Electric Field Paradox: Unraveling the Gravity vs. Repulsion Debate in Physics

[kamil9876]

Ok for the record I havn't studied physics in a year, I'm a math student. But I was reminded of a paradox I came up with once in physics:

Suppose we place a positively charged ball above an infinite plane with positive charge. We know by Gauss law that the electric force on the ball ends up being constant and independent of the height. Suppose that the gravitation force pulling it down is stronger than the repulsion force. Hence the ball falls with constant acceleration all the way.

Now if I was to remove every point in this plane except for the one just below the ball, then clearly the repulsion force is weaker and hence the ball should continue moving. But using the q1q2/r^2 formula we see that the force should eventually become stronger than gravity and so it shouldn't fall down all the way.

?

Back then I resolved this by saying that, "oh a point has zero area hence the charge is 0 hence it falls all the way(if it had some non-zero charge then, assuming uniformity, any subset of the plane with non-zero area would have infinite charge". But still, what if it was just an proton or something like that?

 

[The_Duck]

If you really want to think of the plane as composed of, say, protons then when you get close to the plane you have to abandon this notion of a continuous charge density, and instead model it as a bunch of point charges. When your distance from the plane is comparable to the distance between the protons that make up the plane, the electric field is going to be a lot more complicated than the uniform field you see at large distances. For instance, as you say, if you try to get close to one of the protons the electric field [~1/r^2] blows up and pushes you away.

 

[Phrak]

I believe you're getting this paradox because q1q2/r^2 is an approximation. In your scenario, two point charges are not being brought to contact. The ball, I presume has finite charge, and the plane has finite charge per unit area.

 

[Delta2]

You are gettign the paradox because you are making some "hidden" assumptions which you change them as you change the plane charge to point charge.

You start with an infinite plane that has finite charge density (so from Gauss Law and symmetry u get that E=constant). This means indeed that if you remove all the other points except the point below, the charge left will be=finite charge density x area=finite charge density x 0=0 hence no charge and no electric field left.

But then if you change the assumption and make it to be a point particle (though protons arent elementary point particles) then you consider it to have infinite charge density=finite charge of proton/area occupied by proton=finite/0=infinite.

So you change it from finite to infinite charge density of the plane below that's why u get the paradox.

 

[Phrak]

Never use the word "indeed".

 

[Delta2]

Never use the word "indeed".

If i am wrong correct me but where is the wrong in : if there is finite charge density on a plane then any point of the plane will have zero charge?

 

[Phrak]

If i am wrong correct me but where is the wrong in : if there is finite charge density on a plane then any point of the plane will have zero charge?

No, sorry. It was just an off-topic remark--psychology in this case. By experience, those who use this superfluous word, 'indeed' are either attempting to persuade their audience of a superior intellect they do not possess, or have adopted its use from those that do. I'm sure you are the latter. It's an alarm word-- a beware word. You may be getting conned.

As to your statement,

"You start with an infinite plane that has finite charge density (so from Gauss Law and symmetry u get that E=constant). This means indeed that if you remove all the other points except the point below, the charge left will be=finite charge density x area=finite charge density x 0=0 hence no charge and no electric field left,"

you must mean finite charge per unit area, as the charge density can be infinite and have an associated finite and constant E perpendicular to the plane. If we're not precise we'll confuse each other about these things.

But overall, I'm not sure the paradox completely disappears if we choose things like, say, a finite point charge over a plane of infinite planar charge density and finite volumetric charge density.

 

[kamil9876]

Ok so yeah the explanations of the previous posters is pretty much what I wrote in my initial post.

As for the word "indeed", a good way to use it that I like is: "[Insert statement of Theorem here]. Indeed [Insert proof here]". Indeed it is good :P

 

[Delta2]

As for the word "indeed", a good way to use it that I like is: "[Insert statement of Theorem here]. Indeed [Insert proof here]". Indeed it is good :P

Indeed :).

