Solving Magnetic Field Problem: Reitz-Milford-Cristy 9.15 (4th Ed.)

In summary: Now I see, determining the tangencial component in this case is equivalent to determine the value of ∅ on the boundary surface so uniqueness can be applied.
  • #1
facenian
436
25

Homework Statement


This is a problem from Reitz-Milford-Cristy-Problem 9.15(fourth Ed.) .There is a long wire carrying a current "I" above a plane,inside the plane we have to cases a)μ=∞; and b)μ=0. We must find the field above the plane.

Homework Equations


H=∇∅
Δ∅=0


The Attempt at a Solution


To solve the problem we split the field sources: 1)wire + 2)Magnetic plane. Part 1) is already known and to find 2) we put H=-grad∅ and solve laplace equation Δ∅=0. To find ∅ we consider an image current in case a) it is a parallel current and in case b) it is an antiparallel one.
I understand case b), in this case we have B=0 inside the plane so the normal component of H must vanish so the normal derivate of ∅ must cancell that of "I" and the problem is uniquely solved.
However in case a) we have H=0 and the tangencial component of H must vanish, we can do this with a parallel image however in this case what we know is the tangencial derivate of ∅ and the unicity theorem for laplace equation does not apply.
Can someone shed some light on this?
 
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  • #2
facenian said:
However in case a) we have H=0 and the tangencial component of H must vanish, we can do this with a parallel image however in this case what we know is the tangencial derivate of ∅ and the unicity theorem for laplace equation does not apply.
Can someone shed some light on this?

If the tangential derivative of ∅ vanishes on the boundary plane, what does that tell you about how ∅ varies on the boundary plane?
 
  • #3
TSny said:
If the tangential derivative of ∅ vanishes on the boundary plane, what does that tell you about how ∅ varies on the boundary plane?
The problem is that the tangencial derivative of ∅ does not vanish but it is equal an opposite to the tangencial component of source 1) (the wire) so the field generated by the to sources combined add up to a null tangencial field
 
  • #4
OK. I see. ∅ is the magnetic scalar potential of the image current alone.

From the known tangential component of H on the boundary plane due to the true current, you know the tangential component of H due to the image current. So, as you said, you know the value of the component of the gradient of ∅ in the direction of the tangential component of H, ∇t∅(x,y), at each point (x,y) on the boundary plane.

Taking ∅ = 0 at infinity, can you use ∇t∅(x,y) to find (in principle) the value of ∅ at any point of the plane? Does uniqueness follow?
 
  • #5
TSny said:
Taking ∅ = 0 at infinity, can you use ∇t∅(x,y) to find (in principle) the value of ∅ at any point of the plane? Does uniqueness follow?

Now I see, determining the tangencial component in this case is equivalent to determine the value of ∅ on the boundary surface so uniqueness can be applied. Thank you TSny
 

1. What is the Reitz-Milford-Cristy method for solving magnetic field problems?

The Reitz-Milford-Cristy method is a mathematical approach used to solve for the magnetic field in a given region. It uses Maxwell's equations and boundary conditions to model the behavior of the field and calculate its values at various points.

2. How does the Reitz-Milford-Cristy method differ from other methods of solving magnetic field problems?

The Reitz-Milford-Cristy method is a more rigorous and comprehensive approach compared to other methods. It takes into account all four of Maxwell's equations and applies them in a systematic manner to solve for the magnetic field. It also allows for the inclusion of different boundary conditions and material properties, making it more versatile.

3. What are the main steps involved in using the Reitz-Milford-Cristy method?

The main steps involved in using the Reitz-Milford-Cristy method are: formulating the problem, setting up the equations, applying boundary conditions, solving the equations, and interpreting the results. These steps ensure that the problem is accurately defined and that the solution is obtained through a logical and systematic process.

4. Can the Reitz-Milford-Cristy method be used for any type of magnetic field problem?

Yes, the Reitz-Milford-Cristy method can be used for a wide range of magnetic field problems, including those involving permanent magnets, electromagnets, and magnetic materials. It can also be applied to problems with different geometries and boundary conditions.

5. What are the advantages of using the Reitz-Milford-Cristy method over other numerical methods for solving magnetic field problems?

The Reitz-Milford-Cristy method offers several advantages over other numerical methods. It is a more accurate and precise method, as it takes into account all four of Maxwell's equations. It also allows for the inclusion of material properties, making it more realistic. Additionally, it is a more systematic approach, making it easier to follow and replicate the solution process.

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