How Much Energy Does It Take to Move a Charge to the Apex of a Charged Cone?

[utkarshakash]

Homework Statement

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 A of the cone. The cone has a slope of length L.

Homework Equations

The Attempt at a Solution

Since work done equals change in P.E so I have to find the P.E of the system.
Suppose at a distance x from A I take an elementary cylinder of length dx. Let charge density be σ.
dQ=σdV

Wait! I don't know PE of disc and this seems an extremely time consuming process to me. Is there any other method?

 

[mfb]

The charge is present at the surface of the cone only. For the apex, there is a nice way to parametrize the surface and its distance to the apex.

 

[rude man]

Homework Statement

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 A of the cone. The cone has a slope of length L.

Don't we also need to know the largest diameter?

 

[haruspex]

Don't we also need to know the largest diameter?

No. Consider a circular element at distance r from the apex. If you vary the angle of the cone, that element moves around but stays the same distance r from the apex and carries the same fraction of the total area, hence the same charge. Thus the potential it generates at the apex is independent of the cone angle.

 

[utkarshakash]

The charge is present at the surface of the cone only. For the apex, there is a nice way to parametrize the surface and its distance to the apex.

Can you tell me what is it?

 

[mfb]

Use anything (linear) you like, it will probably work.

 

[utkarshakash]

Use anything (linear) you like, it will probably work.

Ok so I take an elementary ring at a distance x from the apex along the surface of cone. Is it alright?

 

[mfb]

Just go on and test it?

 

[utkarshakash]

Just go on and test it?

But nothing other than slope length L is given. I'm falling short of data. Even the radius is not given.

 

[mfb]

Just introduce additional variables, if you like. If they drop out later (they should do this here), everything is fine again.

 

[Saitama]

But nothing other than slope length L is given. I'm falling short of data. Even the radius is not given.

What is the expression for differential area (dA) ?