Calculating Magnetic Flux in 3D

by OrenKatzen
Tags: density, flux, magnet
 P: 2 Hi everybody, first time poster here. I am working on calculating the force of magnets in a 3 dimensional space. I have found a formula for the magnetic flux density at a distance z from the magnet face at this link http://www.magneticsolutions.com.au/magnet-formula.html, under Flux density at a distance from a single rod magnet. My problem is that I can't find a formula which will relate the magnetic flux density with distances in the x and y directions as well as z. Does anyone know of a formula or way to figure this out? On a similar note, how do I then relate magnetic flux density to the pulling force at that distance? Thanks!
P: 964
Welcome to PF!

 how do I then relate magnetic flux density to the pulling force at that distance?
The force the magnet will exert does not depend only on the magnetic field of the magnet, but also on the object that is pulled/pushed. What is it? Another magnet, or piece of iron?
 P: 2 I am probably pulling a piece of ferrite, but if it's simpler, we can just make it a piece of iron.
P: 964

Calculating Magnetic Flux in 3D

The problem is quite difficult in general. If both pieces are magnetized hard ferrites - magnets (have permanent magnetization), here is what I would do:

0. find out the magnetization ##\mathbf M## of both pieces; in the simplest case, each magnet has uniform magnetization, so just there are just two vectors, one for each magnet;
1. divide both the magnet and the ferrite into small domains (cubes) ##i## with volume ##\Delta V_i##;
2. the magnetic moments ##\mathbf m_i## can be found as ##\mathbf m_i = \mathbf M(i) \Delta V_i##, where ##\mathbf M(i)## is the magnetization at i;
3. there is a formula for the force acting on the moment i due to the moment j:

$$\mathbf F(i) = - \mathbf m_i \cdot \nabla \mathbf B_j(\mathbf x_i)$$

where ##\mathbf B_j(\mathbf x)## is the magnetic field due to the moment j:

$$\mathbf B_j(\mathbf x) = \frac{\mu_0}{4\pi} \frac{3\mathbf n(\mathbf n\cdot \mathbf m_j)- \mathbf m_j}{|x-\mathbf r_j|^3}$$

and ##\mathbf n = \frac{\mathbf x-\mathbf r_j}{|x-\mathbf r_j|}##

4. total force = sum of the forces between all pairs (i,j), where i comes from the first magnet, j comes from the second magnet.

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