# How to distribute the charge between two spheres to minimize the potential?

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1. May 16, 2017

### Buffu

1. The problem statement, all variables and given/known data
We have two metal spheres of radii $R_1$ and $R_2$ placed far apart. Given total amount of of charge $Q$ to distributed between the two spheres, how should you do it so that the potential of given distribution is as small as possible.

2. Relevant equations

3. The attempt at a solution

Let some point $P(x,y,z)$ in the space. Now the distance of $P$ from centre of spheres be $r_1$ and $r_2$.

So, potential at this point would be $\phi = \dfrac{q}{r_1} + \dfrac{Q- q}{r_2} = \dfrac{Q}{r_2} + q\left(\dfrac1{r_1}- \dfrac1{r_2}\right)$

Now how do I minimize this function for $q$ ?, it is a linear function in $q$ it has no minima.

2. May 16, 2017

### BvU

Is 'the potential at some point' the same thing as 'the potential of the given distribution'?

3. May 16, 2017

### BvU

 my mistake. Make an assumption about which of the two r is the biggest and plot V(q). Where is it lowest ?

4. May 16, 2017

### Buffu

I assume $r_1$ to be bigger. but it is still a linear function with not minima, no ?

5. May 16, 2017

### BvU

Did you make the plot ? I agree there is no point where dV/dq=0 but a linear function with nonzero slope over a given domain is not a constant, so ...

6. May 16, 2017

### Buffu

The domain is $(0, Q)$, right ?

7. May 16, 2017

### Buffu

I made a plot with numeric values for $r_1/r_2/Q$.

8. May 16, 2017

### Ray Vickson

You also need to include a "mutual-repulsion" potential that accounts for the forces between the spheres.

Furthermore, if the two spheres are made out of conducting metals, then you need to consider how the charges on one sphere affect the charge distribution on the other sphere. In other words, the expression for potential may be more complicated than what you wrote, even after accounting for mutual repulsion. The problem is not entirely straightforward; its solution is contained in https://arxiv.org/ftp/arxiv/papers/0906/0906.1617.pdf .

However, perhaps the person setting the problem did not realize all that, and would be content with the simpler expression you get after you include mutual repulsion.

9. May 16, 2017

### Buffu

They are not conducting.

I guess we can neglect this because they said that spheres are far apart from one another.

10. May 16, 2017

### BvU

Funny metal ?

11. May 16, 2017

### BvU

Imagine that at first there is a wire between the two spheres. Charge distributes - or not - to minimize potential. Take away the wire and there's your answer ?

 a metal wire (conducting metal )

12. May 16, 2017

### Buffu

I guess so because else it would be very difficult.

Then $\phi (q) = \dfrac{q}{r_1} + \dfrac{Q- q}{r_2} = \dfrac{Q}{r_2} + q\left(\dfrac1{r_1}- \dfrac1{r_2}\right)$ is wrong ?

Even though I understand what you said I don't know how to convert this to "equations" that I can solve.

13. May 16, 2017

### haruspex

Not sure what is being asked. The potential will vary across space, so I suggest it either means the potential difference between the spheres or the potential energy of the system.
Does it really say that? Why would it then say they are metal?
I think the far apart condition can be used to say the distribution on each sphere will be roughly uniform.

14. May 16, 2017

### Buffu

I don't know it just says potential. Now, since you said it, I think it is potential energy :(.

Would it matter if it is conductor or insulator as long as charge is uniformly distributed ?

If it is potential energy then can I use $\dfrac35 \dfrac{Q^2}{R}$ as the potential energy for the sphere that is assembled next or I have to derive a new formula taking the field into account by the sphere that was placed first.

15. May 16, 2017

### Staff: Mentor

I suspect that the problem is looking to minimize the total electric potential energy of two essentially isolated conducting spheres. Investigate the concept of self-capacitance of a conducting sphere and then consider the energy stored in a capacitor given its capacitance and charge.

See, for example, the Hyperphysics entry on the isolated sphere capacitor:
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capsph.html

16. May 17, 2017

### BvU

That was my trigger for post #3 since it leads to the same expression as the one in post #1. But, again: the existence of a lowest value over a domain does not imply the derivative dV/dq is zero.

nevertheless I am struggling with the consequence, as seen in post #11.

I suspect the question is ambiguous, but I can't pinpoint it.

17. May 17, 2017

### haruspex

In post #1 r1 and r2 were defined as distances to some arbitrary point, not the radii of the spheres. And redefining them to be radii gives a slightly different expression (after differentiation).

18. May 17, 2017

### BvU

I don't suppose you mean the $\displaystyle {1\over 4\pi\varepsilon_0}$ ?
However, there is no point in adding the two V from isolated sphere capacitors -- (it gives the same expression, that's all) .
Instead one might add the stored energies (but now I'm robbing Buffu of his exercise IF I am correct AND that is what the exercise composer meant). Capital IF.

19. May 17, 2017

### Buffu

Should I minimize $U(q) = \dfrac{q^2}{R_1} + \dfrac{(Q-q)^2}{R_2}$ ?

Yes can you elaborate how to proceed after finding capcitance between the spheres ? I think I can do that.

20. May 17, 2017

### haruspex

I believe so.