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Calculating Magnetic Flux in 3D

  1. Jan 3, 2013 #1
    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?

  2. jcsd
  3. Jan 3, 2013 #2

    Jano L.

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    Gold Member

    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?
  4. Jan 6, 2013 #3
    I am probably pulling a piece of ferrite, but if it's simpler, we can just make it a piece of iron.
  5. Jan 6, 2013 #4

    Jano L.

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    Gold Member

    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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