# Calculating the magnitude of Eddy current's retarding force?

#### PhiowPhi

I could not find a direct formula calculating the retarding force generated from the eddy currents, when a conductor passes a magnetic field at $v$ ($m/s$). Can someone please help me in figuring that out?

I couldn't find a direct way to derive anything either...
There we're some interesting formula's I found online, like $F = \nu vB^2 \sigma$

$\nu$: Volume of the conductor.
$v$: Velocity of the conductor.
$B$: Magnetic field.
$\sigma$: Conductor's conductivity(I assume electrical, in Ohm.m?).

Is this a valid formula? Or are there any others that are much more accurate?

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#### Jeff Rosenbury

Try I2R. That gives the power used by the eddy current. From that you can figure the torque, force, etc.

It might be problematic depending on how you measured your eddy current. If you want only eddy current and not other effects like hysteresis but you measured your eddy current with those effects it will give a wrong answer. But if you just want your total (electrical) loss torque then total loss current is the way to go. Also I wouldn't swear there won't be some time/phase effects where the force is periodic or delayed. (Of course if the current is accurately measured in time... but it generally is approximated.)

 It occurs to me you might be talking about magnetic damping and not know the eddy current. Here's a discussion if that's the case.

#### PhiowPhi

Well I could use $\epsilon = -vBL$ to calculate the induced voltage? Then use ohms law $V= IR$ to calculate the induced current, then the power dissipation formula?
I only want to focus on the retarding force, so if calculating the eddy currents alone is what's key, so be it.

Edit
: Yes, magnetic damping would be a correct way to phrase it, or even magnetic "breaking". Thanks for the link.

#### stedwards

It's a very general question, phiowphi. Do you have a specific geometry in mind such as a conductive plate of uniform thickness, passing passing through a region of constant magnetic field?

#### PhiowPhi

@stedwards Well, it's more about the principle. I wanted to know how powerful would the Lorentz force be due to the eddy current to act as a magnetic brake.
You could pick any form of conductor you'd like, the principle is what I'm trying to learn. Of course, in a uniform magnetic field.

From my research/study so far, I've found I could apply motional-EMF formula $\epsilon = -vBL$ to find the induced EMF of the eddy currents, then apply Ohm's law, then Lorentz force. I played around with a few examples, the retarding force is huge! But not sure my method of finding that force is correct.

#### stedwards

EMF = vBL doesn't tell you where the current in the conductor is. You need more information. The induced current is not necessarily in the L direction, and in general is usually not. So there is not enough information to calculate resistive loses.

#### PhiowPhi

What kind of information must be provided? Could you possibly give out an example.
Or reference a good source for me to start studying from, because I'm stuck here... all the sources I'm studying don't give a direct method of calculating eddy currents, I know that Lorentz force formula is right, but... finding the magnitude of $I$ of the eddy's is something difficult. I might need to consider Ampere's law...

@jim hardy , @vanhees71 help!

#### vanhees71

Gold Member
I don't remember to have ever seen a full treatment of eddy currents, including retardation. That sounds like a very tough but also interesting problem. Perhaps one can google something.

#### PhiowPhi

I've been doing that for the last 2 days , can't find something direct and clear at all...
The search continues!

#### jim hardy

Gold Member
2018 Award
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Seems to me it'd depend on geometry.

Eddy currents want to flow in circles -

A conductive surface moving past a stationary magnet will have circular electric currents called eddy currents induced in it by the magnetic field, due to Faraday's law of induction. By Lenz's law, the circulating currents will create their own magnetic field which opposes the field of the magnet. Thus the moving conductor will experience a drag force from the magnet that opposes its motion, proportional to its velocity. The electrical energy of the eddy currents is dissipated as heat due to the electrical resistance of the conductor.

Your wire is damped solely by eddy currents ? Or is it a closed loop ? Or a flat disc ? Uniform infinite field or defined by some volume ?

