- #1
wisky40
- 56
- 0
Homework Statement
Find the equipotential surface at the edge of a uniform electric charged disk ?
Homework Equations
[tex]\nabla^2 V= - \displaystyle \frac{\rho}{\epsilon} [/tex]
[tex]\displaystyle V= \iiint_R \frac{ {\rho}r dr d\phi dz}{4 {\pi} {\epsilon}|\vec r -\vec r'|}[/tex]
The Attempt at a Solution
In the case of Poisson's equation, I would've have to solve for PDEs, so I tried to aviod it.
In the integral, the distance of the disk radius was almost similar to an arbitrary point close to the border line, so that I couldn't use [tex]R>>r[/tex] as aproximation.
However, I decided to use symmetry, so that I thought if the electric field lines go radially inwards or ourwards, depending on the charge, then, the equipotential are concentric circles, but some classmates told me that was not enough. Finally, I tried use the triple integral to prove that a displacement of a point charge on the equipotential surfaces must be zero.
[tex]\int_{\phi_1}^{\phi_2 } q \displaystyle \hat e_\phi \frac{1}{a}\displaystyle \frac{ \partial V}{\partial \phi} \cdot \hat e_\phi a d \phi =0 \Rightarrow[/tex], so basically my proof consisted on:
[tex] \displaystyle \frac{\partial V}{\partial \phi} = 0[/tex], but nobody agree with me.
So, my question is this. Is there something wrong with my reasoning? or is there any other way?
Regards, wisky 40