A $$non-conducting$$ right circular hollow cone of base radius R has a uniform surface charge density σ . A part ABP is cut and removed from the cone. The potential due to the remaining portion of the cone at the apex point ‘P’ is xy σR∈$$0$$, where x, y are positive integers and $$co-primes$$. Find the value of y−x ?

Question

A $$non-conducting$$ right circular hollow cone of base radius R has a uniform surface charge density σ . A part ABP is cut and removed from the cone. The potential due to the remaining portion of the cone at the apex point ‘P’ is  xy  σR∈$$0$$, where x, y are positive integers and $$co-primes$$. Find the value of  y−x ?

Infinity Learn · Accepted Answer

Open the remaining cone, to form a part of disc of angle $θ$ .

Potential due to circular non conducting disc, $V_{C} = \frac{σ r}{2 ε_{0}}$ (where r is radius of disc)
So, $V_{C} = (\frac{θ}{2 π}) (\frac{σ \cdot ℓ}{2 ε_{0}})$ $= (\frac{σ ℓ}{2 ε_{0}}) (\frac{5}{3} \frac{π R}{(ℓ)}) (\frac{1}{2 π}) = \frac{5 σ R}{12 ε_{0}}$

Electrostatic potential

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Electrostatic potential

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Open the remaining cone, to form a part of disc of angle θ.Potential due to circular non conducting disc, VC=σr2ε0 (where r is radius of disc)So, VC=(θ2π)(σ⋅ℓ2ε0)=σℓ2ε053 πR(ℓ)12π=5σR12ε0

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