ABC Magnetic Field derivation

  • #1
Can somebody please post the derivation of the ABC magnetic field components whose magnetic field components in the three cartesian directions(B1, B2, B3) are given by the following:
B1=A*sin(z)+C*cos(y)
B2=B*sin(x)+A*cos(z)
B3=C*sin(y)+B*cos(x)
 

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  • #2
Charles Link
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An interesting magnetic field pattern. I'm going to presume this is not a homework problem because it would then belong in the homework section. ## \\ ## The first thing I observed is that ## \nabla \cdot \vec{B}=0 ##. ## \\ ## To analyze what it is, you can set ## B=0 ## and ## C=0 ##, and see what ## A ## does by itself. The ## A ## term causes a magnetic field with constant amplitude, (amplitude =## A ##), in the x-y plane that spirals around as ## z ## is increased. The cycle of the spiral for ## z ## is ## \Delta z=\frac{1}{2 \pi } ##. The ## B ## and ## C ## terms behave similarly, and the results are superimposed, (with ## B ## acting in the y-z plane, and ## C ## in the x-z plane). Perhaps someone else can see something else of interest. One item is it seems to have infinite extent, and does not appear to represent a real physical system.
 
  • #3
True, it is a force-free helical steady-state solution of the Euler's equation in fluid dynamics. It does represent a physical situation as if you set A=1,B=1 and C=1, it represents a kinetic dynamo and this field is present in the solar corona as well. It has a regular and chaotic trajectory after plotting its Poincare map and calculating its Liapunov Exponents. I just need how it is derived from ##\nabla\cross\vec{B}=constant*\vec{B}## and using Euler's Equation
 
  • #4
Charles Link
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Given ## \nabla \times \vec{B}=k \, \vec{B} ##, this differential equation does have an integral solution. Sometimes there are additional homogeneous solutions that need to be added, but it is basically of the Biot-Savart form: Maxwell's equation ## \nabla \times \vec{B} =\mu_o \vec{J} ## has the Biot-Savart solution: ## \vec{B}(\vec{x})=\frac{\mu_o}{4 \pi} \int \frac{\vec{J}(\vec{x'}) \times (\vec{x}-\vec{x'})}{|\vec{x}-\vec{x'}|^3} \, d^3x' ##. ## \\ ## If your last equation in the previous post is correct, it should be possible to at least show that the solution is consistent, i.e. if you replace ## \mu_o \vec{J} ## by ## k \vec{B} ##, with the form as provided, you should get consistency on both sides. ## \\ ## Edit: The better way would be to simply put in the solution of ## \nabla \times \vec{B}=k \, \vec{B} ##, and solve for ## k ##. The ## A ## term satisfies it for ## k=1 ## , and I believe the ## B ## and ## C ## do also. Starting with ## \nabla \times \vec{B}=\vec{B} ##, it really is only necessary to show that your solution satisfies the differential equation. It is unnecessary to derive anything.
 
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