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Calculating Magnetic Flux in 3D 
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#1
Jan313, 04:34 AM

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/magnetformula.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
Jan313, 12:02 PM

PF Gold
P: 1,167

Welcome to PF!



#3
Jan613, 01:28 AM

P: 2

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



#4
Jan613, 10:30 AM

PF Gold
P: 1,167

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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