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

[greypilgrim]

You'd have to calculate the radiation-patterns and predict the radiated-power of the device over the entire frequency spectrum of current flow expected as the capacitor discharges and the current begins to oscillate and decay.

For the moment I'm focusing on the forces acting at very first moment when the left plate is positive and current starts to flow clockwise.

In other words, everything would happen in accord with Newton's laws.

I'm sure it's in accord with conservation of momentum, but is there really a Newtonian 3rd law formulation (i.e. with forces) when it comes to photons? Anyway, that's just semantics.

If that's true, your antenna would accelerate forward like a "photon rocket" until the energy originally stored in the charged capacitor (the "fuel") is expended and the radiation ceases, after which it would coast at constant speed.

Some energy will also dissipated as heat due to the resistance of the wire. Can there be made a a very, VERY rough estimate about the percentages of energy radiated and dissipated as heat in the limit $t\rightarrow\infty$? Or are there calculations/measurements for real antennae?

 

[alan123hk]

If I'm not missing anything, all other forces cancel due to the rigidity of the arrangement. This seems to violate conservation of momentum

My point is that the device will not accelerate in any direction because the law of conservation of momentum cannot be violated. Even from Lenz's law and the Lorentz force, there is no sufficient reason to infer that the entire device is accelerated in any direction due to the forces exerted on the transverse wire.

Note that the wires in this structure are not infinitely long. The direction of magnetic field created by the current flowing through these wires slowly changes with distance and angle.

The direction of the magnetic field generated by the two wires connecting the two plates of the capacitor inside this approximately ring-shaped structure is downward, but this does not prove that this downward magnetic field will extend to evenly wrap the entire transverse wire.

If you think that's roughly the case, how do you prove that the magnetic field generated by the current in the transverse wire does not produce the same reaction force on the two wires connecting the capacitor?

 

[greypilgrim]

The direction of the magnetic field generated by the two wires connecting the two plates of the capacitor inside this approximately ring-shaped structure is downward, but this does not prove that this downward magnetic field will extend to evenly wrap the entire transverse wire.

Not evenly, but enough to produce a force in the direction given. That's how railguns work.

If you think that's roughly the case, how do you prove that the magnetic field generated by the current in the transverse wire does not produce the same reaction force on the two wires connecting the capacitor?

Most forces cancel due to the (anti-)symmetry of the structure, since the currents in the parallel wires go opposite ways.

But yes, maybe the finiteness of the wires leads to non-cancelling contributions that add up go an exact counterforce. I might try and calculate that at some point, but I assume that could become very ugly.

 

[hutchphd]

May I suggest Feynman Lectures II 27 about field momentum. I will not try to recapitulate the master's explanation. As the slider accelerates it creates nonzero field momentum in its wake which then interacts with the rails and capacitor making them "recoil". It is left as an exercise for the interested student to verify Newton. I will rely on physics faith here.

 

[vanhees71]

Also one should note that, when taking into account field momentum to get the conservation laws right, you need to calculate everything relativistically. This gives rise to the somewhat unfortunate notion of "hidden momentum", though it's just momentum done relativistically right. There are nice examples in Griffiths's E&M textbooks and (imho even better) on McDonald's website:

http://kirkmcd.princeton.edu/examples/

The most simple "magnetostatic" example is

http://kirkmcd.princeton.edu/examples/penfield.pdf

 

[renormalize]

I'm sure it's in accord with conservation of momentum, but is there really a Newtonian 3rd law formulation (i.e. with forces) when it comes to photons? Anyway, that's just semantics.

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.

 

[renormalize]

My point is that the device will not accelerate in any direction because the law of conservation of momentum cannot be violated.

When the capacitor discharges, time-varying electric and magnetic fields exist in and around the device. How do you know that the device doesn't thereby emit a pulse of non-isotropic electromagnetic radiation and, in response, it recoils in the opposite direction?

 

[greypilgrim]

As the slider accelerates it creates nonzero field momentum in its wake which then interacts with the rails and capacitor making them "recoil".

