How Can the Magnetic Vector Potential be Numerically Solved in the X-Y Plane?

In summary, the conversation discusses solving the wave equation of the magnetic vector potential numerically in the x-y plane grid. The equation can be simplified using the identity "curl squared = grad div - del squared" and choosing a gauge. The resultant equation is Laplace's equation, which can be solved numerically. The conversation also mentions using a grid for electromagnetics and solving the vector components separately in Cartesian coordinates. Different numerical methods and approaches are also mentioned.
  • #1
merro
6
0
hi everybody
i want to solve the wave equation of the magnetic vector potential numerically in x-y plane grid,
curl curl A= µ J
anyone can help me

thanks in advance
 
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  • #2
Well first you should use the identity "curl squared = grad div - del squared": ∇ x (∇ x A) = ∇ ∇·A - ∇2 A. Then choose a gauge ∇·A = 0. You're left with Laplace's equation, which can be solved numerically.
 
  • #3
thanks, but i have another question , the reusltant equation will be ∇^2 A=-µ J, and A is a vector, not a scalar, this one is not Laplace , is Poisson, or
 
  • #4
because the A is a vector then must be put in links of the grid, not on the nodes, or?
 
  • #5
merro said:
thanks, but i have another question , the reusltant equation will be ∇^2 A=-µ J, and A is a vector, not a scalar, this one is not Laplace , is Poisson, or

But J is a vector too. Solve the vector components separately in Cartesian coordinates and you have three Laplacian equations.
 
  • #6
do you mean that
∇^2 Ax=Jx,
∇^2 Ay=Jy,
∇^2 Az=Jz,
and Ax, Ay, Az lies on the grid nodes
 
  • #7
merro said:
do you mean that
∇^2 Ax=Jx,
∇^2 Ay=Jy,
∇^2 Az=Jz,
and Ax, Ay, Az lies on the grid nodes

As you wish, though there are a few ways to setup the grid for electromagnetics depending on the exact numerical procedure. For example, a common grid for FDTD, the Yee grid, will offset the electric and magnetic fields from the grid points. That may also be appropriate here if you wish to do a second order finite difference approach.
 
  • #8
please , can you show me some examples to discrization of the magentic field potential in x-y-z plan , if anyone has a paper of chapter of book can help me. and which methods could be stable numerically
 

What is magnetic vector potential?

The magnetic vector potential is a mathematical quantity used to describe the magnetic field in a region of space. It is denoted by the symbol A and is related to the magnetic field B through the equation B = ∇ x A.

What are the units of magnetic vector potential?

The units of magnetic vector potential depend on the system of units being used. In SI units, the units of A are joules per ampere-meter (J/A·m). In Gaussian units, the units are ergs per gauss-centimeter (erg/G·cm).

How is magnetic vector potential different from magnetic field?

Magnetic vector potential and magnetic field are related but distinct quantities. The magnetic field B is a physical quantity that can be measured, while the magnetic vector potential A is a mathematical construct used to describe the magnetic field. B is a vector quantity, while A is a vector potential, meaning it has both magnitude and direction.

What are some real-world applications of magnetic vector potential?

Magnetic vector potential has many applications in physics and engineering, including in the design of magnetic circuits, electromagnetic devices, and particle accelerators. It is also used in the study of superconductivity and in the development of magnetic levitation systems.

How is magnetic vector potential related to electric potential?

There is a close relationship between magnetic vector potential and electric potential. Both are vector potentials and can be used to describe electric and magnetic fields, respectively. In fact, in the absence of time-varying electric and magnetic fields, the magnetic vector potential and electric potential can be combined to form the electromagnetic potential, which is a scalar quantity.

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