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Complex current density of sphere confusion

  1. Apr 5, 2013 #1
    Hello,

    I am a bit confused on how to get this into the proper form,

    $$
    \begin{eqnarray}
    \vec{J} &=& \vec{v}\rho \\
    &=& (\vec{\omega} \times \vec{r}')\rho_o \Theta(R-r') \\
    &=& \omega r' \sin(\theta')\rho_o \Theta(R-r')[\cos(\phi')\hat{y} - \sin(\phi')\hat{x}] \\
    &=& \vdots \\
    J_x + i J_y &=& -i \sqrt{\frac{3}{2 \pi}}\frac{q \omega r'}{R^3}\Theta(R-r')Y_{1,1}(\theta', \phi')
    \end{eqnarray}
    $$

    I have tried expanding and such but the algebra does not work out. What am I missing conceptually? I know that the term in the brackets should equate to,

    $$
    [\cdots] = i e^{i \phi'}
    $$

    which I can then do some algebra to get the result into spherical harmonics. How does the term in brackets equate to [itex]i e^{i \phi'}?[/itex] Do I simply multiply the [itex]\hat{y}[/itex] term by i?

    Hope you can shed some light on this matter!

    EDIT

    Yep, turns out I simply multiple [itex] \hat{y} [/itex] term by i. I just missed a minus sign! Sorry for the silly post!
     
    Last edited: Apr 5, 2013
  2. jcsd
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