Calculus - vector potentials

[Niles]

1. The problem statement, all variables and given/known data
Hi all.

Please take a look at:

http://books.google.com/books?id=9p6sUTxUoZ0C&pg=PA417&lpg=PA417&dq=%22seek+a+vector+potential+that+is+parallel&source=web&ots=peoLUp4z_M&sig=vaRVDMazSj10x0k4v4iGKCHni0o

What do they mean by that hint?

3. The attempt at a solution
First I want to find the divergence of the three vectors.

Second, for the vectors which have a divergence equal to zero, I want to find the vector A so the rotation of A = B. But this is the part I am in doubt about - can you help?

Sincerely, Niles.

[Tom Mattson]

1. The problem statement, all variables and given/known data
Hi all.

Please take a look at:

http://books.google.com/books?id=9p6sUTxUoZ0C&pg=PA417&lpg=PA417&dq=%22seek+a+vector+potential+that+is+parallel&source=web&ots=peoLUp4z_M&sig=vaRVDMazSj10x0k4v4iGKCHni0o

What do they mean by that hint?


For static fields, the vector potential \vec{A} is parallel to the current density \vec{J}. And, Ampere's law says that \vec{\nabla}\times\vec{B} is parallel to the current density as well, under the same condition. Since the vector potential and the curl of the magnetic field are both parallel to the same vector, they are parallel to each other.


3. The attempt at a solution
Second, for the vectors which have a divergence equal to zero, I want to find the vector A so the rotation of A = B. But this is the part I am in doubt about - can you help?


Here's a hint: If \vec{A} is parallel to \vec{\nabla}\times\vec{B}, then \vec{A}=k\vec{\nabla}\times\vec{B} for some k.