Solving Laplacian in Ex(r,z) Equation

[korps]

I need to know the steps involved in solving this laplacian.

Ex(r,z) = Eo*e^[-(r/ro)^2]*e^[-ibz]

the laplacian \/^2*Eo = ?
Eo is a vector
\/ is laplacian symbol

any help would be appreciated.

Thanks in advance.

 

[quasar987]

you probably meant \/^2*Ex, not \/^2*Eo.

Well, the problem is that there are polar and cartesian coordinates mixed up in the expression of Ex. So either transform r in cartesian and use the cartesian laplacian or transform z in polar and use the polar laplacian.

 

[HallsofIvy]

No, that's in "cylindrical coordinates" which is perfectly fine- just use polar coordinates with z appended.

The Laplacian of Y in cylindrical coordinates is
$$\frac{\partial^2 Y}{\partial r^2}+ \frac{1}{r}\frac{\partial Y}{\partial r}+ \frac{1}{r^2}\frac{\partial^2 Y}{\partial \theta^2}+ \frac{\partial^2 Y}{\partial z^2}$$