Magnetic field of an irregularly shaped permanent magnet?

In summary, the conversation discusses the calculation of the magnetic field of an irregularly shaped permanent magnet. The equation for the field of a magnetic point dipole is provided, with the dipole moment and vector to the point of evaluation as variables. The person has attempted to use a field of moments for a solid to calculate the magnetic field, but the result was incorrect. They suspect that a "moment element" analogue of a volume element is needed, but are unsure how to approach it. They have also looked for resources on this type of calculation.
  • #1
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Homework Statement



How do I calculate the magnetic field of an irregularly shaped permanent magnet?

Homework Equations



Field of a magnetic point dipole:

[tex]
\vec{B}(\vec{m},\vec{r}) = \frac{(\vec{m}.\vec{r})\vec{r}-\vec{m}\left\|\vec{r}\right\|^{2}}{\left\|\vec{r}\right\|^{5}}
[/tex]

where

[tex]\vec{m}[/tex] is the dipole moment (in Teslas, I think) and
[tex]\vec{r} = \vec{p} - \vec{b}[/tex] is the vector that points from the base [tex]\vec{b}[/tex] of the dipole to the point where the field is being evaluated

The Attempt at a Solution



I integrated this point dipole over a field of moments for a solid. The result was wrong because the integration treats the dipole field itself as a derivative.

The solid is a 1x1x1cm cube.

The moment field consists of unit vectors everywhere pointing along the Z-axis (only as a test, the field must be allowed to vary arbitrarily).
The surface field strength is supposed to be around 1.0 Tesla (within an epsilon), but the integration gave 5.7714e+10 Tesla, which tells me I did something very wrong indeed.

Note: I am using numerical integration without physical constants (because free space permeability is too small for machine precision), so I don't expect the results to be exact, just within a scalar multiple of the desired answer. I have adjusted the predicted results accordingly.How can I use a field of moments for a solid to calculate the correct magnetic field?

I believe I need a "moment element" analogue of a volume element, but I have no idea how to make it.
 
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  • #2
I have read some papers on the subject, but they all assume that the solid is a sphere (which is not the case here).I would also appreciate any tips or resources for learning more about this kind of calculation!
 
  • #3


I would suggest using a different approach to calculate the magnetic field of an irregularly shaped permanent magnet. Instead of using a point dipole model, which assumes a uniform magnetic field, you can use a finite element method (FEM) to accurately model the magnetic field of the magnet. This method involves dividing the magnet into smaller elements and solving for the magnetic field at each element using numerical methods. This approach takes into account the varying magnetic field strength at different points on the magnet and can provide more accurate results compared to the point dipole model.

Additionally, you can also use a magnetic field mapping technique to experimentally measure the magnetic field of the magnet. This involves using a magnetic field sensor to map out the magnetic field at different points around the magnet and then using interpolation techniques to calculate the field at points where it was not directly measured.

Overall, while the point dipole model can provide a rough estimate of the magnetic field of an irregularly shaped permanent magnet, using numerical methods such as FEM or experimental techniques such as magnetic field mapping can provide more accurate results.
 

1. What is a magnetic field?

A magnetic field is a region in space where magnetic forces are present. It is produced by a magnetic object, such as a permanent magnet, and is characterized by its strength and direction.

2. How does the shape of a permanent magnet affect its magnetic field?

The shape of a permanent magnet can affect its magnetic field in terms of its strength and direction. Irregularly shaped magnets may have a less uniform distribution of magnetic poles, resulting in a weaker magnetic field compared to a regularly shaped magnet with the same magnetic material. The shape can also determine the direction of the magnetic field lines, which can impact its interaction with other magnetic objects.

3. How is the magnetic field of an irregularly shaped permanent magnet measured?

The magnetic field of an irregularly shaped permanent magnet can be measured using a magnetometer, which is a device that detects and measures the strength and direction of a magnetic field. The magnetometer can be moved around the magnet to map out the magnetic field lines and determine its overall strength.

4. Can the magnetic field of an irregularly shaped permanent magnet be changed?

Yes, the magnetic field of a permanent magnet can be changed by applying an external magnetic field or by physically altering its shape. However, this change is usually temporary and the magnet will return to its original magnetic field once the external influence is removed or the shape is restored.

5. What are some real-world applications of irregularly shaped permanent magnets?

Irregularly shaped permanent magnets have various applications, such as in magnetic sensors, electric motors, generators, and magnetic levitation systems. They are also commonly used in magnetic therapy to treat certain medical conditions, and in consumer electronics like speakers and headphones.

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