Question

A cone made of an insulating material has a total charge Q spread uniformly over its sloping surface calculate the energy required to bring a small test charge q from infinity to the apex of the cone the cone has a slope length L​

Answer 1

given, charge Q on the cone is uniformly spread over its sloping surface.

Let $\sigma$ be the surface charge density. the charge on an elementary area of cone of side length dx at a distance x from the apex A[ as shown in figure. ] is given by,

$dq=\sigma dA$

= $2\pi\sigma xsin\alpha dx$

now the potential at the apex due to this element is given by, $V=\frac{dq}{4\pi\epsilon_0x}$

V = $\frac{2\pi\sigma xsin\alpha dx}{4\pi\epsilon_0x}$

= $\frac{\sigma sin\alpha x}{2\epsilon_0}\int\limits^L_0dx$

= $\frac{\sigma sin\alpha L}{2\epsilon_0}$

now, total charge on the cone is given by, Q = $\sigma A$

= $\pi L^2sin\alpha\sigma$

so, V = $\frac{Q}{2\pi\epsilon_0 L}$

now, workdone to bring a small test charge q from infinity to the apex of the cone is W = qV = $\frac{Qq}{2\pi\epsilon_0L}$

Answer 2

Answer:

Please do have a look at my method.....

Hope this helps you!

Do let me know if this is incorrect :)