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 * 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 x 10^-7 T/A. The magnetic field B(y) above the magnet was measured to be B(y) = B_0 e^-y^2/a with a = 0.00237 m^2. In the experiment, the frog levitated at a distance y0 = 0.011 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

so we want Fmagnet = mg

mg = m chi B/(d uo) dB/dy

g = chi B/(d uo) dB/dy

where, dB/dy = Bo e(-y^2/a) * - 2y/a

9.81 = 8.8*10-6* Bo*e^(-0.0112/0.00237)* Bo*e^(-0.0112/0.00237)* 2*0.011/(1000*4*pi*10-7*0.00237)

solve for Bo, we get

B = 12.93 T

12.93/[45*10-6] = 287290.5

The magnet is 287290.5 times stronger.

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