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

[Buffu]

Homework Statement

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.

Homework Equations

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.

 

[BvU]

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

 

[BvU]

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

 

[Buffu]

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

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

 

[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 ...

 

[Buffu]

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 ...

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

 

[Buffu]

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

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

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.

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.

 

[Buffu]

if the two spheres are made out of conducting metals

They are not conducting.

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

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

 

[BvU]

They are not conducting

Funny metal ?

 

[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 ?

[edit] a metal wire (conducting metal )

 

[Buffu]

Funny metal ?

I guess so because else it would be very difficult.

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 ?

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.

 

[haruspex]

so that the potential of given distribution is as small as possible

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.

They are not conducting.

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.

 

[Buffu]

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.

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.

 

[gneill]

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

 

[BvU]

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

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.

 

[haruspex]

it leads to the same expression as the one in post #1.

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).

 

[BvU]

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).

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.

 

[Buffu]

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.

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

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

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

 

[haruspex]

Should I minimize U(q)

I believe so.

 

[Buffu]

I believe so.

$q = Q\dfrac{R_1}{R_2 + R_1}$ ??

 

[haruspex]

$q = Q\dfrac{R_1}{R_2 + R_1}$ ??

Right, which is different from the equation in post #1, no? Check the signs.

 

[Buffu]

Right, which is different from the equation in post #1, no? Check the signs.

Yes it is different there I took potential at some point due to the spheres' field, here I took potential energy by each sphere.

I get positive sign, should it be negative ,$q = -Q\dfrac{R_1}{R_2 + R_1}$, ?

 

[haruspex]

Yes it is different there I took potential at some point due to the spheres' field, here I took potential energy by each sphere.

I get positive sign, should it be negative ,$q = -Q\dfrac{R_1}{R_2 + R_1}$, ?

No, sorry, I've confused you. In post #22 I thought I was replying to BvU, who seemed to think the equation you should get happens to be identical to the one you had in post #1, even though the r variables mean something else. It is almost the same but there is a sign difference. (BvU, Buffu... Easy mistake... like a sign difference.)

Q and q must have the same sign. Your post #21 is correct.

 

[Buffu]

No, sorry, I've confused you. In post #22 I thought I was replying to BvU, who seemed to think the equation you should get happens to be identical to the one you had in post #1, even though the r variables mean something else. It is almost the same but there is a sign difference. (BvU, Buffu... Easy mistake... like a sign difference.)

Q and q must have the same sign. Your post #21 is correct.

Can you help me with
https://www.physicsforums.com/threa...-point-charge-above-a-plane-conductor.914829/ ? please. Nobody answered it .