point charge

[Aiezsedai]

What is a point charge? I read that it is hypothetical an it hasn`t got any dimensions, length, etc. The article even mentioned that for that reason its density is infinite. I am really confused. Could you please explain to a beginner? Is it a charge smaller than an electron and what do they use it for?

[swansont]

Coulomb's law assumes point charges — this allows you to do a single calculation for the force between any two charges. Having to worry about the spatial extent of a charge complicates calculations and in most cases the effect is small, which means the effect can often be ignored. In the case of the electron, as far as one can tell when trying to determine if there is a charge distribution, it is a point charge.

[Bignose]

It is primarily done to make the math easier.

 

In real life, there is no such thing as a point-anything. Point charge, point mass, point source of heat, or energy, etc.

 

But, the math is a lot easier when you approximate an object as a point, rather than taking is physical dimensions into consideration.

 

The example I like to use is in launching a probe to Mars, the influence of Pluto's gravity is significant. It isn't much, but if you don't add it in, the probe won't go in the right direction. But, the models can treat Pluto as a point-mass because Pluto is far enough away. The gravitational different between a slightly more dense Northern hemisphere versus a slightly less dense Southern hemispehre is too small to worry about. (I actually don't know if the hemispheres of Pluto have different densities, I just made that up as an example.) So, the mode treats Pluto as a point mass so that Pluto's gravitational influence is included in the model, but the details aren't super important.

 

Whether something can be treated as a point or not depends a lot on the application and the desired accuracy. For example, the ideal gas laws can be derived treating gas molecules as points. And, under the right circumstances (A gas like Argon under high temperatures and pressures), the Ideal Gas Law is very, very accurate. But, when the gas molecule starts getting more complex, and under low temperatures and pressures, the Ideal Gas Law doesn't work as well -- the modifications take into account the "non-point-like" of the molecules. The difference also comes down to how accurate do you need to be. I.e. if you only need a "ballpark" figure, or rough guess, the ideal gas law will usually get you reasonably close. If you need to calculate the exact pressure to make sure that your vessel won't experience pressures that are higher than it was safely designed for, you'd probably want to use the most accurate gas laws you know of for that situation.

 

In that regard, treating an object as a point is a "rough estimate". You will get an answer that is close to correct. And, if all you needed was a "close" estimate, then you are done. If you wanted to be more accurate, then you have to start taking into account the structure of the item.

[big314mp]

I'm just nitpicking (as the post was quite good), but I believe the ideal gas law breaks down under high pressures or low temperatures. In the former, the volume of gas particles becomes significant. In the latter, the attractive forces between the molecules becomes significant.

[swansont]

The example I like to use is in launching a probe to Mars, the influence of Pluto's gravity is significant. It isn't much, but if you don't add it in, the probe won't go in the right direction. But, the models can treat Pluto as a point-mass because Pluto is far enough away.

 

Right answer, AFAIK, but the reasoning is incomplete. Gauss's law tells us we can treat extended masses and charges as points, but there are exceptions: if the distribution isn't uniform, or we aren't outside of the volume. So a mass or charge distribution, as long as it's uniform, behaves just like a point for r>R.

 

When you have a nonuniform distribution is when you have to have to worry about the spatial extent. That's when you make the assessment of size of effect vs distance, because it drops off like r^-3, i.e. faster than the force.

[Bignose]

The "reasoning", swansont, is that Pluto is so far away from Mars, and the mass of the probe is so small, that the difference between using the actual spatial nonuniform distribution of the Mass of Pluto versus just using a point-mass with the total mass of Pluto centered in the mathematical center of Pluto's location is virtually negligible.

 

It isn't due to Gauss' law or anything like that. It is just a matter of distance and small influence.

 

It is a simplification that can be made because the difference between the simplification (treating it as a point source) and real life (the actual non uniform mass distribution) leads to answers that are similar to many significant figures. Now, if the probe was to travel near Pluto, then the simplification won't be as accurate -- it may become so inaccurate as to give bad predictions -- but for launching a probe from Earth to Mars, treating the mass of Pluto as a point source is good enough.

 

None of the planets are perfectly uniform -- there are probably very, very few truly uniform objects in existence anywhere. But, treating them as uniform can often be close enough, and it is done in practice fairly often.

 

Ultimately, the error in treating Pluto as a point source versus the complete treatment of Pluto is probably just in the noise of all the other errors. For example, where the asteroids are is probably of a much larger influence than the difference between the point source simplification and reality in terms of the probe from Earth to Pluto. When differences are small, there is little point in using the more complicated procedure to calculate with, it is just a waste of time and computer resources. If two answers are the same within an acceptable % of error, then you might as well use the point source approximation.

[swansont]

I think you are mixing two effects here, or at least not sufficiently differentiating them. Pluto (or any massive body) can be treated as a point to the extent that it's homogeneous, regardless of distance from the surface. It's the effect of the inhomogeneities that either can or cannot be ignored, depending on the distance.

 

IOW, it can be treated as homogeneous because it's so far away. It can be treated as a point if it is homogeneous. Two separate things.