Auxiliary Field H Problem, Asks for Magnetic Field etc

[laz0r]

Homework Statement

Homework Equations

int[H*dl] = I fenc

∇×H = cr^2 [Khat]

∇×H = Jf

The Attempt at a Solution

The textbook I have been assigned (griffiths) only deals with currents that are uniformly distributed and I'm not sure how to go about calculating the free enclosed current for the problem. If I knew how to do that, this problem would be very easy.

Using an amperian loop that extends the radius of the cylinder

∫(closed)H*dl = I fenc

|H|[2*pi*s] = I fenc
H = I fenc / [2*pi*s] [phi hat direction]

Then I would use the cross product ∇×H = Jf in order to calculate Jf, sub in Jf = cr^2 and then solve for c, thus solving part a. I'm just not sure about how to calculate the free enclosed current. if anyone knows the proper formula for this problem I would be grateful to see it.

 

[TSny]

Hi laz0r. Welcome to PF!

Regarding finding the proportionality constant c: If you integrate J_free over the cross-sectional area (a<r<b) what should that equal?

 

[laz0r]

Ah ok, I understand now. INT[J*dA] = Ifree, then you can use that for your H field later on and find the expression for c immediately by rearranging.

 

[TSny]

Ah ok, I understand now. INT[J*dA] = Ifree, then you can use that for your H field later on and find the expression for c immediately by rearranging.

You should be able to get c from just ∫J$\cdot$dA = I_o