Inverse Square Law

[michel123456]

In the theory of electricity and magnetism, which follows from Coulomb's inverse square law, the magnetic field and force arises from moving charge due to considerations of special relativity and different moving frames of reference.

 

Therefore, since Newton's and Coulomb's laws are identical in form (both inverse square), with the exception of the constants involved, and the fact that Coulomb's law allows for charge and force to have the same or opposite signs, why is there NO analog to magnetism in the theory of gravity? Why doesn't a moving mass (or "mass current") give rise to another velocity-dependent force on a nearby test-mass, as a moving charge does to a nearby test-charge? Why do Maxwell's equations have no analog for gravity, when the fundamental laws - Coulomb's Law and Newton's law - are identical in form?

Very interesting question.

 

At my knowledge there is no clear answer.

 

Some speculate that gravity relates to electricity but the concept has been debunked many times.

 

My own quest is focused on the "square law".

 

Why are we obliged to use the square of the distance in both laws?

The answer is:

in order to make results coincide with observation.

 

So "the-distance-that-we-observe" is only the square root of "the-measure-that-we-have-to-use-in-the-equation".

And with an outstanding, an incredible, an unbelievable characteristic: the unit of the observed distance is squared.

 

So, the "square law" does not say that you multiply a distance by itself and obtain a larger distance, no no.

 

The "square law" says that you multiply the distance by itself and obtain a SURFACE that you DO NOT observe.

 

If the "square law" acts the same for Coulomb's law and for Newton's law, to me it means that the "square law" has something to do with spacetime.

[ajb]

The inverse square law follows from some vector calculus in 3-d. Specifically, if we have a vector field that is the gradient of some function, then the inverse square law corresponds to the divergence being zero outside the sources.

 

You get in general a [math]1/r^{n-1}[/math] law in n-dimensions.

[michel123456]

The inverse square law follows from some vector calculus in 3-d. Specifically, if we have a vector field that is the gradient of some function, then the inverse square law corresponds to the divergence being zero outside the sources.

 

You get in general a [math]1/r^{n-1}[/math] law in n-dimensions.

Our world is 4D.

[ajb]

Our world is 4D.

 

Ok, but we have make a space-time cut into space and time here, so the important thing is that we have 3 space and one time. The same applied in higher dim, I only refer to space here and not space-time.

[michel123456]

 

Ok, but we have make a space-time cut into space and time here, so the important thing is that we have 3 space and one time. The same applied in higher dim, I only refer to space here and not space-time.

I don't get it.

You talked about a vector field in 3D. Without time?

[swansont]

I don't get it.

You talked about a vector field in 3D. Without time?

 

A spatial vector field. The inverse-square equations are time-independent.

[michel123456]

 

A spatial vector field. The inverse-square equations are time-independent.

The inverse square equations are about force. And force means time.

I mean both Newton's and Coulomb's are expressed in Si unit of Newton where 1 N = 1 kg.m/s²

 

Without the s² it does not work.

[swansont]

The inverse square equations are about force. And force means time.

I mean both Newton's and Coulomb's are expressed in Si unit of Newton where 1 N = 1 kg.m/s²

 

Without the s² it does not work.

 

The force itself is not time dependent, it is position dependent. Motion of a particle in the field is time dependent, because position and time are coupled when you talk about motion, but the field itself is static.

[MigL]

Forces propagate or act through space not through time, although a force acting through time would be extremely interesting ( and causality violating ).

 

The 1/r^(n-1) in n spatial dimensions can be easily seen in three dimensions, geometrically, by the area subtended by radial lines on two concentric spheres. Doubling the radius spreads the 'lines of force' over four times the area, effectively reducing the force by 1/r^2.

[michel123456]

Forces propagate or act through space not through time, although a force acting through time would be extremely interesting ( and causality violating ).

 

The 1/r^(n-1) in n spatial dimensions can be easily seen in three dimensions, geometrically, by the area subtended by radial lines on two concentric spheres. Doubling the radius spreads the 'lines of force' over four times the area, effectively reducing the force by 1/r^2.

(emphasis mine)

Yes, but isn't that an explanation based on wishful thinking? Who said that a force is applied on a surface?

A force is applied on masses (or charges). Newton showed that a force is applied as if all the mass was on a single point. Where is the surface?

 

The force itself is not time dependent, it is position dependent. Motion of a particle in the field is time dependent, because position and time are coupled when you talk about motion, but the field itself is static.

That is understandable. But still weird since there is clearly an acceleration involved in the concept of force.

