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Homework Help: Griffith's Electrodynamics 6.8

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data
    A long cylinder of radius R carries a magnetization [tex]\vec{M}=Ks^{2}\hat{\phi}[/tex] where k is a constant, s is the distance from the axis, and [tex]\hat{\phi}[/tex] is the usual azimuthal unit vector. Find the magnetic field due to [tex]\vec{M}[/tex] for points inside and outside the cylinder.


    2. Relevant equations
    [tex]\vec{J}_{b}=\nabla\times\vec{M}[/tex]
    [tex]\vec{K}_{b}=\vec{M}\times\hat{n}[/tex]
    Formula for the Vector potential That I can't seem to get to work in tex
    3. The attempt at a solution
    [tex]\vec{J}_{b}=3k\sqrt{x^{2}+y^{2}}\hat{z}[/tex]
    [tex]\vec{K}_{b}=3k(x^{2}+y^{2})\hat{z}[/tex]

    Truthfully my biggest annoyance right now is formulating the separation vector (I know I'm lame) then a little plug and chug and I'm done but I wouldn't mind someone checking my work (probably wrong...)
     
    Last edited: May 3, 2010
  2. jcsd
  3. May 3, 2010 #2
    You're told that it is a cylinder, why are you using Cartesian coordinates?

    [tex]
    \vec{J}_b=\vec{\nabla}\times\vec{M}=\frac{1}{s}\,\frac{\partial}{\partial s}\left(s\,ks^2\right)\hat{z}=3ks\hat{z}[/tex]

    Similarly for [itex]\vec{K}_b[/itex]:

    [tex]\vec{K}_b=\vec{M}\times\hat{n}=-kR^2\hat{z}[/tex]

    You can then use Ampere's Law to find the magnetic field.
     
    Last edited: May 3, 2010
  4. May 3, 2010 #3
    You might want to double check your bound surface current. Also, try to keep everything in cylindrical coordinates. Also, Ampere's law will be real useful for this problem.

    EDIT: Seems jdwood983 beat me to it :(
     
  5. May 3, 2010 #4
    Don't have Ampere's law to work with and not allowed to use cylindrical coordinates for some reason.
     
  6. May 3, 2010 #5
    Does it specifically say you can't use either of those? And what kind of question uses cylindrical coords and asks you afterwards not to use them.
     
  7. May 3, 2010 #6
    Ampere's law for polarized objects is about 3 sections ahead of where the problem is located in the book (not that that would usually stop me...).
     
  8. May 3, 2010 #7
    I have griffith's and Ampere's law can be found in section 5.3.3. Long before your problem.
     
  9. May 3, 2010 #8
    This problem is very similar to a long straight current carrying wire. Except the current density is radially dependent and you have a negative surface current.
     
  10. May 3, 2010 #9
    If I'm not mistaken there is some reason not to use that statement of ampere's law in cases of magnetization.
     
  11. May 3, 2010 #10
    If you are determined not to use Ampere's law (which I advise against) then you will have to use the Biot-Savart law. But you will just end up with the same answer.
     
  12. May 3, 2010 #11
    I'm pretty sure if I were allowed to use it in cases of magnetization there wouldn't be a section "6.3.1 Ampere's Law in Magnetized Materials"
     
  13. May 3, 2010 #12
    Well you aren't using H when solving for Ampere's law in this problem. When applying Ampere's law for this problem you only worry about bound currents and treat it as the total current. But as I said before you can always use the Biot-Savart law, just be very careful when solving it.
     
  14. May 3, 2010 #13
    Well I see what you mean since there is a problem in 6.3 that is roughly the same and asks you to solve it using the method in 6.2 for part (a) and using Ampere's law for part (b). I seriously don't think you will be deducted points for using ampere's law early. But you can always solve the magnetic fields using integrals just in case. Just be very careful :)
     
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