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

[facenian]

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?

 

[TSny]

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?

 

[facenian]

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

 

[TSny]

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?

 

[facenian]

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