# Parallel plates capacitor

1. Dec 6, 2015

### alejandrito29

1. The problem statement, all variables and given/known data

Capacitor parallel plates separation $$d$$, potencial diference $$V_0$$, at the center there is a semi sphere of radius $r_0$.

Find the potencial as function of position if $$d>>r_0$$
2. Relevant equations
I think that relevant equation are Gauss Law

3. The attempt at a solution
Mi proposed of solution is

$$V_{plates}= \sigma /\epsilon_0 z$$

for $$r<|r_0|$$

Gauss Low

$$E 4/3 \pi r^2 = \sigma 4/3 \pi r_0^2$$

$$E= \sigma r_0^2 /\epsilon_0 r^{-2} \hat{r}$$

$$V_{sp} = \sigma r_0 /\epsilon_0 ( r_0/r-1) = \sigma r_0 /\epsilon_0 ( r_0/\sqrt{x^2+y^2+z^2}-1)$$

And for $$r>|r_0|$$

$$V=V_{plates}+V_{sp}$$

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2. Dec 7, 2015

### BvU

Hi Alex,

Can't really follow. It looks to me as if you assume $\sigma$ is a constant, but why would it be constant ? The plates are conducting, so the electric field has to be perpendicular to the surface at all places. I don't think you potential satisfies that.

3. Dec 7, 2015

### alejandrito29

Tanks

Some suggest for solving my problem?

my adittion out of radius $$r_0$$ is because there is two field electrics at zone $$r>|r_0|$$, then two potential . I dont am sure of this.

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4. Dec 7, 2015

### BvU

I think I understand what you mean, and I tried to bring across that the complete field is not a simple addition obtained from two plates and half a sphere. You will have to find something that satisfies the field equations and the boundary conditions as well. Not so trivial, I grant you. Do you know about the image charge method ?