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**1. Find the electric field at the tip of a cone of height and radius R with uniform surface charge density [tex] \sigma [/tex].**

I get that the field diverges at the tip, which is puzzling because it's not as though there's a point charge at the tip. I thought this sort of thing can't happen when you treat charge as smeared over a surface.

I get that the field diverges at the tip, which is puzzling because it's not as though there's a point charge at the tip. I thought this sort of thing can't happen when you treat charge as smeared over a surface.

**2. Homework Equations**

The field from a hoop of radius z, charge q, at height z above the hoop center is

[tex] E_{hp} = \frac{q}{4\pi\epsilon_0} \frac{1}{2\sqrt{2}z^2} [/tex].

## The Attempt at a Solution

I break the cone into hoops of variable radius. Because it's a right cone, the distance from the tip to each differential hoop equals the radius -- very convenient. The charge on each hoop is

[tex] dq = 2\pi z \sigma ds = 2\pi z \sigma \sqrt{2} dz [/tex]

where [tex] ds = \sqrt{2} dz [/tex] is a differential arc length along the side of the cone.

Each hoop contributes to the field

[tex] dE = \frac{dq}{4\pi\epsilon_0} \frac{1}{2\sqrt{2}z^2} [/tex]

[tex] = \frac{2\pi z \sigma \sqrt{2} dz}{4\pi\epsilon_0} \frac{1}{2\sqrt{2}z^2} [/tex]

[tex] = \frac{\sigma}{4\epsilon_0} \frac{dz}{z} [/tex],

and the field is then

[tex] \int dE = \int_0^R \frac{\sigma}{4\epsilon_0} \frac{dz}{z} = \left . \frac{\sigma}{4\epsilon_0} \ln z \right |_R^0 [/tex],

which blows up at 0.

Correct? If so, what to make of it? Special surfaces can mimic point charges?