A charge q is located at the tip of hollow cone (such as an ice cream cone without the ice cream) with surface charge density $σ$ . The slant height of the cone is L, and the half-angle at the vertex is $θ .$ If the top half of the cone is removed and thrown away (see Figure), then the charge q experiences a force due to the remaining part of the cone? For what angle $θ$ is this force maximum?

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$\tan^{- 1} (2)$

$45^{0}$

$60^{0}$

$30^{0}$

answer is A.

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Detailed Solution

Let an element between slant height x and x+dx
$d F = (k) (q) \frac{[σ 2 π x \sin θ d x] (x \cos θ)}{x^{3}}$
$= \frac{k q σ 2 π}{x} \sin θ \cos θ d x$
$d F = \frac{q σ}{2 E_{0}} \sin θ \cos θ \frac{d x}{x}$
$∴ F = \frac{q σ}{2 E_{0}} \sin θ \cos θ \int_{L / 2}^{} \frac{d x}{x}$
$= \frac{q σ}{2 E_{0}} \sin θ \cos θ \ln (2)$
Therefore for maximum force $θ = 45^{0}$