- #1
bigplanet401
- 104
- 0
Hi,
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!
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!