel-plate capacitor of area $A$ and separation $d$, charged to voltage $V$, that energizes the antenna.
• Conductors with no resistive losses.

Then assume, upon capacitor discharge, that the antenna emits a unidirectional burst of radiation carrying energy $E_{\text{rad}}$ and momentum $E_{\text{rad}}/c$, and the whole device of mass $M$ recoils in the opposite direction with speed $v$.

Ignoring the capacitor's fringing fields, its capacitance is $C=\epsilon_0\frac{A}{d}$ and it stores an energy $E_{\text{cap}}=\frac{1}{2}C V^2$. By conservation, the total energy and linear-momentum is the same before and after capacitor discharge, so:$$E_{\text{cap}}=E_{\text{rad}}+\frac{1}{2}Mv^{2},\quad0=Mv-\frac{E_{\text{rad}}}{c}\tag{1,2}$$
(Here I've used the non-relativistic expressions for the energy and momentum of the device because I anticipate that $v\ll c$.) Solving the conservation eqs.(1,2) gives:$$\frac{v}{c}=\frac{E_{\text{rad}}}{Mc^{2}}=\sqrt{1+2\left(\frac{E_{\text{cap}}}{Mc^{2}}\right)}-1\approx\frac{E_{\text{cap}}}{Mc^{2}}\text{ for }\frac{E_{\text{cap}}}{Mc^{2}}\ll1\tag{3}$$Now it's time to put in your numbers (slightly adjusted): $A=1 \text{ m}^2,d=1\text{ }\mu\text{m},V=10\text{ kV},M=1\text{ gm}$. (Note that I've reduced your distance $d$ by $10^{-6}$ to boost the capacitance $C$ and lightened your mass $M$ by $10^{-3}$ to help the device "fly"!) Plugging these into (3), we find:$$C=8.85\:\mu\text{F},\:E_{\text{rad}}\approx E_{\text{cap}}=443\text{ J},\:Mc^{2}=9\times10^{13}\text{ J},\:v=1.48\text{ mm/s},\:\frac{1}{2}Mv^{2}=1.09\times10^{-9}\text{ J}$$So this idealized device achieves a speed on the order of just a millimeter per second. And any real test device will almost certainly be slower due to a much poorer directivity of the radiated momentum, as well as conductor losses. But maybe you could detect the effect by suspending the test device in a vacuum and looking for minute movements?

 

[alan123hk]

Thank you for introducing this useful and informative calculation process.

However, I noticed that doing this experiment can be very difficult and expensive.
Capacitor parameters $~~~A=1 \text{ m}^2,d=1\text{ }\mu\text{m},V=10\text{ kV},M=1\text{ gm}$
If I calculated correctly,
the force between the two plates is 445174 Ton-Force
In order to avoid dielectric breakdown of the capacitor, the dielectric material between the two plates of the capacitor can only be a perfect vacuum.. https://en.wikipedia.org/wiki/Dielectric_strength
So what material should we use to separate and secure the two parallel plates of the capacitor while keeping the weight to only 1gm?

 

[renormalize]

Thank you for introducing this useful and informative calculation process.

However, I noticed that doing this experiment can be very difficult and expensive.

Agreed, my idealized device is hardly practical. So let's make a more realistic example by increasing the total mass and storing energy in a commercial capacitor. For example, on Amazon you can find this $1\text{ gm}$ Torong HV ceramic chip capacitor:

characterized by $C=10\text{ nF},V=30\text{ kV}$. Coupling this to a lightweight antenna structure, we could perhaps put together a device of, say, $M=50\text{ gm}$ total. Energy-momentum conservation then yields:$$E_{\text{rad}}\approx E_{\text{cap}}=4.5\text{ J},\:Mc^{2}=4.5\times10^{15}\text{ J},\:v=300\text{ nm/s},\:\frac{1}{2}Mv^{2}=2.25\times10^{-15}\text{ J}$$So the recoil speed $v$ is only on the order of hundreds of nanometers per second, and this is an upper limit due to resistive losses and the reduced directivity of a real antenna. Nevertheless, I can imagine that a sensitive lab experiment in vacuum might be able to detect the motion.

 

[alan123hk]

So the speed v is only on the order of hundreds of nanometers per second, and this is an upper limit due to resistive losses and the reduced directivity of a real antenna. Nevertheless, I can imagine that a sensitive lab experiment in vacuum might be able to detect the motion.

Agree that this movement should be detectable with advanced scientific instruments.