## 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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