I'd look for scholarly articles on eddy current brakes and see what equations they use, and visit their references..

http://scholar.lib.vt.edu/theses/available/etd-5440202339731121/unrestricted/CHAP2_DOC.pdf

http://deepblue.lib.umich.edu/bitstream/handle/2027.42/109373/me450w10project16_report.pdf?sequence=1

http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/e%26m/Experiments_eddy_currents.pdf [Broken]

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#### stedwards

As vanhees, I doubt if there is an analytical solution for an arbitrary shaped conductor, with or without constant resistivity. In any case, the Lorentz force as you've stated it, is insufficient. It's unlikely to be useful, but you would use the force density Lorentz Force equation $f = \rho E + J \times B$. $f$ is the force per unit volume. $\rho$ is the charge density, and $J$ is the current density. You can find this, similarly given, on the Lorentz Force Wikipedia page.

#### PhiowPhi

@jim hardy It certainly is a matter of geometry and their circular flow, is it only limited to the surface of the conductor? Not throughout?
Most likely a closed loop, if it we're... would that mean there is both eddy currents and induced EMF & induced current flow? Certainly in a uniform magnetic field focused to a certain volume of the conductor. Thanks for the references I will need them!

@stedwards I agree, because so far I haven't found a direct one... also the Lorentz force formula I considered is a terrible use I think, the values are extremely large and unreasonable , I've started to consider that formula you provided from wiki. over it, and looking into other formula's too.

I guess the reason behind my conflict, would be the resistance Vs. the volume of conductor placed in a magnetic field. Assume a small volume of a conductor would pass through a powerful magnetic field, based on the resistance of the conductor(throughout all volume of the conductor) the current induced(even though a small volume of the conductor is in the magnetic field) is enormous, due to the low resistance of the conductor's large volume... but, it makes no sense, the numbers don't.

Let me consider an example to show my point:
A rectangular copper conductor that is 10cm wide x 10cm long x 0.2cm thick($20cm^3$), has a resistance of ($0.00000843 Ohms$), and the conductor is passing the magnetic field($1T$) with a constant($20m/s$) from an exterior mechanism , and only 5cm x 5cm x 0.2cm of the conductor will be in the magnetic field, not all the conductor will enter it at once, half of it first, then the rest. The method I had in mind or used earlier was like so:

Calculate the induced V(on the surface of the conductor):

$\epsilon = -vBL$ = $-2V$

From that, I used Ohms law to find the current:

$I$ = $\frac{V}{R}$ Which is an enormous amount of current for that volume inside the magnetic field, and from that I used Lorentz force formula(ILxB) to find the retarding force, this not logical or valid , the numbers make no sense...

But then again, that is true if the conductor was connected to a circuit the Induced EMF is correct and induced current will flow. Like a generator, wow the retarding force would be massive. But I don't think the Eddy current's magnitude is in that order...?

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#### Nidum

Gold Member
The whole subject of induced currents in conductors passing through magnetic fields and the forces experienced by them has been extensively researched in electric motor and generator technology .

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#### Jeff Rosenbury

@jim hardy It certainly is a matter of geometry and their circular flow, is it only limited to the surface of the conductor? Not throughout?
Most likely a closed loop, if it we're... would that mean there is both eddy currents and induced EMF & induced current flow? Certainly in a uniform magnetic field focused to a certain volume of the conductor. Thanks for the references I will need them!

@stedwards I agree, because so far I haven't found a direct one... also the Lorentz force formula I considered is a terrible use I think, the values are extremely large and unreasonable , I've started to consider that formula you provided from wiki. over it, and looking into other formula's too.

I guess the reason behind my conflict, would be the resistance Vs. the volume of conductor placed in a magnetic field. Assume a small volume of a conductor would pass through a powerful magnetic field, based on the resistance of the conductor(throughout all volume of the conductor) the current induced(even though a small volume of the conductor is in the magnetic field) is enormous, due to the low resistance of the conductor's large volume... but, it makes no sense, the numbers don't.

