# Magnetic Field and Magnetic Induction in a Cylindrical Bar Magnet

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This Demonstration shows the quantitative magnetic field and magnetic induction for a cylindrical bar magnet. The magnetization is assumed to be in the direction and uniform within a magnet of diameter and length . Colors on a logarithmic scale show field intensity and arrows show its direction. The field and induction outside the magnet satisfy , so the field patterns are identical (the same color is used for both fields). They are different inside the magnet, since there . While vectors go from the north pole to the south pole (top to bottom) in all regions, vectors return to the north pole through the magnetized body.

Contributed by: Y. Shibuya (September 2013)
Open content licensed under CC BY-NC-SA

## Details

The magnetic field can be regarded as originating from two parallel, oppositely charged monopolar disks situated at the ends of the magnet. Since the effective surface magnetic charge densities are equal to , the magnetic field of any point in the - plane, a 2D vector, can be calculated by integrating on those surfaces:

.

Mathematica finds a function after integrating with respect to (i.e. in radial direction) that can be written as:

.

Snapshot 1: example of field; all vectors go from north pole to south pole

Snapshot 2 : example of field; vectors circulate between north pole and south pole

Snapshot 3 : example of field of thin bar magnet; the field resembles that of two point charges