Calculating Magnetic Flux in 3D

[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!

 

[Jano L.]

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?

 

[OrenKatzen]

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

 

[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.

 

[sardar]

Hello,

Calculating magnetic flux in 3D can be a complex task, as it involves taking into account the distance and orientation of the magnet in all three dimensions. The formula you have found for magnetic flux density at a distance from a single rod magnet is a good starting point, but as you have mentioned, it only takes into account the distance in the z direction.

To calculate the magnetic flux density in the x and y directions, you can use the mathematical concept of vector addition. This means that you need to calculate the vector components of the magnetic field in each direction and then add them together to get the total magnetic field at that point.

As for relating magnetic flux density to the pulling force, you can use the formula F = B x I, where F is the pulling force, B is the magnetic flux density, and I is the current. This formula is known as the Lorentz force law and it describes the force exerted on a charged particle in a magnetic field. In your case, the current can be considered as the strength of the magnet.

I hope this helps. Good luck with your calculations!