Finding minimum electrostatic energy

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Homework Help Overview

The discussion revolves around determining the ratio of charges on two small metal balls for minimizing electrostatic energy when they are placed at a large distance apart. The context involves concepts from electrostatics, particularly focusing on self-energy and the implications of charge distribution.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the assumption that mutual electrostatic energy is negligible due to the distance between the spheres. There is a focus on minimizing the total energy, which is expressed as the sum of self-energies. Questions arise regarding the correct ratio of charges and how to derive it under the constraint that the total charge remains constant.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some have provided insights into the relationship between the charges and the total energy, while others express difficulty in finding the correct ratio, indicating a need for further exploration and clarification.

Contextual Notes

Participants note the constraint that the total charge of the system must remain constant, which is central to the problem. There is an emphasis on the need for a clear understanding of the self-energy of metallic spheres and how it relates to the charge ratio.

x00m_x00m
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Homework Statement


There are 2 small metal balls of radius r1 and r2 and are kept at very large distance , what should be ratio of charges on them for electrostatic energy to be minimum

Homework Equations


E=Q^2/(8πϵR)
Self Energy

The Attempt at a Solution


In this problem, electric potential must be zero since the balls are at a very large distance from each other, so I was considering the self energy of metallic spheres which is E . How to find the charge ratio for minimum electrostatic energy?
 
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Well, if the spheres are far away from one another, it's safe to assume their mutual electrostatic energy will be negligible. The total energy of the system will then simply be the sum of their self energies, you just need to find the minimum of that energy. Note that the total charge of the system must be constant, q_{1}+q_{2}=Q=const., so the expression you obtain for the total energy should be a single variable function (in this case, a function of charge) for which the minimum is easy to find.
 
kontejnjer said:
Well, if the spheres are far away from one another, it's safe to assume their mutual electrostatic energy will be negligible. The total energy of the system will then simply be the sum of their self energies, you just need to find the minimum of that energy. Note that the total charge of the system must be constant, q_{1}+q_{2}=Q=const., so the expression you obtain for the total energy should be a single variable function (in this case, a function of charge) for which the minimum is easy to find.

yes i know that already but failed to find the correct ratio , what would it be ?
 
x00m_x00m said:
yes i know that already but failed to find the correct ratio , what would it be ?
Pls post your working.
 
x00m_x00m said:

Homework Statement


There are 2 small metal balls of radius r1 and r2 and are kept at very large distance , what should be ratio of charges on them for electrostatic energy to be minimum

Homework Equations


E=Q^2/(8πϵR)
Self Energy

The Attempt at a Solution


In this problem, electric potential must be zero since the balls are at a very large distance from each other, so I was considering the self energy of metallic spheres which is E . How to find the charge ratio for minimum electrostatic energy?

Electrostatic energy = work needed to bring q1 from infinity to r away from q2. What is that work?
As someone else pointed out, your constraint is that q1 + q2 = Q = constant.
 
x00m_x00m said:
yes i know that already but failed to find the correct ratio , what would it be ?

It comes out this way I tried myself
 

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