Computing for Electric Field given cylindrical coordinates of v.

[jhosamelly]

Homework Statement

If the scalar electric potential v in some region is given in cylindrical coordinates by
$v (r, \phi, z) = r^2 sin \phi e^{\frac{-3}{z}}$, what is the electric field $\vec{E}$ in that region?

Homework Equations

$E = -\nabla v$

The Attempt at a Solution

So, first I need to change the cylindrical coordinates to cartesian coordinates.

$v (r, \phi, z) = r^2 sin \phi e^{\frac{-3}{z}}$

$v (r, \phi, z) = (x^2 + y^2) \frac{y}{r} e^{\frac{-3}{z}}$

$v (r, \phi, z) = (x^2 + y^2) \frac{y}{\sqrt{x^2 + y^2}} e^{\frac{-3}{z}}$

$v (r, \phi, z) = y e^{\frac{-3}{z}} \sqrt{x^2 + y^2}$

** so is this already the cartesian coordinates? can I perform the gradient now?

 

[vela]

That's one way to do it. It would probably be simpler to use the gradient in cylindrical coordinates. See, for example, equation (32) on http://mathworld.wolfram.com/CylindricalCoordinates.html.

 

[jhosamelly]

That's one way to do it. It would probably be simpler to use the gradient in cylindrical coordinates. See, for example, equation (32) on http://mathworld.wolfram.com/CylindricalCoordinates.html.

ow, I see. We were not given that formula though. So I think I need to do it in cartesian coordinates. Thanks.