E/M Vector Potential of finite Wire

[shannon]

Homework Statement

Consider a finite wire which lies on the z-axis and extends from the point z=-Λ to the point z=+Λ. The vector potential in the xy plane a distance s from the wire is:
Λ->∞
A=(µₒI/2π) ln (2Λ/s) k̂

An equally good vector potential is given by A'=A+∆λ, where λ is any scalar field we wish to choose. Determine a suitable choice for λ which has the property that A' remains finite in the limit Λ->∞.
(Let ∆ be rotated 180 to be the gradient.)

Homework Equations

I found the curl of A=(µₒI/2π) ln (2Λ/s) k̂ to find the magnetic field, and I got:
A*= -(µₒI/2πs)

The Attempt at a Solution

Ok, here is what I did:
I figured that I had to find a value of λ that gave way to a A' that when you took the curl of it would yield A*. At first, I just tried to find a value of λ that was involving ln (...)k̂,
but then I saw that A' would then depend (s), and it would be tricky to find a value. Then I saw the hint that it was better if λ didn't depend on (s).

So, if I just want A, can I just choose a random constant for λ? Like 4 or something?

Please Help!

 

[Pythagorean]

Homework Statement

Consider a finite wire which lies on the z-axis and extends from the point z=-Λ to the point z=+Λ. The vector potential in the xy plane a distance s from the wire is:
Λ->∞
A=(µₒI/2π) ln (2Λ/s) k̂

An equally good vector potential is given by A'=A+∆λ, where λ is any scalar field we wish to choose. Determine a suitable choice for λ which has the property that A' remains finite in the limit Λ->∞.
(Let ∆ be rotated 180 to be the gradient.)

Homework Equations

I found the curl of A=(µₒI/2π) ln (2Λ/s) k̂ to find the magnetic field, and I got:
A*= -(µₒI/2πs)

The Attempt at a Solution

Ok, here is what I did:
I figured that I had to find a value of λ that gave way to a A' that when you took the curl of it would yield A*. At first, I just tried to find a value of λ that was involving ln (...)k̂,
but then I saw that A' would then depend (s), and it would be tricky to find a value. Then I saw the hint that it was better if λ didn't depend on (s).

So, if I just want A, can I just choose a random constant for λ? Like 4 or something?

Please Help!

For the derivative in the curl of A, remember the chain rule (derivative of the outside times the derivative of the inside).

As for the star question, since z goes to infinite perhaps you could view this in terms of an angle instead of a distance.