What is the Magnetic Vector Potential for a Given Magnetic Field?

[beth92]

Homework Statement

For the magnetic field B=k/s³ z determine the magnetic vector potential A. For simplicity, assume that A does not have a component in the s direction.

(I don't know if this is relevant but this was a follow up question to one in which I was required to find the induced current for a bar moving along a semicircular loop of wire - like a slide wire generator bent into a semicircular shape - and then the torque on the bar due to the magnetic force and then the position ∅ at which the bar comes to rest.)

Homework Equations

Curl of A = B

Divergence of A = 0

The Attempt at a Solution

The z component of the curl of A in cylindrical coordinates is:
1/s[d(sA_∅)/ds - d(A_s)/d∅]

The B field we are considering has only a z component so the s and ∅ components of the curl of A can be disregarded. Also, we are told in the problem that A_s = 0 so the only surviving term, equal to B, is:

1/s d(A_∅)/ds = k/s³

Separating the variables:

∫ d(sA_∅) = ∫ (k/s²) ds

After integration, we get:

A_∅ = -k/s² + C/s

Where C is the integration constant.

This is as far as I got...I'm not sure how how to find out what C is. Any tips would be appreciated!

 

[tiny-tim]

hi beth92!

vector potential is like scalar potential …

only the potential difference matters, so you can add any constant vector you like!

i'd add 0 ​