Okay, if the field momentum is fully absorbed by the rails and capacitor, then it obviously won't accelerate.
I think the piece I'm missing here is why part of the field momentum can't just fly away as radiation, not interacting with rails and capacitor. Then global momentum would still be conserved.

Actually, isn't that what happens in a directional antenna? The Poynting vector is non-isotropic, so there is momentum carried away from the antenna and doesn't interact with it again. So there must be a steady recoil force acting on the antenna.
Or even a simple flashlight ...
Is there a particular reason that this doesn't happen here?

 

[hutchphd]

The presence of the capacitor and the rails means that this is not a magnetic dipole in free space . It is, in fact, what it is. The exact solution is beyond my ability to intuit. Therefore my fundamental beliefs (carried out within the doctrinal strictures of lorentz invariance) give me solace. Examination of less ambiguous situations serve to shape these beliefs over (nearly) a lifetime of struggle. No net momentum for the system. (and alas No EM drive......which is similar)

 

[greypilgrim]

The presence of the capacitor and the rails means that this is not a magnetic dipole in free space . It is, in fact, what it is. The exact solution is beyond my ability to intuit. Therefore my fundamental beliefs (carried out within the doctrinal strictures of lorentz invariance) give me solace.

So, would you say a directional antenna – where there's also capacitors present! – does not experience recoil?

 

[hutchphd]

No I would not be foolish enough to answer such an ill-defined question. It is certainly true that a flashlight will accelerate (if turned on) in space.
Did you not read my answer?

The exact solution is beyond my ability to intuit.

 

[renormalize]

The presence of the capacitor and the rails means that this is not a magnetic dipole in free space . It is, in fact, what it is. The exact solution is beyond my ability to intuit. Therefore my fundamental beliefs (carried out within the doctrinal strictures of lorentz invariance) give me solace. Examination of less ambiguous situations serve to shape these beliefs over (nearly) a lifetime of struggle. No net momentum for the system. (and alas No EM drive......which is similar)

@hutchphd, with a time-varying current this device is a (possibly very inefficient) antenna that will radiate power at some (likely very low) power. If the radiation pattern is directional, in free-space the device will recoil (probably at a glacial acceleration) like a photon rocket. This is completely consistent with Poynting's theorem and Lorentz invariance.

 

[greypilgrim]

Isn't it conversely very hard to imagine a device with (non-isotropically) accelerated charges that would completely reabsorb all radiation and hence not be accelerated. This sounds like the special, probably unrealistic, case to me, not the other way around.

 

[hutchphd]

The radiation (far-field) effects will be tiny compared to the "recoil" from the "rail gun". It is like reducing the recoil of a 12 gauge shotgun by taping a flashlight to the barrel.

 

[alan123hk]

When the capacitor discharges, time-varying electric and magnetic fields exist in and around the device. How do you know that the device doesn't thereby emit a pulse of non-isotropic electromagnetic radiation and, in response, it recoils in the opposite direction?

Because the original description of OP seems to use simple concept of magnetic force on a current-carrying conductor in the approximate DC model to infer that the device will accelerate, and suspects that the law of conservation of momentum is violated, and there is no mention of AC and radiation, so I discuss it on this basis.

Electromagnetic waves and photons have momentum. When an object emits electromagnetic waves, of course there will be a reaction force. According to the law of conservation of momentum, the object itself inevitably accelerates. But it's not bound to happen. For example, even if it is not an isotropic antenna, as long as the electromagnetic waves emitted by the antenna in any direction are symmetrical and the net force on the antenna itself is zero, the antenna will not accelerate in free space. But of course, perfect symmetry almost never happens in reality.

Measuring the recoil of a glowing object is difficult for the average amateur science enthusiast. For example, I once held a 120,000 lumens flashlight and didn't feel any recoil when I turned it on.

 

[hutchphd]

Isn't it conversely very hard to imagine a device with (non-isotropically) accelerated charges that would completely reabsorb all radiation and hence not be accelerated. This sounds like the special, probably unrealistic, case to me, not the other way around.

I think this is a good point, although there are some symmetries here. Most of the interactions of consequence are in the near-field regions where the fields are larger. The rigidity of the structure is itself electromagnetic and so I cannot pretend to intuit this. It will likely oscillate and possibly rotate! Any accelerations produced by radiation will be miniscule (as you mention) and not require reviewing Newton, IMHO. I do appreciate the question.....

