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 was held vertically and the magnetic force was strong enough, the object could levitate against the gravitational force. In 1997, researchers where able to produce a strong enough magnet that was able to levitate a small frog. They reasoned that since water is a diamagnetic material, with susceptibility X = -8.8 Times 10-6, and that a frog is mostly water, the frog itself would be diamagnetic. They found that the magnetic force on the frog depended on the distance y above the magnet pole, where m is the frog's mass, the density of water rho = 1000 kg / m3, and the permeability of free space mu q = 4pi Times 10-7 T m/A. The magnetic field B(y) above the magnet was measured to be, B(y) = B0e-y2/a with a = 0.00234 m2. In the experiment the frog levitate at a distance y0 = 0.0090 m above the magnet. What was the magnet's strength, B0, in their experiment? The Earth's magnetic field is typically around 45 microteslas, how many times stronger is the magnet?

Explanation / Answer

so we want Fmagnet = mg

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

dB/dy = B0 e^(-y^2/a) * - 2y/a

9.81= 8.8E-6* B*e^(-0.008^2/0.00237)/(1000*4*pi*10^(-7))* B*e^(-0.008^2/0.00237)* 2*0.008/0.00237
solve for B

B=14.8 T

14.8/45E-6=328889

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