Off-axis magnetic field due to a current loop in cartesian coordinates

In summary, the conversation discusses the derivation of the magnetic field of a current loop in different coordinate systems. The speaker has tried to transform the expressions into cartesian coordinates but has been unsuccessful. They are considering deriving the equations again or using the spherical harmonic addition and translation theorems. The conversation also mentions a paper where the authors claim to have derived the cartesian result from the spherical expression.
  • #1
johnpatitucci
6
0
Hi there,


a few days ago I derived the probably well-know expression for the magnetic field of a current loop including elliptic integrals of the first and second kind (it can be seen here http://plasmalab.pbworks.com/f/bfield.pdf" [Broken] ). As I'd like to rotate and shift the position of the current loop I tried to transform the expressions for the fields [tex] B_r, B_z [/tex] into cartesian coordinates but failed because i also need to transform the elliptic integrals and I don't know how to do that.

Do you think I have to derive the whole thing again (starting with the current density's in x-,y- and z-direction but now strictly in cartesian coordinates which is very tedious) or is there a way to get the wanted cartesian expression from the one's in cylindrcal coordinates which I have already written down.

Thanks for your comments !
 
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  • #2
I would instead express the field in spherical coordinates (see Jackson's Classical Electrodynamics, or everywhere on the web). Then use the spherical harmonic addition theorem for rotations, and spherical harmonic translation theorem for translations, to position your loop.

You can find these theorems in Steinborn and Ruedenberg, Rotation and Translation of...Spherical Harmonics, Advances in Quantum Chemistry, v. 7 (1979), and certainly elsewhere.
 
  • #3
Thanks @marcusl. That is a pretty good idea and I going to try it now.


Yesterday, I found a paper (you can see it here: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20010038494_2001057024.pdf" ) where the current loop expressions are written down in spherical as well as cartesian coordinates. On the first page the authors claim to have derived the cartesian result from the spherical expression but I don't know how. Anybody got a clue how one could do that ?
 
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1. What is an off-axis magnetic field?

An off-axis magnetic field is a type of magnetic field that is not aligned with the axis of a current loop. It is created when a current-carrying wire or loop is positioned at an angle to an external magnetic field.

2. How is an off-axis magnetic field calculated?

The strength and direction of an off-axis magnetic field can be calculated using the Biot-Savart law, which takes into account the current in the loop, the distance from the loop, and the angle between the loop and the external magnetic field.

3. What are the applications of off-axis magnetic fields?

Off-axis magnetic fields are used in various applications, including magnetic resonance imaging (MRI), particle accelerators, and electromagnetic motors. They can also be used to control the motion of charged particles in a specific direction.

4. How does the off-axis magnetic field change with distance from the current loop?

The strength of an off-axis magnetic field decreases as the distance from the current loop increases. This is because the magnetic field follows an inverse square law, meaning it weakens by a factor of the distance squared.

5. Can off-axis magnetic fields be shielded?

Yes, off-axis magnetic fields can be shielded using materials with high magnetic permeability, such as iron or mu-metal. These materials redirect the magnetic field lines, reducing the strength of the field outside the shielded area.

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