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A cone made of insulating material has a...

A cone made of insulating material has a total charge Q spread uniformly over its sloping surface. Calculate the work done in bringing a small test charge q from infinity to the apex of the cone. The cone has a slope length L.

Text Solution

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The correct Answer is:
` (Qq)/( 2 pi varepsilon_0 L)`
`dQ = Q/(piRh)- 2 piydx` …(1)
put ` R= L sin theta , h = L cos theta` and ` y =x` tan theta in equation ...(1)
` dQ = (2Q)/(L^2 sin thetacos theta 0 x tan theta dx`
` (##NG_PHY_C01_E01_125_S01.png" width="80%">
`dQ = (2Q)/l^2 sec^2 theta x dx` …(2)
`dQ = (KdQ)/(x sec theta)` ...(3)
From (2) and (3)
`dV = K (2Q)/(L^2) sec^2 theta x dx xx 1/(x sec theta) = (2 kQ)/(L^2) sec theta dx`
` V= (2 kQ)/l^2 sec thea theta int_0^(L cos theta) dx = (2 KQ)/L rArr V= 1/( 2 pi in_0 ) Q/L`
`W= qV = (Qq)/(2 pi in_0 L)`.
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