# Electric Potential Difference on a Cone

by grindfreak
Tags: cone, difference, electric, potential
 P: 39 1. The problem statement, all variables and given/known data I'm working out of Griffith's "Intro to Electrodynamics" and the problem states: "A conical surface (an empty ice-cream cone) carries a surface charge $$\sigma$$. The height of the cone is h as is the radius of the top. Find the potential difference between points a (the vertex) and b (the center of the top). 2. Relevant equations and Attempt at a solution So, since this is the chapter that I'm in, I'm going to use $$$V(R)=\frac{\sigma }{4\pi \varepsilon _{0}}\int _{S}\frac{da{}'}{R}$$$. Now since a is at the vertex I chose $$$\vec{a}=0$$$ and $$$\vec{b}=h\hat{z}$$$. Thus the equation would become $$$V(\mathbf{b})-V(\mathbf{a})=\frac{\sigma }{4\pi \varepsilon _{0}}\int _{S}\left [ \frac{{da}'}{\sqrt{(h-{z}')^2+{s}'^2}} -\frac{{da}'}{\sqrt{{z}'^2+{s}'^2}}\right ]$$$ Now da' is what I was having a little trouble attaining, so I thought the best place to start would be with the surface area of the cone: $$$a'=\pi s\sqrt{s^2+z^2}$$$ but since the radius s is equal to the height z in our case the formula becomes $$$a'=\pi s\sqrt{s^2+s^2}=\sqrt{2}\pi s^2$$$. Now since fractions of this area can be represented by multiplying in terms of the angle that determines the fraction of area, $$$\frac{\theta }{2\pi }$$$. Thus $$$a'=(\sqrt{2}\pi s^2)\cdot (\frac{\theta }{2\pi })=\frac{\sqrt{2}}{2}s^2\theta$$$ and if I consider the angle to be small $$$a'=\frac{\sqrt{2}}{2}s^2d\theta$$$. Now to find the differential area I should subtract to get $$$da'=\frac{\sqrt{2}}{2}(s+ds)^2d\theta -\frac{\sqrt{2}}{2}s^2d\theta=\frac{\sqrt{2}}{2}d\theta(s^2+sds+ds^2-s^2)=\frac{\sqrt{2}}{2}sdsd\theta$$$ since ds^2 is to small to matter. The main equation then becomes: $$$V(\mathbf{b})-V(\mathbf{a})=\frac{\sigma }{4\pi \varepsilon _{0}}\int _{S}\left [ \frac{{da}'}{\sqrt{(h-{z}')^2+{s}'^2}} -\frac{{da}'}{\sqrt{{z}'^2+{s}'^2}}\right ]=\frac{\sigma }{4\pi \varepsilon _{0}}\int _{S}\left [ \frac{1}{\sqrt{(h-{s}')^2+{s}'^2}} -\frac{1}{\sqrt{{s}'^2+{s}'^2}}\right ](\frac{\sqrt{2}}{2}{s}'{ds}'{d\theta}' )$$$ $$$=\frac{\sqrt{2}\sigma }{8\pi \varepsilon _{0}}\int_{0}^{2\pi }\int_{0}^{h}\left ( \frac{{s}'}{\sqrt{(h-{s}')^2+{s}'^2}}-\frac{\sqrt{2}}{2} \right ){ds}'{d\theta}'$$$ $$$=\frac{\sqrt{2}\sigma }{4\varepsilon _{0}} [(-hln({s}'-h)+\frac{{s}'^3}{3}+{s}')|_{0}^{h}-\frac{\sqrt{2}}{2}h]$$$ but the above does not converge when evaluated so I'm at a loss. This isn't for a class or anything, I'm just self studying so answer at your convenience.
 Sci Advisor HW Helper P: 6,590 Try slicing the cone along the vertical axis into rings of area $dA = 2\pi s dz$ where s = radius of the ring at height z, which is a linear function of z. So each ring carries a charge that is proportional to z. That should be easy to integrate. AM

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