ABC Magnetic Field derivation

  • #1

Main Question or Discussion Point

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)
 

Answers and Replies

  • #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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