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Torque in an MRI scanner

by dhigger
Tags: scanner, torque
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Apr26-12, 07:54 AM
P: 1
There are (at least) two issues regarding the presence of ferromagnetic materials around an MRI scanner (of strength e.g. 3 Tesla).
(i) that something (e.g. scissors in a health worker's pocket) becomes a projectile, and
(ii) that an implant may experience a twisting force (e.g. ferromagnetic shrapnel or bullets or a needle or other implanted medical device with ferromagnetic parts).

I am trying to determine the equation relating the torque τ experienced by an object, to
the magnetic susceptibility of a material χ
the static magnetic field strength B
the volume of the material V
the angle it makes with B, θ

I am stuck because my derivation doesn't agree with the textbooks I have. My working is:

The magnetisation of a sample which is not a permanent magnet is given by M=χH, where χ is the (dimensionless) magnetic susceptibility of the sample. We may also write B=μ[itex]_{0}[/itex](1+χ)=μ[itex]_{0}[/itex]μ[itex]_{r}[/itex]H=μH, where μ0 is the (constant) permeability of free space (units of henries per metre, or newtons per ampere squared), μ[itex]_{r}[/itex] is the (dimensionless) relative permeability of the sample to μ[itex]_{0}[/itex], μ is the magnetic permeability of the sample, B is the magnetic field strength as flux density (Tesla) and H is magnetic field strength given in ampere-turns per metre. We may write, then, M=χB/μ.

So, torque (τ) is given by τ = m x B (where m denotes the magnetic dipole moment, which equals the magnetisation of a sample (M) multiplied by the volume of the sample (V)). Thus, |τ| = mBsinθ where θ is the angle between the magnetic moment and the applied magnetic field. In terms of the magnetisability of a sample (χ), |τ| = χVB[itex]^{2}[/itex]sinθ/μ (using M=χB/μ and m=MV).

One textbook says |τ| [itex]\propto[/itex] χ[itex]^{2}[/itex]VB[itex]^{2}[/itex].
It seems that either I am missing one χ or my textbook has one χ too many.
Another says F[itex]_{rot}[/itex] [itex]\approx[/itex] χ[itex]^{2}[/itex]VB[itex]^{2}[/itex]/2Lμ[itex]_{0}[/itex], where L is the length of the object.

Thanks for any pointers.

Edit: sorry, probably this should be in the homework forum. I'd move it if I could.
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