What is the effect of temperature on electrostatic-gravitational balance?

Question

We have two identical massive metal spheres at the same temperature at rest in free space. Both have an identical charge and the Coulomb force [plus the black-body radiation pressure if the temperature is non-zero] exactly counteracts the gravitational force between them, resulting in no net forces on either object. They are electrostatically levitating at rest in space. It is my understanding that the charge distribution in each sphere exists only at the surface, and should be concentrated on the side facing away from the other sphere.

Now heat one of the spheres uniformly with an external energy source. What happens in the instant following?

Could it be: The increased mass-energy of the hot sphere increases the gravitational force, and the cold sphere starts to fall inward.

Or else: The increased temperature modifies the charge distribution on the hot sphere by giving it more variance and bringing it closer on average to the cold sphere, increasing the Coulomb force. The cold sphere starts to fall away. [Carl's answer says there is no effect on the charge distribution like I describe, but DarrenW points out that there will be an increased black-body radiation pressure between the spheres, similarly causing an outward force.]

Which direction is correct, and what is the best explanation? Does it depend on the amount of temperature change or the initial conditions?

Answer 1

First thing to say is that this system is unstable. Though the simple gravitational and electrostatic forces may be balanced, there is an additional effect for the electrical interaction that won't exist for gravity: the charge of each sphere polarizes the other. The charge distributions on each sphere adjust themselves for minimal energy, and that leads to a feeble but non 1/r^2 attractive force. If anything at all brings the spheres closer together even the tiniest bit, with the plain vanilla Coulomb repulsive force effectively nulled by the attractive gravity (at any distance as both are 1/r^2), this polarizing effect will lead to attraction. We could try to avoid that by tweaking the charges up for a wee little more repulsion, just enough for balance. But the system is unstable, like a pencil balanced on its point, since we're balancing forces that follow different power-of-r laws.

Having a nonzero temperature implies random fluctuations. Even at zero temperature, QM says there will be fluctuations. The net effect of fluctuations over all location on one sphere will lead to random fluctuations in the electric field sensed by the other sphere, and not long you will have to wait for a random extra bit of attraction to start pulling the spheres together. Will random attractions be cancelled by an equal number of random repulsions? Short answer: no. This is like the Van der Waals force between molecules.

Making one sphere hotter will add to its time component of energy-momentum, and increase the size and vigor of those random fluctuations. That would increase the rate at which the random fluctuations and induced polarization do their work. That, and increased gravity, both increase attraction.

OTOH, a hotter sphere would emit more thermal radiation - which would push the other sphere away. How much hotter? The Stefan-Boltzman law says we can get a lot of bang for the buck, so to speak. Without a particular physical system to discuss, I imagine that a big increase in temperature would lead to overall repulsion, the thermal radiation winning out over the other effects. But a small temperature increase? That might take calculation with particular numbers.

The question asks about metal spheres. So What if the spheres were made of an insulator like ceramic or rubber?

(I probably stupidly left off some additional physical effects. Have at it, commenters!)

Answer 2

Heat has no effect on E&M until you get hot enough that the spheres are modified in shape, or hot enough that the charges are able to leave the sphere. So before these sorts of things happen, yes, the increased gravitational attraction will increase the attraction.

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In any conceivable practical implementation, these two effects will occur long, long before the gravitational force is changed sufficient to take them significantly out of balance.

Practical example: 1000kg spheres, 0.5m in radius, nearly touching. The gravitational attraction is $G(1000)^2/1m^2 = 6.67428 \times 10^{-5}$ Newtons, say at 300K.

To compute the energy change of 1 degree Kelvin, we need to know the substance. Let's use iron so that an increase of 1 degree Kelvin (at 25 °C) gives $25.10$ Joules per mole. At 55.845 grams per mole, 1000kg has 17900 moles so each degree Kelvin increases the energy by about 45,000 Joules. Using $E=mc^2$ so $E/c^2=m$ this gives a mass increase of $5.0\times 10^{-12}$ kg, or about $5\times 10^{-15}$ change in mass. Assuming that both masses are heated up, the change in force is twice this, or $1.0\times 10^{-14}$ per degree Kelvin.

Current measurements of gravity using spheres gives the gravitational constant to an accuracy of 4 or 5 digits, for example, $6.693(27)(21)\times 10^{-11}$ in a recent measurement. In order to get an error in the smallest (6th) digit of 6.67428 you need to have a temperature change of about $10^8$ degrees Kelvin. At that temperature, you will have to include fusion (and plasma) effects.