Complex current density of sphere confusion

1. Apr 5, 2013

Shinobii

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 $i e^{i \phi'}?$ Do I simply multiply the $\hat{y}$ term by i?

Hope you can shed some light on this matter!

EDIT

Yep, turns out I simply multiple $\hat{y}$ term by i. I just missed a minus sign! Sorry for the silly post!

Last edited: Apr 5, 2013