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Electric Potential Difference on a Cone 
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#1
Dec2610, 01:54 PM

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 icecream cone) carries a surface charge [tex]\sigma[/tex]. 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 [tex]\[V(R)=\frac{\sigma }{4\pi \varepsilon _{0}}\int _{S}\frac{da{}'}{R}\][/tex]. Now since a is at the vertex I chose [tex]\[\vec{a}=0\][/tex] and [tex]\[\vec{b}=h\hat{z}\][/tex]. Thus the equation would become [tex]\[V(\mathbf{b})V(\mathbf{a})=\frac{\sigma }{4\pi \varepsilon _{0}}\int _{S}\left [ \frac{{da}'}{\sqrt{(h{z}')^2+{s}'^2}} \frac{{da}'}{\sqrt{{z}'^2+{s}'^2}}\right ]\][/tex] 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: [tex]\[a'=\pi s\sqrt{s^2+z^2}\][/tex] but since the radius s is equal to the height z in our case the formula becomes [tex]\[a'=\pi s\sqrt{s^2+s^2}=\sqrt{2}\pi s^2\][/tex]. Now since fractions of this area can be represented by multiplying in terms of the angle that determines the fraction of area, [tex]\[\frac{\theta }{2\pi }\][/tex]. Thus [tex]\[a'=(\sqrt{2}\pi s^2)\cdot (\frac{\theta }{2\pi })=\frac{\sqrt{2}}{2}s^2\theta \][/tex] and if I consider the angle to be small [tex]\[a'=\frac{\sqrt{2}}{2}s^2d\theta \][/tex]. Now to find the differential area I should subtract to get [tex]\[da'=\frac{\sqrt{2}}{2}(s+ds)^2d\theta \frac{\sqrt{2}}{2}s^2d\theta=\frac{\sqrt{2}}{2}d\theta(s^2+sds+ds^2s^2)=\frac{\sqrt{2}}{2}sdsd\theta\] [/tex] since ds^2 is to small to matter. The main equation then becomes: [tex]\[V(\mathbf{b})V(\mathbf{a})=\frac{\sigma }{4\pi \varepsilon _{0}}\int _{S}\left [ \frac{{da}'}{\sqrt{(h{z}')^2+{s}'^2}} \frac{{da}'}{\sqrt{{z}'^2+{s}'^2}}\right ]=\frac{\sigma }{4\pi \varepsilon _{0}}\int _{S}\left [ \frac{1}{\sqrt{(h{s}')^2+{s}'^2}} \frac{1}{\sqrt{{s}'^2+{s}'^2}}\right ](\frac{\sqrt{2}}{2}{s}'{ds}'{d\theta}' )\][/tex] [tex]\[=\frac{\sqrt{2}\sigma }{8\pi \varepsilon _{0}}\int_{0}^{2\pi }\int_{0}^{h}\left ( \frac{{s}'}{\sqrt{(h{s}')^2+{s}'^2}}\frac{\sqrt{2}}{2} \right ){ds}'{d\theta}' \][/tex] [tex]\[=\frac{\sqrt{2}\sigma }{4\varepsilon _{0}} [(hln({s}'h)+\frac{{s}'^3}{3}+{s}')_{0}^{h}\frac{\sqrt{2}}{2}h]\][/tex] 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. 


#2
Dec2710, 10:24 AM

Sci Advisor
HW Helper
P: 6,653

Try slicing the cone along the vertical axis into rings of area [itex]dA = 2\pi s dz[/itex] 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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