Finding minimum electrostatic energy

[x00m_x00m]

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?

 

[kontejnjer]

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.

 

[x00m_x00m]

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 ?

 

[haruspex]

yes i know that already but failed to find the correct ratio , what would it be ?

Pls post your working.

 

[rude man]

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.

 

[anshul_agrawal]

yes i know that already but failed to find the correct ratio , what would it be ?

It comes out this way I tried myself