1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Curl of the vector potential produced by a solenoid

  1. Aug 27, 2015 #1
    1. The problem statement, all variables and given/known data / 2. Relevant equations
    I was looking at Example 5.12 in Griffiths (http://screencast.com/t/gGrZEPBpk0) and I can't manage to work out how to verify that the curl of the vector potential, A, is equal to the magnetic field, B.

    I believe my problem lies in confusion about how to apply the equation for curl in cylindrical coordinates (given the final vector potential in equation 5.71).

    3. The attempt at a solution
    I was unable to make progress while trying to apply the formula for curl in cylindrical coordinates as per https://upload.wikimedia.org/math/1/d/c/1dc228ba3864e64bff1d49fd0984a80f.png

    Am I using the wrong equation? If not, how should I be going about doing it? Any assistance is appreciated.
     
  2. jcsd
  3. Aug 27, 2015 #2

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You're doing fine. Griffiths wants to avoid confusion between r, R and ##\rho## so he uses s. In the curl expression that is ##\rho##.
     
  4. Aug 27, 2015 #3
    Ok thank you.

    So then do I write:

    [tex]A_{\phi} = \frac{\mu_0nIR^2}{2\rho}=\frac{\mu_0nI(\rho^2 + z^2)}{2\rho}[/tex]

    Then after simplification as all other terms are zero [tex]\bigtriangledown \times A = -\frac{\partial A_{\phi}}{\partial z}\hat{\rho}+\frac{1}{\rho}\left(\frac{\partial (\rho A_{\phi})}{\partial \rho}\right)\hat{z}[/tex]
     
    Last edited: Aug 27, 2015
  5. Aug 27, 2015 #4

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I don't think ##R^2 = \rho^2 + z^2##. I think it's the radius of the solenoid. Simplifies things !
     
  6. Aug 27, 2015 #5
    Sorry I don't think I'm following, I'm still not sure what I'm supposed to have for [tex]A_{\phi}[/tex] then. I'm thinking that I need to get [tex]\mu_0 n I \hat{z}[/tex] out of this as the result but I've tried many different things for [tex]A_{\phi}[/tex] with no progress. What should the value of [tex]A_{\phi}[/tex] be please.

    Ugh, don't know how to format the latex, sorry.
     
    Last edited: Aug 27, 2015
  7. Aug 27, 2015 #6

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    For ##A_\phi ## you have 5.70 for inside and 5.71 for outside.
    e.g. 5.70 :
    $$A_\phi = {\mu_0 nI\over 2} s\hat \phi $$doesn't depend on z, so:

    For the curl then the only nonzero is ## \displaystyle \frac{1}{\rho}\left(\frac{\partial (\rho A_{\phi})}{\partial \rho}\right)\hat{z}## , in this case ##\displaystyle\frac{1}{s}\left(\frac{\partial (s A_{\phi})}{\partial s}\right)\hat{z}## . Bingo !
     
  8. Aug 27, 2015 #7
    But then for the outside wouldn't that mean that [tex]\frac{1}{\rho}\left({\frac{\partial (\rho A_{\phi})}{\partial \rho}}\right)\hat{z}=\frac{1}{\rho}\left({\frac{\partial (\rho\frac{ \mu_0 n I R^2}{2 \rho})}{\partial \rho}}\right)\hat{z}=\frac{1}{\rho}\left({\frac{\partial (\frac{ \mu_0 n I R^2}{2})}{\partial \rho}}\right)\hat{z} = 0[/tex]

    Which isn't equal to [tex]B = \mu_0 n I[/tex]?
     
  9. Aug 27, 2015 #8

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    And are you unhappy about that ? You just found the same result as 5.57 in example 5.9 ! Bravo !
     
  10. Aug 27, 2015 #9
    Oh... *sigh*

    Thank you for your assistance BvU, it was much appreciated!
     
  11. Aug 27, 2015 #10

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You're welcome. I learn too (haven't worked with vector potential since university).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Curl of the vector potential produced by a solenoid
  1. Vector potential (Replies: 1)

Loading...