No permanent magnets, or other field generation, Just the magnetic field, resulting from the current in the loop, is there a force on a flexible conductor? If so, does the force tend to open the loop?
Consider a circular planar loop traversed ba current I. The magnetic field created by I at a point M in the loop's axis : B=m0 I (sinq)^3/2R , B is vertical ascending.
The force exerced by B is then F=(m.grad)B where m=I pi R^2 n is the magnetic moment of the loop; n is unitary vector diriged in the loop axis.(see attached fig.).
F=m0 pi ( I^2)(R^2)( n.grad) (sinq)^3/2R (where d q/dz= -(sinq)^2/R)
after a simple calculation, we find F=-3/2 m0m I cos(q)=-3/2 m0 I^2 pi R^2 cos(q)
Thus, the applied force is diriged vertically and can't open the loop It only can move it vertically in the top or down sens!!
This is a simple example!
Mohamed,
I respectfully question the analysis, that you present, It disagrees with Fleming's left hand rule for motors, which has B [magnetic flux], I [Current], and F [Force, or motion], in mutually perpendicular directions. As I read your picture, it has B and F in the same direction.
Thank you for your good attention and your rigor. I apologize for the confusion about my explanation.
You are quite right: the magnetic force exerted on the loop is actually given by Laplace law dF= IdlxB (where dl is an oriented element of the loop). Its driection and sense are quitly given by the usual right hand rule (see the new attached figure, I hope that is exact now!). Thus, the force is radial and repulsive (along the radial er vector): dF= I B R d(alpha) er
Consequetly, your question still legitim but this force must be compared to the other mechanical forces that maintain the loop cohesion!
Remark : My previous explanation can be reviewed if we consider a little loop (z>>R) assimilated to a magnetic dipole (with a magnetic moment m) and placed in an inhomegenous field B, so F=(m.grad)B
Although this seems to be a simple problem at first glance, and I agree with the solution (radial force), a closer look reveals something surprising (at least to me; perhaps to others this is a well known curiosity?):
If asked to sketch the field around a radial cross-section of the current loop, I had drawn the field lines in such a way that the point with B = 0 coincides approximately with the center of the cross-section of the conductor. Then the non-vanishing force can be explained by the fact that the field is stronger on the inside of the loop than on the outside.
But if we (in a thought experiment, not feasible in reality, of course) shrink the area of the cross-section of the conductor to zero, i. e. to just one current filament with zero volume, we can expect the force to remain the same, at least qualitatively. To back this expectation, we can replace the ring by a polygon approximating a circle: The field of a single straight section excerts no total force on this same section, but the field of the rest of the polygon is not zero at the position of the section under consideration, and this field has the same direction as inside the loop. As a consequence, the circle defined by B = 0 has to have a slightly larger radius than the circle of the current filament!
But wait, that's impossible because in the static case, curl H = J (J being the current density). And at B = 0, there is certainly curl H != 0 but J = 0. Where is the error in my line of thought? Probably, it starts with the fact that shrinking the cross-section of the current to zero would increase the strength and energy of the field to infinity; but I feel this cannot be the whole story ...
Come to think of it, does not the Biot-Savart law "project" curl B (and curl H, at least in an isotropic, uniform medium) along the axis of the respective differential element of wire? And for an arbitrary point in the plane defined by a circle of current, outside the circle there are always two differential elements whose axes touch the point under consideration. Far from the circle, both curls add up virtually to zero, but near the circle the sum is non-vanishing.
With realistic areas of cross-section, I assume that B = 0 is still located within the conductor.
The force is in such a direction as to expand the loop and so reduce the total energy of the magnetic field. In rail guns this force is used to propel one side of a rectangular loop along the length of the gun.
Well depending on coil or loop geometry a force of significant magnitude can be present- think railgun. One long and thin current carrying loop -with the allowance for one part of the loop (projectile) to move freely along the sides while still maintaining electrical contact is proof enough.The force tends to open the loop - and in the case of the railgun, it CAN but only on one direction (because of railgun design constraints, you would not want it to ..spread but to ,push the projectile forward...). So to sum up - yes there is a force, yes it appears in any kind of powered loop and is one of the reasons the coils on motors and electromagnets seem over-designed mechanically..This force is geometry dependent , proportional to the current in the loop and.. yes it is a result of the Lorentz force. Small currents and usual loop geometries are the reasons this force is small enough to ignore in day-to-day applications- normal low power coils in analog meter readouts (remember those?) and small and not-so-small motors are coated in various types of resin that helps insulate and stabilize. For small enough loops/currents the friction with the loop support where this exists or the elastic nature of the wire are perhaps enough to totally nullify the effects of this force so we don't normally see one wire loops straightening out due to the current that passes through them. But given enough current and a good geometry one can build a nice railgun for example.
I conducted an experimental verification, and an extremely weak force was generated.
