Magnetic field of a hollow cylinder

[toomuchfries]

Homework Statement

Find an expression for the magnetic field of a hollow finite solenoid carrying uniform current with n turns (there is no top or bottom surfaces and the turns a very closely winded) at a point arbitrary P both inside and outside. Let's just say windings around a hollow pvc pipe.

Homework Equations

I can use either biot savarts law and/or magnetic vector potential for surface currents. I'll use griffiths notation 3rd edition equations 5.39 and 5.64 respectively.

The Attempt at a Solution

I'll attempt this in cylinderical coordinates. Since the current is uniform. I'll use the

[itex]\vec{A}\left(\vec{r}\right)=\frac{1}{4\pi}\int\frac{\vec{K\left(r'\right)}}{\tau}da'[/itex]

Current density [itex]\vec{K}[/itex] is in the [itex]\hat{\phi}[/itex]

so [itex]\vec{K}=\frac{nI}{z'}\hat{\phi}[/itex] ? Since current is uniform.
[itex]\tau=\left|r-r'\right|[/itex]

[itex]r=s\hat{s}+s\phi\hat{\phi}+z\hat{z}[/itex]
[itex]r'=s\hat{s}?[/itex]

[itex]da'=sd\phi'dz'[/itex]

Where the limits of phi' is 0->2*pi and z' is 0 -> L (length of cylinder)

 

[sandy.bridge]

Do you have an image of the particular set-up? Or could you reference to one in Griffiths?

 

[toomuchfries]

Hi Sandy, I just attached an image file its just turns of wire wrapped around a pvc. I looked up a few things I think I have to use elliptic integrals... My first aim is to write an expression for the current density and the position and source vectors r and r'.

 

[sandy.bridge]

Are you neglecting fringing? If you use Biot-Savart, have you considered analyzing a single loop, and then applying superposition to n loops?

 

[toomuchfries]

I've thought about that since Jackson's has a solution to a single loop of wire. But I'm going to be inserting a ferromagntic material inside at some stage so I might need to take fringing fields into account?

 

[sandy.bridge]

Well, I am not really in a position to help you with the fringing effects as I only have experience at the undergraduate level thus far.