# Calculating Magnetic Flux in 3D

• OrenKatzen
In summary: This is a sum of forces between all pairs of magnets, not just between the two in the problem.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?In summary, to calculate the force of a magnet, you need to know its magnetization (which is a vector), the magnetic field it produces (which is a vector), and the distance between the magnets.
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!

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?

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

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.

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!

## 1. What is magnetic flux?

Magnetic flux is a measure of the number of magnetic field lines passing through a given area. It is represented by the symbol Φ and is measured in units of webers (Wb).

## 2. How do you calculate magnetic flux in 3D?

To calculate magnetic flux in 3D, you will need to use the formula Φ = B ⋅ A ⋅ cosθ, where B is the magnetic field strength, A is the area of the surface, and θ is the angle between the magnetic field and the surface normal. This formula takes into account the three dimensions of the magnetic field and the surface.

## 3. What is the unit of measurement for magnetic flux?

The unit of measurement for magnetic flux is the weber (Wb). It is equivalent to one volt-second and is named after German physicist Wilhelm Eduard Weber.

## 4. How does magnetic flux differ from magnetic flux density?

Magnetic flux and magnetic flux density are related but different concepts. Magnetic flux is a measure of the total number of magnetic field lines passing through a given area, while magnetic flux density is a measure of the strength of the magnetic field at a specific point. In other words, magnetic flux is a scalar quantity, while magnetic flux density is a vector quantity.

## 5. What factors affect the calculation of magnetic flux in 3D?

The calculation of magnetic flux in 3D is affected by factors such as the strength and direction of the magnetic field, the size and shape of the surface, and the angle between the magnetic field and the surface normal. Additionally, the presence of any nearby magnetic materials can also affect the calculation.

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