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

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

    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

    [tex] 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) [/tex]

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

    Ampere's law: [tex] \oint B \cdot dl = \mu_0 I [/tex]

    3. 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

    [tex] \oint B \cdot dl = BL = \mu_0 I N L \rightarrow B = \mu_0 N I \hat{z} [/tex]

    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?
  2. jcsd
  3. kreil

    kreil 641
    Gold Member

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