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 Summary

The problem of a uniformly magnetized cylinder of finite length is of much interest. There are basically two methods of solution for the magnetic field
## \vec{B} ##
that arrive at the same result: The magnetic pole method and the magnetic surface current method.
A very important problem in magnetostatics is the uniformly magnetized cylinder of finite length. Permanent cylindrical magnets can be modeled as having approximately uniform magnetization, and it is of much interest, given such a uniformly magnetized cylinder, to be able to calculate the magnetic field both outside and inside the magnetized cylinder.
The solution of the magnetic field ## \vec{B} ## can be computed by both the magnetic pole method and by the magnetic surface current method. Both methods yield identical results for the vector ## \vec{B} ##. We will summarize the two methods below:
1) Magnetic pole method: Magnetic pole density (fictitious) is given by ## \rho_m=\nabla \cdot \vec{M} ##. The result for uniform magnetization ## \vec{M}=M_o \hat{z} ## is simply a magnetic surface charge density at the end faces of the cylinder given by ## \sigma_m=\vec{M} \cdot \hat{n} ## resulting in + and  poles at the end faces of the cylinder. The vector ## \vec{H}## is then computed everywhere as ## \vec{H}(\vec{x})=\int \frac{ \sigma_m (\vec{x}\vec{x}')}{4 \pi \mu_o \vec{x}\vec{x}'^3} \, dA ## and ## \vec{B}=\mu_o \vec{H}+\vec{M} ##. This last equation takes some work to prove in detail, but we will simply use it in the solution for ## \vec{B} ##. It has in fact been proven.
2) Magnetic surface current method: Magnetic surface current density ## \vec{J}_m=\frac{\nabla \times \vec{M}}{\mu_o} ## . For a uniform ## \vec{M}=M_o \hat{z} ## the result is a surface current density per unit length ## \vec{K}_m=\frac{\vec{M} \times \hat{n}}{\mu_o} ## on the outer surface of the cylinder, in the same geometry as a solenoid. The magnetic field ## \vec{B} ## is then found everywhere inside and outside the cylinder using BiotSavart as ## \vec{B}(\vec{x})=\frac{\mu_o}{4 \pi} \int \frac{ \vec{K}_m \times (\vec{x}\vec{x}')}{\vec{x}\vec{x}'^3 } \, dA ##, where this ## dA ## is over the cylindrical surface, unlike the previous ## dA ##, where the integration is over the end faces. ## \\ ##
This is an important problem in magnetostatics, and it is hoped that the E&M (electricity an magnetism) students at the upper undergraduate level and higher find it of interest.
The solution of the magnetic field ## \vec{B} ## can be computed by both the magnetic pole method and by the magnetic surface current method. Both methods yield identical results for the vector ## \vec{B} ##. We will summarize the two methods below:
1) Magnetic pole method: Magnetic pole density (fictitious) is given by ## \rho_m=\nabla \cdot \vec{M} ##. The result for uniform magnetization ## \vec{M}=M_o \hat{z} ## is simply a magnetic surface charge density at the end faces of the cylinder given by ## \sigma_m=\vec{M} \cdot \hat{n} ## resulting in + and  poles at the end faces of the cylinder. The vector ## \vec{H}## is then computed everywhere as ## \vec{H}(\vec{x})=\int \frac{ \sigma_m (\vec{x}\vec{x}')}{4 \pi \mu_o \vec{x}\vec{x}'^3} \, dA ## and ## \vec{B}=\mu_o \vec{H}+\vec{M} ##. This last equation takes some work to prove in detail, but we will simply use it in the solution for ## \vec{B} ##. It has in fact been proven.
2) Magnetic surface current method: Magnetic surface current density ## \vec{J}_m=\frac{\nabla \times \vec{M}}{\mu_o} ## . For a uniform ## \vec{M}=M_o \hat{z} ## the result is a surface current density per unit length ## \vec{K}_m=\frac{\vec{M} \times \hat{n}}{\mu_o} ## on the outer surface of the cylinder, in the same geometry as a solenoid. The magnetic field ## \vec{B} ## is then found everywhere inside and outside the cylinder using BiotSavart as ## \vec{B}(\vec{x})=\frac{\mu_o}{4 \pi} \int \frac{ \vec{K}_m \times (\vec{x}\vec{x}')}{\vec{x}\vec{x}'^3 } \, dA ##, where this ## dA ## is over the cylindrical surface, unlike the previous ## dA ##, where the integration is over the end faces. ## \\ ##
This is an important problem in magnetostatics, and it is hoped that the E&M (electricity an magnetism) students at the upper undergraduate level and higher find it of interest.