Energy of magnetic field created by magnetic dipoles in a shphere.

In summary, the speaker is researching genetic algorithms to solve a problem involving the magnetic orientation of particles in a hollow sphere. They have successfully solved the problem in 2 dimensions and are seeking advice on how to calculate the magnetic field energy in 3 dimensions. They suggest defining the orientation of each magnetic dipole moment with spherical coordinates and then transforming it into cartesian coordinates.
  • #1
sotnet
3
0
Dear everyone,

I am researching genetic algorithms and at the moment I am trying to solve a problem how would magnetic particles (dipoles) orient themselves in a thin hollow sphere.

Suppose that I have N magnetic dipoles placed in the sphere. There is no external magnetic field. The dipoles create their own magnetic field, thus the system has some magnetic energy. I am writing a genetic algorithm to minimize this energy.
I did the problem in 2 dimensions (on a plane) already, and it worked perfectly!

The problem is that I am not sure how to calculate the magnetic field energy created by dipoles in 3 dimensions.

The formula for calculating magnetic field created by one dipole with magnetic moment [tex]\vec{m}[/tex] at point [tex]\vec{r}[/tex] in SI system is:
[tex]\vec{B}(\vec{r}) = \frac{\mu_{0}}{4\pi}(\frac{3(\vec{m}\hat{r}) - \vec{m}}{r^{3}})[/tex]

To find the energy, I just sum over all dipole moments [tex]\vec{m}[/tex] and multiply by [tex]\vec{B}[/tex] created by others (with minus sign).

In 2D I set the problem so that all the dipoles had [tex]\vec{m}[/tex] and [tex]\vec{r}[/tex] perpendicular, that is [tex]\vec{m}\vec{r} = 0[/tex] (magnetic dipole moment [tex]\vec{m}[/tex] was perpendicular to the 2d plane the dipoles were on).

In 3D I have the [tex]\vec{m}\vec{r}[/tex] term, which I am not sure how to calculate.

Any advice how to calculate this term to find energy in 3 dimensions?



Stan
 
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  • #2
You can define orientation of each magnetic dipole moment with two angles in spherical coordinates (theta,fi). Then you transform this vector into cartesian coordinates:

mx=m*sin(theta)*cos(fi)
my=m*sin(theta)*sin(fi)
mz=m*cos(theta)

And the product of r and m is:

mx*x+my*y+mz*z

where (x,y,z) is a vector from the source dipole to the second dipole.
 
  • #3
ley

Dear Stanley,

Thank you for sharing your research on genetic algorithms and magnetic particles with us. It sounds like a fascinating problem to solve. I can offer some insights on calculating the energy of magnetic fields created by dipoles in a sphere in 3 dimensions.

In order to calculate the energy, you are correct in using the formula for magnetic field created by a single dipole at a point in space. However, in 3 dimensions, the orientation of the dipoles will play a crucial role in determining the total energy of the system. This is because the distance between dipoles and their orientation will affect the strength and direction of the magnetic field they create.

To calculate the energy in 3 dimensions, you will need to take into account the orientation of each dipole in relation to the others. This can be done by using vector algebra to find the dot product between the dipole moments and the distances between them. The dot product will give you the component of the dipole moment that is parallel to the distance vector, which can then be used in the formula for magnetic field.

Additionally, you may need to consider the distance between the dipoles in your calculations. In a thin hollow sphere, the distance between dipoles may vary depending on their location within the sphere. This could affect the overall energy of the system and should be taken into account in your calculations.

I hope this helps in your research. Best of luck in solving this interesting problem.


 

1. What is a magnetic dipole?

A magnetic dipole is a pair of equal and opposite magnetic poles separated by a certain distance. It can be visualized as a tiny magnet with a north pole and a south pole.

2. How is the energy of a magnetic field created by magnetic dipoles in a sphere calculated?

The energy of a magnetic field created by magnetic dipoles in a sphere can be calculated using the formula E = -m*B, where E is the energy, m is the magnetic moment of the dipole, and B is the magnetic field strength.

3. What factors affect the strength of the magnetic field created by magnetic dipoles in a sphere?

The strength of the magnetic field created by magnetic dipoles in a sphere is affected by the distance between the dipoles, the orientation of the dipoles, and the strength of the individual dipoles.

4. How does the energy of a magnetic field change when the distance between magnetic dipoles in a sphere is increased?

As the distance between magnetic dipoles in a sphere is increased, the energy of the magnetic field decreases. This is because the magnetic field strength decreases as the distance between dipoles increases.

5. Can the energy of a magnetic field created by magnetic dipoles in a sphere be manipulated?

Yes, the energy of a magnetic field created by magnetic dipoles in a sphere can be manipulated by changing the strength of the individual dipoles or by changing the orientation of the dipoles. Additionally, the energy can be changed by altering the distance between dipoles.

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