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I've been going through some equations and such trying to determine the work done by a solenoid on a ferromagnetic object. I have the following:

Magnetic field due to solenoid:

$\vec{B} = \langle0,0,\mu_0nI\rangle$

(Assuming coils are on xy-plane and current is counter-clockwise)

Force of magnetic field:

$ F = q\vec{v} \times \vec{B} $

Work:

$ W = \int F \cdot dl $

Work of Magnetic Field:

$ W = \int_c(q\vec{v} \times \langle0,0,\mu_onI\rangle) \cdot d\vec{r} $

For one, this seems to indicate a work of 0 if the object is not charged, which I have seen in some places but just doesn't seem right. Also, this does not take into account the properties of the object, such as relative permeability, which I guess could have some effect with the charge value. I'm trying to calculate the acceleration of a ferromagnetic object from a magnetic field, is there a better way to do this? I've thought about the following:

$ \vec{a} = \frac{q\vec{v} \times \vec{B}}{m} $

However, this is where I started running into the charge issue and thought to calculate it from the work done.

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2 Answers

up vote 1 down vote accepted

You can apply the Lorentz force equation $F=qv\times B$ at the microscopic level, since the magnet is made out of charged particles. However, it's not practical to do this for a ferromagnetic object.

The electrons in the ferromagnetic material also have intrinsic spin 1/2 and an intrinsic dipole moment, and they therefore experience an additional force in a field gradient. This force is not described by the Lorentz equation.

A complete, realistic calculation is going to be extremely difficult. You could solve Maxwell's equations numerically, putting in the correct permeability. Note that you're going to have hysteresis effects.

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Thanks! I looked over Maxwell's equations a bit before making this post and had determined that it would be easier to do the Lorentz force equation, but I guess that's not really feasible. I'm assuming you're speaking of the maxwell version of Faraday's law: $\vec{E} \times \nabla = -\frac{\partial\vec{B}}{\partial t}$? I'll try to work it out to a general solution and if not, I will write a simulation program. Let me know if you have any other ideas. Also, I know your name from somewhere. Are you on physicsforums.com? –  danielu13 Jun 20 '13 at 1:46
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I think you're right to consider the magnetic permeability of the object. However, I don't think the equation for force you have is valid here as the object may neither be charged (q=0) nor initially moving (v x B = 0), conditions which according to the equation would result in zero force.

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+1 Yes, you are right about the equation not being applicable, but you should not write "neither be charged nor moving initially", instead "either be uncharged or stationary" because even if it is moving, but if it is uncharged, the force is still going to be 0. –  Dimensio1n0 Jun 18 '13 at 13:06
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