- #1

bigplanet401

- 104

- 0

A conical surface (an empty ice-cream cone) carries a uniform surface charge [tex] \sigma[/tex]. The height of the cone is h, and the radius of the top is R. Find the potential difference between points

**a**(the vertex) and

**b**(the center of the top.)

I've tried integrating over the conical surface (zenith [tex]\phi[/tex] fixed):

[tex]

V(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \int d\mathbf{a} \frac{\sigma(\mathbf{r}^\prime)}{|\mathbf{r} - \mathbf{r}^\prime|} \quad \rightarrow \quad

\frac{1}{4\pi\epsilon_0} \int r^{\prime 2} dr^\prime \, d\theta^\prime

\frac{\sigma}{\sqrt{1 - r^{\prime 2} \cos^2 \phi}} \, ,

[/tex]

but I think that's wrong. Next I tried building up from a series of rings with charge density [tex]\lambda[/tex]:

[tex]

V_{\text{ring}} = \frac{\lambda}{2 \epsilon_0} \frac{R}{\sqrt{R^2 + z^2}} \, ;

[/tex]

unfortunately, I don't know how to set up the integration for this. Any help is appreciated,hopefully sooner than later--my written qualifier is ~3 weeks away!