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How much force would a ~10 Tesla magnet exert on a weakly magnetic C13 isotope? If I made a molecule of diamond with $N$ C13 atoms, how large would $N$ need to be for me to pull on it with something like a ~1 pico-newton force?

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Interesting. This sounds like something you might see on a grad school qualifying exam. Or maybe as extra credit on a upper division E&M final. You have at least two things to look into here. Maybe three or more. (1) Do you know the transitive force on a magnetic dipole in the external field, and how depend on the geometry of the field and the orientation of the dipole? (2) What free orientations are to be expected of the C-13 in a diamond? (3) Does the field affect the orientations, and if so how? (4) How many such dipoles are in your sample? And so on. – dmckee Aug 17 '12 at 13:57

The strength of the magnet isn't important when it comes to magnetic force, what is important is the flux that the dipole experiences, i.e the rate of change of the magnetic field.

Using an example of the Stern-Gerlash experiment its shown that the Bohr-Magneton provides an energy of:

$ U = \mu_B B $.

Using: $ \frac{dU}{dx} = F $ then $ F = \frac{d(\mu_B B)}{dx} $

or $ F = \mu_B \frac{\partial B}{\partial x} $

where $ \mu_B = \frac{e \hbar}{2m_{e}} $

On an estimate I would say that for a pico-newton force

$13 \times \frac{e \hbar}{2m_{e}} \times \frac{\partial B}{\partial x} > 1 \times 10^{-9} N $

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