# Calculate the energy required

1. May 5, 2013

### utkarshakash

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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?

2. May 5, 2013

### Staff: Mentor

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.

3. May 5, 2013

### rude man

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

4. May 5, 2013

### haruspex

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.

5. May 6, 2013

### utkarshakash

Can you tell me what is it?

6. May 6, 2013

### Staff: Mentor

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

7. May 8, 2013

### utkarshakash

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

8. May 8, 2013

### Staff: Mentor

Just go on and test it?

9. May 9, 2013

### utkarshakash

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

10. May 9, 2013

### Staff: Mentor

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

11. May 9, 2013

### Saitama

What is the expression for differential area (dA) ?

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