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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 pointsa(the vertex) andb(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!

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# Homework Help: Potential difference/cone

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