# Buoyancy of a positively charged object inside positively charged fluid

An object with a density of $x$ inside a liquid with a density greater than $x$ would float. If we assume that both of these are positively charged and the object is in the middle of the liquid:

1. Would the liquid touch the object?

2. Would the object still float up because of buoyancy?

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Technically, nothing ever "touches". But I guess it depend on the liquid and the volume of it, and the magnitude of the charge. Have you done any calculations? Or what do you think would be the case? – jerk_dadt Mar 6 '14 at 0:55
Calculations on what? And I don't know, the object can stay on the same place or it can move up? I don't know where I would even need to start to do calculations. – user2715411 Mar 6 '14 at 1:07

This is a interesting question. This is how I reasoned it out.

I think there are three situations here (assuming rotationally invariant):

1) The object is very deep where it cannot overcome the electrostatic force from all the positive charge above it, with only its Buoyant force.

2) The object is close enough to surface to where the buoyant force and the electrostatic repulsion has a net upward force and it accelerates up to the surface.

3) There is a sweet spot where the downward electrostatic repulsion exactly cancels the upward buoyant force.

1) Assuming each particle of the fluid has a net positive charge, then the direction of the electrostatic repulsive force is dependent on the amount of liquid above and below the object. In this case (1) there is so much fluid above it (and hence positive charge) than below it the ball will actually sink!

2) If the ball is close to the surface than there will be more positive charge below it and hence contribute to the net upward force (buoyant and electrostatic repulsion) and thus accelerate (non-linearly) to the surface and perhaps float above the surface (if $mg<F_{electrostatic}$)

3) If there so happens to be enough liquid (and charge) above the ball so that it perfectly cancels the buoyant force it will stay suspended at that level. However, this equilibrium is unstable for any perturbation would cause it gain a net forc down or upward and accelerate.

Remember I assumed that it was perfectly in the center of the pool or glass of bucket.

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Like many things in electrostatics, this class of problem will come out with an unintuitive answer.

Would the liquid touch the object?

The thinking of the question is that the fluid is positively charged, so the fluid is attracted to the object.

For the bulk fluid, this would be true. By that logic, it would be somewhat safe to say that the electric repulsion decreases the pressure around the craft. However, since we're in a gravitational field (and probably Earth), it can't possibly decrease enough to cause cavitation. But even if it were, there's another issue...

This is a fluid. That means that there's mobility of the molecules within it. This fluid could be a conductor, or it could not be. It doesn't matter - the ions will migrate to the edges. The field within the fluid will be zero. This is the strange way that electrostatics works.

But it's even more complicated than that. The movement of free ions (which again, we know are free on some time scale because it's a fluid) will make the field approximately zero even against the added field of the object.

But wait you say, that will require that the fluid arranges dislike charges around the craft, even though the fluid has a surplus of like charges. This is correct. Even if the charged object did not exist, the area in the center of the fluid would be relatively devoid of any charge imbalance. The charges to move to make the fluid more equipotential. This is the lower-energy state.

So if the object is charged, and it is surrounded by dislike charges... that draws the fluid closer to the craft. The fluid will touch the craft. If electrical connection is possible, this will cause the object to lose its charge, and that charge will merge with the rest of the fluid's charge. But the object may be covered in electrical insulation for all I know. If that's the case, it will be squeezed by the fluid close by with dislike charges. But that doesn't change the fact that it pushes against the charges far away on the boundary of the fluid.

I wouldn't venture any guess as to an impact on buoyancy. The ion movement isn't affected by gravity except for in some negligible sense (although our sun's interior is a different story). Buoyancy is caused by the gravitational properties of matter, and I don't see how that should be affected.

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