Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Spherical capacitor and vector potential

  1. Feb 7, 2014 #1

    ShayanJ

    User Avatar
    Gold Member

    There is a spherical capacitor with inner and outer radii of a and b respectively,has a dielectric material of small conductivity [itex] \sigma [/itex] between its concentric conducting spheres.Calculate vector potential and magnetic field arising from this configuration.
    This is how I did it:
    [itex] \nabla^2\phi=0 \Rightarrow \phi=-\frac{C}{r}+B [/itex]
    [itex] R=\int_a^b \frac{dr}{4\pi \sigma r^2}=\frac{1}{4\pi\sigma}(\frac{1}{a}-\frac{1}{b}) [/itex]
    [itex] I=\frac{\Delta \phi_{ab}}{R}=4\pi\sigma C[/itex]
    [itex] \vec{J}=-\frac{\sigma C}{r^2}\hat{r} [/itex]
    [itex] \vec{A}=\frac{\mu_0}{4\pi}\int_a^b \int_0^\pi \int_0^{2\pi} (-\frac{\sigma C}{r'^2}\hat{r})\frac{ r'^2 \sin{\theta'}d\varphi' d\theta' dr' }{ \sqrt{ r^2+r'^2-2rr' \cos{\gamma} } }
    [/itex]
    Then using Legendre functions and their addition theorem and some manipulations,I've reached to the following:
    [itex]
    \vec{A}=-\frac{\sigma\mu_0 C}{r} \sum_{n=0}^\infty \sum_{m=-n}^n \frac{1}{2n+1}\frac{1}{r^n} Y^m_n(\theta,\varphi) \int_a^b r'^n dr' \int_0^\pi \int_0^{2\pi} \hat{r} Y^{m*}_n(\theta',\varphi')\sin{\theta'} d\varphi' d\theta'
    [/itex]
    Then solved the integral above to get:
    [itex]
    \vec{A}=\frac{\sigma \mu_0 C(b^2-a^2)}{12r^2} \sin{\theta} e^{i\varphi}[(\hat{x}-i\hat{y})+2\hat{z}]
    [/itex]
    I'm just skeptical about the result.I tried to find a similar calculation somewhere but found nothing.
    Is it correct?
     
    Last edited: Feb 7, 2014
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Spherical capacitor and vector potential
  1. Spherical capacitor (Replies: 2)

  2. Vector potential (Replies: 3)

  3. Vector Potential (Replies: 1)

  4. Vector potential (Replies: 5)

Loading...