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Show that w is solenoidal having spherical polar coordinates

  1. May 7, 2015 #1
    1. The problem statement, all variables and given/known data
    The gradient, divergence and curl in spherical polar coordinates r, ∅, Ψ are
    nablaΨ = ∂Φ/∂r * er + ∂Φ/∂∅ * e 1/r + ∂Φ/∂Ψ * eΨ * 1/(r*sin(∅))
    nabla * a = 1/r * ∂/∂r(r2*ar) + 1/(r*sin(∅)*∂/∂∅[sin(∅)a] + 1/r*sin(∅) * ∂aΨ/∂Ψ
    nabla x a =

    |er r*e r*sin(∅)*eΨ |
    |∂/∂r ∂/∂∅ ∂/dΨ |
    | 1/r2*sin(∅) |ar r*a r*sin(∅)*aΨ |

    Where a = ar * er + a*e + aΨ*eΨ

    Suppose that the vector field w has the form w = wΨ(r, ∅) eΨ Show that w is
    solenoidal and find wΨ(r, ∅), when w is also irrotational. Find a potential for w in this case.
    hah..PNG
    2. Relevant equations


    3. The attempt at a solution
    I started by finding vector potential. however it seems too simple. (Or most likely id id it wrong). In addition i am not sure, if it is even the right way to about it. I can't put in working out of the vector potential i have, as i have trouble with coding it on the forum(My handwriting is not legible). So you can assume i have got it wrong. Any help will be greatly appreciated.
     
  2. jcsd
  3. May 7, 2015 #2
    First.

    To show that ω is solenoidal implies that the divergence of the vector field is 0. Thats easy to show:

    $$∇·\omega = \frac{1}{r\sin\theta}\; \frac{\partial\omega_\phi(r,\theta)}{\partial\phi}$$

    and since the φ component of ω does not depend on φ, it's partial derivative equals 0.

    So the vector field is solenoidal.

    Second.

    We must impose that ×ω=0.
    $$∇×\omega=\begin{vmatrix} ê_r & rê_\theta &r\sin\thetaê_\phi \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta}& \frac{\partial}{\partial \phi} \\ 0&0&r\sin\theta\omega_\phi\end{vmatrix} \;=\; ê_r \left[ \frac{\partial\omega_\phi}{\partial \theta}r\cos\theta \right] - ê_\theta \left[ \frac{\partial\omega_\phi}{\partial r}r\sin\theta \right] $$

    And this is 0 ∀r,φ if ωφ is not really dependent on θ and r. So ωφ=C (is constant on all variables).

    And now I am stuck... I really doubt on this second result. I'm sorry, but I'll wait until someone else come check it. Maybe this help you on finding a path.
     
  4. May 7, 2015 #3
    20150507_201842.jpg
    I have figured out part 2, very similar to part one. Cross multiply the matrix, and we get 0. I think I am getting somewhere on part three as well. Thank you. And your LaTeX skills are too good xD
     
  5. May 7, 2015 #4
    MMMM.... I think you got that rotational wrong. You cannot cancel those partial derivatives inside the determinant.

    Those vectors aren't zero, but their associated components.

    Also I think you haven't understood the exercice. It doesn't demand you to show that is irrotational, but to assume it is in order to find out what is the function of the phi component.
     
  6. May 7, 2015 #5
    Of course. I rushed to conclusion without reading question fully.....

    thanks for the help thought, i would have had it completely wrong otherwise.
     
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