 

[greypilgrim]

Note that the wires in this structure are not infinitely long. The direction of magnetic field created by the current flowing through these wires slowly changes with distance and angle.

The direction of the magnetic field generated by the two wires connecting the two plates of the capacitor inside this approximately ring-shaped structure is downward, but this does not prove that this downward magnetic field will extend to evenly wrap the entire transverse wire.

If you think that's roughly the case, how do you prove that the magnetic field generated by the current in the transverse wire does not produce the same reaction force on the two wires connecting the capacitor?

Just an update on this: I sat down to calculate the magnetic forces on the three wires (without the capacitor) mentally ready for some ugly vector integrals. But everything turned out way easier than I thought: There is a closed-form expression for the magnetic field of a finite straight wire (google it) that even simplifies in this case, and what is even better, the field is still concentric everywhere, even beyond the wires. That surprised me at first (as it is very different for the electric field of a finite line charge), but actually follows directly from Biot-Savart.

This makes it straightforward that the Lorentz forces on the parallel wires exactly cancel (since the currents are oppositely directed), and there's even a closed-form expression of the Lorentz force on the transverse wire (well, at least Mathematica says so), that's of course nonzero and exactly in the direction of the green arrow.

Now of course this is not physical as it violates the continuity equation, that's why I added the capacitor in the first place.

 

[renormalize]

Now of course this is not physical as it violates the continuity equation, that's why I added the capacitor in the first place.

@greypilgrim can you answer the question I asked in post #36 about the scale of your hypothesized configuration, i.e., order-of-magnitude values for: the area of, distance between, and maximum voltage on the charged capacitor plates, as well as the total mass of your device? (I want to make a crude estimate of the possible momentum imparted by the emission of radiation.)

 

[alan123hk]

Just an update on this: I sat down to calculate the magnetic forces on the three wires (without the capacitor) mentally ready for some ugly vector integrals. But everything turned out way easier than I thought: There is a closed-form expression for the magnetic field of a finite straight wire (google it) that even simplifies in this case, and what is even better, the field is still concentric everywhere, even beyond the wires. That surprised me at first (as it is very different for the electric field of a finite line charge), but actually follows directly from Biot-Savart.

This makes it straightforward that the Lorentz forces on the parallel wires exactly cancel (since the currents are oppositely directed), and there's even a closed-form expression of the Lorentz force on the transverse wire (well, at least Mathematica says so), that's of course nonzero and exactly in the direction of the green arrow.

You are right, I didn't notice before that according to Biot-Savart's law, the magnetic field produced by a finite length of wire remains symmetrical and concentric along the axis even beyond the length of the wire segment.
Yes, if we extrapolate from this magnetic field spatial distribution, it seems almost certain that the upper transverse current conductor will experience an upward force and the entire device should accelerate in the direction of the green arrow.

But this conclusion violates the law of conservation of momentum. I admit that in this case it is natural to think of performing a comprehensive time-varying electromagnetic analysis to find the answer. In fact, this device can be thought of as a closed loop circuit consisting of a capacitor, an inductor and a resistor in series. When the resistance is small and the switch is closed, high-frequency current oscillation will occur, producing radiation over a period of time. Therefore, this irregularly shaped structure would experience a reaction force from the electromagnetic wave emission, so it should accelerate or start rotating in a certain direction.
But this is a very complex problem that may only be solved through advanced simulation software.

However, I personally believe that with this device, the effects of radiation reaction forces are minimal in most cases. I still believe it is possible to solve this problem without considering radiation. There may be some details we haven't thought of yet. But to be honest, I still have no clue.

 

[greypilgrim]

The capacitor was the only thing that came to my mind where the terminals are separated over a distance with nothing (except an E-field) in between. But it complicates things since the voltage doesn't remain constant.

Is there a voltage source (or EMF) imaginable with constant voltage (for any nonzero amount of time) where the plus and minus terminals are physically separated such that there is no internal flow of charges (such as there would be in a battery)?

 

[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?