When a diamagnetic object is placed near one pole of a strong magnet, the object

Question

When a diamagnetic object is placed near one pole of a strong magnet, the object is repelled away from the pole by a magnetic force. If the magnet were held vertically and the magnetic force were strong enough, the object could be levitated. In 1997, researchers were able to produce a magnet strong enough to levitate a small frog. They reasoned that since water is a diamagnetic material, with susceptibility X = -8.8 times 10^-6, and since a frog is mostly water, the frog itself would be diamagnetic. They found that the magnetic force on the frog depended on the frog's distance y above the magnet by F_magnet = m X/rho mu_0 B(y) dB(y).dy where m is the frog's mass and rho is the density of water (rho = 1000 kg/m^3). The permeability of free space is mu_0 = 4pi times 10^-7 T middot m/A. The magnetic field B(y) above the magnet was measured to be B(y) = B_0e^-y^2/a with a = 0.00237 m^2. In the experiment, the frog levitated at a distance y_0 = 0.0090 m above the magnet. What must have been the magnet's strength, B_0, in this experiment? If the Earth's magnetic field is typically around 45 microteslas, how many times stronger was the magnet?

Explanation / Answer

B(y) = Boe^(-y^2/a)
Fmag = [mXB(y)]/(rho)Uo[dB(y)/dy] = mg [for levitation]
[8.8*10^-6*Bo*e^(-0.0090^2/0.00237)]/1000*4*3.14*10^-7 * [2*0.0090*Bo(e^(-0.0090^2/0.00237))/0.00237] = 9.8
0.0532129*0.9339*Bo^2 = 9.8
Bo = 14.03469 T

Times stroner than earths field = Bo/45*10^-3 = 311.882 times

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