Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

Force between 2cyl

For two cylindrical magnets with radius  R and height  t , with their magnetic dipole aligned, the force can be well approximated (even at distances of the order of  t ) by,


F(x) = frac{pimu_0}{4} M^2 R^4 left[frac{1}{x^2} + frac{1}{(x+2t)^2} - frac{2}{(x + t)^2}
ight]

where M is the magnetization of the magnets and x is the distance between them. In disagreement to the statement in the previous section, a measurement of the magnetic flux density very close to the magnet  B_0 is related to M by the formula


B_0 = mu_0 M

The effective magnetic dipole can be written as


m = M V

Where V is the volume of the magnet. For a cylinder, this is V = pi R^2 t.

When  t << x , the point dipole approximation is obtained,


F(x) = frac{3pimu_0}{2} M^2 R^4 t^2frac{1}{x^4} = frac{3mu_0}{2pi} M^2 V^2frac{1}{x^4} = frac{3mu_0}{2pi} m_1 m_2frac{1}{x^4}

which matches the expression of the force between two magnetic dipoles.

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