I have 2 meters of AWG 30 wire wrap wire, hung from close end terminals. From the center of the wire, hangs a 10 gram weight. 5 amps of current increases spacing between out and return, half a meter down, by about 1.5 millimeter. I turn the current on-and-off, with a switch. The displacement tracks the current every time. I think that the force must be in micrograms.
Can any of the authors, citing rail guns, tell me the proportionalities that make the rail gun so much different than my experiment?
Well sure, the railguns in question use thousands of kiloamps or even more current.The two long rails that give the railgun its name are much closer to one another. This current is delivered in pulses (although DC , pulsed DC is more like a very powerful square wave in the ideal case with very abrupt fronts). Also there is significant difference between the closed loop that has a weight on it and the two railroad-like rails with the projectile carrying mobile sliding armature that moves almost freely on them.
Wikipedia has a very nice article on them, from which I quote
"For single loop railguns, these mission requirements require launch currents of a few million amperes, so a typical railgun power supply might be designed to deliver a launch current of 5 MA for a few milliseconds. As the magnetic field strengths required for such launches will typically be approximately 10 Tesla (100 kilogauss), most contemporary railgun designs are effectively "air-cored", i.e., they do not use ferromagnetic materials such as iron to enhance the magnetic flux. However, if the barrel is magnetic, i.e., produces a magnetic field perpendicular to the current flow, the force is augmented." - so says Wikipedia.
There you have it in a nutshell your power source was a just a few....orders of magnitude too small... 5 mega-amps. Retry with..a much stronger loop and a huge switched capacitor bank or something that generates a million times the current and you will succeed to impress some tens of mega joules of energy to the projectile or really deform that loop if you are more...pacifist.
To finally convince you take a look at this picture of the prototype in question, again from Wikipedia.(link below). The part around which the four gentlemen are discussing is all just the ...power connector of the thing. I could not guess the gauge of those wires but they're power plant big.
.
https://upload.wikimedia.org/wikipedia/commons/5/53/US_Navy_101210-N-7676W-032_Charles_Garnett%2C_left%2C_Naval_Surface_Warfare_Center_Dahlgren_Division_Electromagnetic_Railgun_project_manager%2C_briefs_Vi.jpg
The inductance of two parallel wires joined at the ends is mainly determined by their length (Wikipedia) and is mu0 x length/pi x loge(separation/radius). The energy in an inductor is half the product of the inductance and the square of the current. From this, for two parallel wires 1 m long with the separation about 6 times the radius the inductance is about equal to 8E-7 H or 800 nH. The energy for 1 amp is about 400 nJ. When the length is increased the inductance goes up but the current drops in proportion, so the drop in energy has the slope about 8E-7 J/m or 800 nN. This is for 1 Amp, and is proportional to current squared. For a million amps it would reach nearly a million Newtons, which is quite respectable.
@Newt
A bit of information on railguns is to be found here: https://en.wikipedia.org/wiki/Railgun#U.S._navy_tests
I wouldn't call the approach "efficiency" - more like "brute force". We're talking about 2-digit MJs - per shell! I've seen similar data in some wikipedia articles about the latest U.S. destroyers, one stating that some destroyer class had hardly the electrical power available to install a railgun.
Looking at this again, I now see the point in the question. The parts of the magnetic field inside the current loop should be repelling each other. Generally speaking, the current in the loop should have an inflationary effect on the circuit, although I've never heard reports of this happening.
The opposite effect is used to generate very high magnetic fields - a high current is set up in a loop, then the loop is explosively compressed, so that the flux trapped in the loop is also compressed and the flux density increases.
See https://pdfs.semanticscholar.org/f20b/94c86b9620166bdb82556b5be93261c80c64.pdf for example.
I'm beginning to think that the answer should be "yes", but I've never heard of it being tested. Malcolm White's answer above is further evidence that the answer should be "yes". It's a question of getting a strong enough electric current through a narrow flexible loop, such as that the expansion effect could be detected.
An old classic in electricity. It has been proved to be a relativistic effect.
Pages 130-132 / 330 of "La microphysique que l'on vous conte est-elle bien la bonne ?"
http://jacques.lavau.deonto-ethique.eu/Physique/Microphysique_contee.pdf
In french, but the mathematical notations are international.
also on researchgate : https://www.researchgate.net/publication/313309517_La_microphysique_que_l%27on_vous_conte_est-elle_bien_la_bonne_La_physique_quantique_transactionnelle_expliquee_pour_tous
page 119-121
Book La microphysique que l'on vous conte est-elle bien la bonne ...
Since the question aroused interest, some of you might be interested in causation for the question. One of humanities problems is cleaning up the mess that we are leaving in space. I imagine a sweeper satellite that has some thrust available from stored energy. The sweeper slowly passes a piece of junk, the throat of a towed net is opened, by passing a current through it, thus the piece of junk is captured. The net is closed by stopping the current and accelerating.
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