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#1 2015-06-13 17:16:23
Minimise the potential energy
Given three charges q, 2q and 8q, not necessarily in that order, how do I place them on a line at most 9 m long so that I can minimize there potential energy?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
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#2 2015-06-14 08:22:55
- Bob
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- Registered: 2010-06-20
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Re: Minimise the potential energy
hi
I had to look up potential energy of a charge.
where k is Coulomb's constant, Q1 and Q2 are the charges and r is the distance between them.
So if the charges are at x1, x2, and x3
k and q^2 are constants in this expression so we just need to minimise the rest by suitable choice of x1, x2, and x3. No idea at the moment but I'll work on it. 
EDIT: seems best to keep the two larger charges as far apart as possible, ie. x2 = 0 and x3 = 9, and then position x1 to minimise:
My computer is playing up so I'm going to leave it there. use calculus.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#3 2015-06-14 11:04:22
Re: Minimise the potential energy
Hi bob;
Thanks!
My main problem is: What determines which charge goes in the middle? What if there are more charges?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#4 2015-06-14 20:14:12
- Bob
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- Posts: 10,163
Re: Minimise the potential energy
hi Agnishom,
I'm not definitely claiming that's the answer. But, as no one else came in with a suggestion I thought I'd have a go. I've never done a problem like it, so I was making it up as I went along.
I put the two larger charges as far apart as possible so that their contribution is minimised. Then it's just a case of finding the best position for the lowest charge. By trial, it's roughly x1 = 3.
With more charges I'm not sure. There probably is a method to get the optimum positions but I don't know what it would be. I'll keep thinking ... maybe something will occur to me.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#5 2015-06-14 20:44:05
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Minimise the potential energy
Hi;
But, as no one else came in with a suggestion I thought I'd have a go.
True, but that does not mean that this problem was not looked at and solved in at least 3 different ways by EM. All of solutions produced vindicated your idea of separating the two largest and placing the smallest.
To produce an analytical solution requires that your equation,
be modified slightly and then the method of Lagrangian multipliers used. Obviously, the problem will get tougher as more charges are added, and someone is busy attempting to compute the answer for 100 charges.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#6 2015-06-14 21:01:13
- Bob
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- Registered: 2010-06-20
- Posts: 10,163
Re: Minimise the potential energy
Thanks bobbym,
I'll look up Lagrangian multipliers.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#7 2015-06-14 21:09:30
Re: Minimise the potential energy
Hi bob, bobbym;
That is definitely the answer. Thank you
I plotted three graphs of potential energy for the cases where we had 8, 2 and 1 in the middle and chose the one with the lowest minima.
But then, I am just wondering what is a general efficient algorithm for the following problem
INPUT: An array of N real numbered charges (maybe positive or 0 or negative) (q1, q2, q3 ... qN) and a length l.
OUTPUT: A linear arrangement of the charges such that the distance between any two charges is atmost l and the potential energy of the system is the minimum.
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#8 2015-06-15 01:05:15
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Minimise the potential energy
Someone else was wondering too and had she succeeded it almost certainly would have been posted here as a challenge.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#9 2015-06-15 04:12:38
Re: Minimise the potential energy
Challenge?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#10 2015-06-15 04:14:47
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Minimise the potential energy
Yes, a nah nah na nah nah.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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