Let me consider an example to show my point:
A rectangular copper conductor that is 10cm wide x 10cm long x 0.2cm thick($20cm^3$), has a resistance of ($0.00000843 Ohms$), and the conductor is passing the magnetic field($1T$) with a constant($20m/s$) from an exterior mechanism , and only 5cm x 5cm x 0.2cm of the conductor will be in the magnetic field, not all the conductor will enter it at once, half of it first, then the rest. The method I had in mind or used earlier was like so:

Calculate the induced V(on the surface of the conductor):

$\epsilon = -vBL$ = $-2V$

From that, I used Ohms law to find the current:

$I$ = $\frac{V}{R}$ Which is an enormous amount of current for that volume inside the magnetic field, and from that I used Lorentz force formula(ILxB) to find the retarding force, this not logical or valid , the numbers make no sense...

But then again, that is true if the conductor was connected to a circuit the Induced EMF is correct and induced current will flow. Like a generator, wow the retarding force would be massive. But I don't think the Eddy current's magnitude is in that order...?
The currents will be somewhat limited to the surface. But that is a frequency effect, and should be minor at this thickness/frequency. You might want to check that though. Calculate the skin depth which is the depth at which the power drops 1 Neeper (≈8dB).

I would check your resistance calculation. As I recall, resistance goes down with area, but up with length. Since both will be variable in this case, the calculation isn't simple. The current path will vary with distance from the "center" which I suspect will be the moment of the field/plate interaction zone. Also include the skin depth if it's relevant. Finally the number you get will likely be a first order approximation.

However, expect a very large force as a result. I seem to recall this principle is used to brake rollercoasters.

#### jim hardy

Gold Member
2018 Award
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It certainly is a matter of geometry and their circular flow, is it only limited to the surface of the conductor? Not throughout?
I think that's a function of frequency - skin depth et al.
In iron i think you'll find the effect more complex becausee the eddy currents cause their own magnetic field with which they interact, sorta a real world recursive effect.... if you can find a copy of "Ferromagnetism" by Bozorth you'll have one of the most thorough books i know of. It contains a lot of high math that i'm sure you will understand better than i.

Most likely a closed loop, if it we're... would that mean there is both eddy currents and induced EMF & induced current flow?
Hmmm what would be free path of a charge not constrained by a wire?
It'll try to curve but in a wire it runs into the insulation.
So assuming the wire is a lot longer than it is wide or thick, eddy currents don't amount to much .

here's an old PF link that has some interesting references posted by other folks

from my old 1901 electric machinery book

#### Gordianus

If memory serves well, Smythe's book "Static and Dynamic Electricity" computes eddy currents in some systems (brake disk for example)

#### PhiowPhi

If memory serves well, Smythe's book "Static and Dynamic Electricity" computes eddy currents in some systems (brake disk for example)
Found an electronic copy, reading that chapter about Eddy currents now, thanks for that.
One thing a lot of sources agree upon(or hit at) is that it's one difficult task to do, calculating the Eddy currents... I assume the force at least, would be a lot more easier in comparison.

#### PhiowPhi

One question, just wanted to be sure if my initial thoughts are correct on this: A conductor is connected to a circuit(closed loop) and passes a magnetic field(example: generator), would the conductor induce both Eddy currents & induced EMF$(\epsilon)$ and induced current to the circuit? Or is it just induced EMF & induced current to the circuit, that would then induce a magnetic field and a drag-force to resist the change?

Still digging into a proper model of calculating Eddy currents, if I do having something I'll share it.

#### Jeff Rosenbury

A changing magnetic field induces an electric field. That electric field induces a magnetic field and a current (in a conductor) which in turn induces a magnetic field ... ad infinitum.

However, our models reduce most of this to simple values. Your problem is finding the right model which should take these ground facts into account since they would have been derived from experimental evidence. So hopefully you can ignore doing some kind of infinite Taylor series in three dimensions.

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