Using Ampere's law to find B just outside finite solenoid

[quasar_4]

Homework Statement

We have a solenoid of radius a, length L, with ends at z = +/- L/2. The problem is to use Ampere's law to show that the longitudinal magnetic induction just outside the coil is approximately

$$B_z (\rho=a^+, z) \approx \left(\frac{2 \mu_0 N I a^2}{L^2} \right) \left(1+ \frac{12 z^2}{L^2}- \frac{9 a^2}{L^2} + \ldots \right)$$

(This is part b of problem 5.5 in Jackson 3rd edition).

Homework Equations

Ampere's law: $$\oint B \cdot dl = \mu_0 I$$

The Attempt at a Solution

I'm pretty sure I'm thinking about this too simplistically, which is why I'm stuck. For an infinite solenoid, the magnetic induction outside is 0. Since we've got a finite solenoid, there are obviously fringing effects of some sort and I guess we can't expect the field to be 0 outside anymore (though it should be reasonably small compared to the field on axis in the center of the solenoid).

I can't figure out what to do differently with Ampere's law though. Does my amperian loop enclose the coils over the full length L of the solenoid or just a short bit? What I was initially thinking was that if my amperian loop encloses all of them, then

$$\oint B \cdot dl = BL = \mu_0 I N L \rightarrow B = \mu_0 N I \hat{z}$$

but this is the result you get with the infinite solenoid inside the solenoid, and is clearly not what I need to have since there's nothing to expand on... so I guess I'm not sure how to set this up. Can anyone help?

 

[kreil]

end of page 6/beginning of page 7 is relevant.

http://www.contrib.andrew.cmu.edu/~bsauerwi/Problems/2005EM1HW8.pdf

 

[Chino_br]

Hey guys, how do you know that for an infinite solenoid, the magnetic induction outside is 0?