# Calculating Magnetic Flux in 3D

1. Jan 3, 2013

### OrenKatzen

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!

2. Jan 3, 2013

### Jano L.

Welcome to PF!

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?

3. Jan 6, 2013

### OrenKatzen

I am probably pulling a piece of ferrite, but if it's simpler, we can just make it a piece of iron.

4. Jan 6, 2013

### Jano L.

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.