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# Magnetic fields within hollow wires

## Magnetic fields within hollow wires

(OP)
Does anyone know a way to calculate the magnetic field acting on a wire, when this wire is placed in the center of a bigger hollow wire?

I have the outer and inner diameter of the hollow wire, the diameter of the wire in the center (this two values define the radial distance between the wires), the current flowing through the hollow wire and the current flowing through the wire in the center, both wires having the same length, assuming either infinite or limited length.

Calculation of the field produced by two wires versus distance is easy (see link) but I couldn't find any calculation for the hollow wire configuration.

http://hyperphysics.phy-astr.gsu.edu/Hbase/magnetic/wirfor.html#c1

Field distribuition in simulation applets show a homogeneous concentric field vector geometry when a conductor is placed within a hollow wire, but no indication for the calculation of the field intensity.

Replies continue below

### RE: Magnetic fields within hollow wires

Superposition.  You have an outer hollow conductor.  Within the center of that you have an inner solid conductor (correct).

Field due to the outer hollow conductor carrying uniform axial current is 0 in the center.  To convince yourself look at tangential and radial components within the hollow conductor center:

First tangential component.  Integral Htangential dot dL = Ienclosed.  Draw the countour of the integral as a circle.  There is no Ienclosed (remember we are looking only at field from outer conductor)

By symmetry, the radial flux inside the conductor must be uniform (doesn't depend on angle theta).  But flux must flow in loops, and the radial symmetry doesn't allow any return path for flux.  So radial flux must be 0.

Above we have shown the outer hollow conductor creates no flux in it's center. So the flux that the inner conductor is exposed to is only the flux that it creates.... which is a simple textbook problem... maybe the one linked above.

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### RE: Magnetic fields within hollow wires

(OP)
Yes,there will be no field inside the hollow conductor, except the field generated by the inner conductor. Actually, what I'm interested is to calculate the magnitude of the forces acting on the inner wire.

The attached screenshot shows the absence of magnetic field within the hollow wire on left side of the picture, on the right side you can see the force acting on the inner wire. The force direction is always concentric (regarding the outer conductor) either attracting or repelling, depending on the direction of the current flow, but independent of the inner wire's position.

But if the current of the hollow conductor doesn't contribute to the generated field at all, then I don't understand what's generating the force between the two conductors. It may be due to my insufficient understanding of electromagnetics, but can a current conducting wire exhibit a force on a conductor which does NOT have current?

### RE: Magnetic fields within hollow wires

1) A current carrying conductor does not exhibit a force on a non current carrying conductor.  A force requires the interaction of two magnetic fields.

2)The fields of both the hollow and interior conductor extend to infinity so they can interact.

3) Is the net force on the inner conductor always zero?

### RE: Magnetic fields within hollow wires

(OP)
sreid, the grafic simulation shows a force acting on the inner conductor (http://www.falstad.com/emwave2/index.html). All I know is that the force is either repelling or attracting, depending on the current flow direction, and that the force vectors are always concentric or excentric, respectively.

Does the fact that the two fields can interact confirm that the current in the outer conductor has an influence on the force acting on the inner wire?

If the outer conductor is removed, then the simulation shows no net force on the wire.

### RE: Magnetic fields within hollow wires

Assume linearity. (if not linear, conclusions change)

Fi = Ii Li  Btotali
where Fi is force on inner conductor
Ii is current from innter conductor
Btotali is total flux at location of inner conductor
Btotali = Boi + Bii
where Boi is B at inner location from outer current
Bii is B at inner location from inner current

Boi = 0 as mentioned.

Btotali = Bii
Fi = Ii Li Bii

The force on the inner conductor is the force created by itself.  A wire cannot create a net non-zero force on itself by principle of equal/oppsotie reaction.   So...

Fi = 0

Very straightforward imo.

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### RE: Magnetic fields within hollow wires

#### Quote:

on the right side you can see the force acting on the inner wire."
How do we see that? I can't tell anything looking at the graphic.

My discussion above has assumed uniform permeability throughout.  If your outer ring was highly permeable, than an inner current carrying conductor off-center would be attracted to the closest part of the ring (regardless of current flowing in the outer ring).  Just as a dc current carrying conductor above a steel plate is attracted to the plate.

I assume we are talking dc, by the way?

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### RE: Magnetic fields within hollow wires

Is current constrained to be uniformly distributed within these conductors?  Or can current redistribute?

If the latter, perhaps the inner current positioned assymetrically within the outer cylinder changes the current distribution in the outer cylinder.  Non-uniform current in outer cylinder destroys the conclusion that the outer cylinder produces no field in the hollow inner portion.  Might allow a force between the conductors... I have to think about that some more.

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### RE: Magnetic fields within hollow wires

sreid, the force on a current carrying wire is due to the local interaction between an externally generated magnetic induction and the moving electrical charges in the wire: so, if the magnetic induction inside the wire is zero, then there is non force.
mwemag, if you use correctly that (remarkable) applet, you'll see that the conclusions of electricpete are correct: zero magnetic field inside a hollow wire and zero net force on a current carrying wire placed inside.

prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes for fun rides

### RE: Magnetic fields within hollow wires

(OP)

#### Quote (electricpete):

... the force acting on the inner wire. - How do we see that? I can't tell anything looking at the graphic.
If you look at the simulation with a closer view (see the new attachment below) you can see that in the force display mode the white arrows are always ponting towards the center of the hollow conductor (you preferably open the image in full resolution to see the arrows).

Despite the obvious fact of the zero magnetic field inside the hollow wire, there seems to be definitely a force interaction between the two conductors (assuming the performance of the applet is correct). However it isn't clear whether this force is rather strong or marginal (the white color of the arrows stands for a stronger force than the green one, but this is only the relative force).

#### Quote (electricpete):

... would be attracted to the closest part of the ring (regardless of current flowing in the outer ring).  Just as a dc current carrying conductor above a steel plate is attracted to the plate.
That's another astonishing point: The forces do always repel in the simulation, unlike the attraction of a current carryer to a metal plate.

I guess that the device shows dc conditions since any frequency source is turned of in the shown approach.

Introducing AC current could actually give a different explication of the force, e.g. diamagnetic effects caused by oscillating fields, a principle used in levitation melting processes:

"... If a piece of metal is placed in an alternating magnetic field with a high frequency current, electric currents will be induced in the surface of the metal, interacting with the magnetic field to produce a Lorentz force which can support the metal against gravity; at the same time the eddy currents induced in the metal are usually strong enough to melt it due to the heat produced by Joule dissipation."

"The frequency of the alternating current is typically of order 10^4 to 10^5 Hz and at such high frequencies the metal behaves as a perfect conductor, confining the field penetration to a thin surface layer; the metal is in effect supported by the magnetic pressure distribution over its surface."

"A conductor can be levitated above an electromagnet with a alternating current flowing through it. This causes any regular conductor to behave like a diamagnet, due to the eddy currents generated in the conductor. Since the eddy currents create their own fields which oppose the magnetic field, the conductive object is repelled from the electromagnet."

"This effect requires non-ferromagnetic conductive materials like aluminium or copper, as the ferromagnetic ones are also strongly attracted to the electromagnet (although at high frequencies the field can still be expelled)."

This could at least explain why the force repels the inner wire towards the center ( If the current were AC and the magnetic induction inside the wire is not zero).

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