Cylindrical Permanent Magnet Viewed as a Solenoid

In this animation we view the sum of all the atomic dipole moments in a given cross section of the cylinder as a single current loop of radius equal to that of the cylinder and current equal to $qomega$ where $q$ is the electron charge and $omega$ is its rotation rate in radians per second. If the magnetic tiny current loops have radius $b$ and current $qomega$ then each dipole contributes a dipole moment of

$bbm=qomegapib^2$

If we have a long cylindrical magnet polarized along its axis, then the magnetic dipole moment of the cylinder is just the sum of the moments due to all the tiny dipoles

$bbM_m=(b^2qomega)piint_0^Rn(r)rdr$

where

$n(r)$ is the number of dipoles per unit area at radius

$r$ on the end of the cylinder. Now we must compare the expression for

$bbM$ to the dipole moment of a macroscopic wire wrapped solenoid. The magnetic field at macroscopic distance r from the solenoid is described by the Biot-Savart Law as:

$B(r)=mu_0int((dbblxxbbr'))/r^3$

where

$i$ is the current and the integral over

$dbbl$ is around the circumference of the cylinder. In a much simpler way we can write the magnetic moment of a single layer solenoid for radius

$R$ as

$bbM_s=NipiR^2bbhata$

where

$N$ is the number of loops in the solenoid,

$i$ is the current in each loop, and

$bbhata$ is a unit vector along the solenoid axis. This is to be compared with the result for

$bbM_m$ .

$bbM_m/bbM_s=((pib^2qomega)int_0^Rn(r)rdr)/(NipiR^2bbhata)$

The important thing to note is that most of the current due to the orbitals cancel out and we are left with large rings of current shown here in the clockwise direction. In fact it is possible to show that the total magnetic moment of the solenoid large current rings is the analog of the moments of the electron orbitals. The integral $int_0^Rpin(r)rdr$ in the $bbM_m$ expressions is the analog of the $piR^2$ in the $bbM_s$ expression. Therefore, it is a good analogy to view the permanent magnet as a cylindrical solenoid.

It has been shown that a current loop's magnetic field forces can be converted to electic field forces by a special relativity transformation. Since the permanent magnet is the analog of a current loop, we can also apply the special relativity transformation to it and its forces then become electric field forces.

Single Step