Yes well the other case we didnt examine is to consider the point a small sphere of radius r1. Then if we hold this assumption and go back to the field above the plane (which is made of small spheres now) the symmetry is breaking as we get closer and closer to the plane (in distances comparable to r1) so E will not be constant.

 

[stevenb]

By experience, those who use this superfluous word, 'indeed' are either attempting to persuade their audience of a superior intellect they do not possess, or have adopted its use from those that do.

You've overlooked the case where the person really is intellectually superior to the fool he's talking too. In such cases, the word is appropriate.

Clearly, that's not a case in this thread, but we see some crazy people come to PF and assert conceptual nonsense, and then defend it vehemently even when given 10 valid arguments why they are wrong. Those are cases where the person should be "indeeded" to death. After all, they are just stupid enough to fall for it, and sometimes it works when all else has failed.

 

[Delta2]

Damn i am exposed, my indeed trick didnt convince you of my intellectual superiority. These are hard times , people are at least as smart as me... Perhaps i am the fool here... NOOOOOOOOOOOOOOO (LOL).

 

[stevenb]

Damn i am exposed, my indeed trick didnt convince you of my intellectual superiority.

At first it did work, indeed.

Actually, you do appear to be good at slipping the word "indeed" into the sentence to give almost a subliminal effect. Unfortunately for you, Phrak seems particilarly tuned to detect the word. I myself didn't notice it on the first read, and was puzzled by my strange feeling of inferiority. Even after Phraks admonition, I struggled to see the word in your statement when reviewing it again, and almost slipped into suicidal depression with feelings of worthlessness. Thanks to Phrak's early warning that resulted in my seeing the issue, I'm ok now.

It's good to see two masters at work.

 

[Delta2]

I could never imagine when i used the word indeed (iirc i used it as synonym to "in agreement with what you(kamil9876) already have said") that his thread would go like this. I was thinking that it would go to questions like "what is proton" "what is the charge distribution of a proton" "are there really point charges in reality". But this is life, no mater what ones does, he/she gets surprises big or small ones.

 

[stevenb]

But this is life, no mater what ones does, he/she gets surprises big or small ones.

Indeed !

 

[Delta2]

I wonder if this (surprises come as result of uncertainty, big or small) relates somehow to the uncertainty principle and to the wave function concept. If the uncertainty we see in the macroscopic world as social/economic/political uncertainty relates somehow to the uncertainty and the probability waves of the microscopic world.

 

[Dickfore]

Now if I was to remove every point in this plane except for the one just below the ball, then clearly the repulsion force is weaker

Please justify this statement.

 

[kamil9876]

Please justify this statement.

Well in both cases the repulsion force is vertically upwards. (can be justified by symmetry in the initial case). Since all the points are below the ball, the vertical component of the repulsion force contributed by each individual point is upwards. Less such points means less force.

 

[Dickfore]

you do realize that the small piece of the plane cannot hold the same charge as the whole plane itself, right? And that, no matter how small this piece is, once the distance between the point charge and the piece becomes comparable to the linear dimensions of this piece, the point charge approximation, and, thus Coulomb's Law, fails.

 

[kamil9876]

Yes that is an explanation of a variation of my paradox:if you remove everything from the plane except for some "small piece". However I asked what happens when that small piece is just a single point, hence coloumb's law applies everywhere in that case (and that paradox has a different solution as discussed previously).

 

[Dickfore]

Yes that is an explanation of a variation of my paradox:if you remove everything from the plane except for some "small piece". However I asked what happens when that small piece is just a single point, hence coloumb's law applies everywhere in that case (and that paradox has a different solution as discussed previously).

What is the charge on that point? The charge density is a finite continuous function and the area of the point tends to zero.

 

[kamil9876]

Yes precisely, which is what we all agreed on earlier.

 

[Dickfore]

Yes, you even have it in the op. The short answer would be that Classical Electrodynamics works on distances that are much larger than the interatomic distances and the fields we consider are a spatial average over volumes that contain a huge number of atoms.

EDIT:

As a mathematician, you may be surprised to encounter a term such as 'physically infinitesimal quantity'.