[swansont]

(emphasis mine)

Yes, but isn't that an explanation based on wishful thinking? Who said that a force is applied on a surface?

A force is applied on masses (or charges). Newton showed that a force is applied as if all the mass was on a single point. Where is the surface?

Nobody said the force is applied on a surface. MigL certainly did not.

 

That is understandable. But still weird since there is clearly an acceleration involved in the concept of force.

But force is not acceleration, and more importantly, neither is it velocity nor position. Because acceleration is a derivative WRT time, a time-varying velocity can have a time-independent acceleration. Position can vary in time with no force at all.

[michel123456]

Then what was MigL's explanation?

[swansont]

Lines of (force, or whatever) is a standard way to describe a field, just as one might describe an electric field. You draw a certain number if them a a representation, and the surface density drops (number per unit area) with distance, representing that the magnitude of the field is reduced. The field exists at all points, so the field strength per unit area is a perfectly reasonable description of the situation, and does not imply a force is exerted over an area..

[michel123456]

"The field exists at all points"

O.K.

So, taking only one point of the field, we observe that the magnitude is a function of the distance squared.

 

And if I understand clearly, in laymen terms, the explanation is that the strength of the force is distributed over an area and somehow "waisted".

 

At the other points of the field at the same distance, there exist a "potential-force-of-same-reduced-strength" that does nothing when there is nothing there but anyway takes something from the original strength.

Is that it?

[swansont]

"The field exists at all points"

O.K.

So, taking only one point of the field, we observe that the magnitude is a function of the distance squared.

That would require looking at more than one point, but yes.

 

 

And if I understand clearly, in laymen terms, the explanation is that the strength of the force is distributed over an area and somehow "waisted".

No, the force is not distributed over an area, and I don't understand the reference to "waisted"

 

The field can be viewed in terms of a strength, in terms of field lines per unit area, as a conceptual aid.

 

 

At the other points of the field at the same distance, there exist a "potential-force-of-same-reduced-strength" that does nothing when there is nothing there but anyway takes something from the original strength.

Is that it?

No, nothing takes away from the force by existing somewhere else. The earth's gravitational pull on the moon is not reduced one iota when a meteor or comet passes nearby. The field simply tells you what the strength of the force is at a point, should a mass be present at that point.

[MigL]

Thanks for pinch-hitting swansont. I would have used the example of Faraday's lines of force concept or an analogy to intensity of light.

[swansont]

Yep. There are lots of examples of the concept.

[michel123456]

That would require looking at more than one point, but yes.

 

 

 

No, the force is not distributed over an area, and I don't understand the reference to "waisted"

 

The field can be viewed in terms of a strength, in terms of field lines per unit area, as a conceptual aid.

 

 

 

No, nothing takes away from the force by existing somewhere else. The earth's gravitational pull on the moon is not reduced one iota when a meteor or comet passes nearby. The field simply tells you what the strength of the force is at a point, should a mass be present at that point.

I must express myself badly. We say the same thing.

"The field simply tells you what the strength of the force is at a point, should a mass be present at that point."

Yes.

Doesn't that mean that the force arises when a mass is there, and when there is no mass there is no force (that's what I mean by "waisted"). The field strength is always there though.

That's what I ment when saying: "At the other points of the field at the same distance, there exist a "potential-force-of-same-reduced-strength" that does nothing when there is nothing there but anyway takes something from the original strength."

 

 

When you say :

nothing takes away from the force by existing somewhere else

 

Yes.

Distance takes away from the force.

Anyway, since the force varies with distance, "something" must take something from the force. Otherwise the force would remain constant.

[swansont]

I must express myself badly. We say the same thing.

"The field simply tells you what the strength of the force is at a point, should a mass be present at that point."

Yes.

Doesn't that mean that the force arises when a mass is there, and when there is no mass there is no force (that's what I mean by "waisted"). The field strength is always there though.

That's what I ment when saying: "At the other points of the field at the same distance, there exist a "potential-force-of-same-reduced-strength" that does nothing when there is nothing there but anyway takes something from the original strength."

 

 

When you say :

Yes.

Distance takes away from the force.

Anyway, since the force varies with distance, "something" must take something from the force. Otherwise the force would remain constant.

 

What you say here are correct things. In the previous post, there were things that weren't clear to me (and of it's not clear, I can't be sure it's right).

[michel123456]

I must be the only one to find all this mind blowing.

 

What is in the distance that makes a force to drop? (MigL will say ; 'geometry", I guess)

 

Something that an object cannot do because as you said, adding objects elsewhere do not change the force.

Something more than adding mass to the object (or to the Earth), because mass is not squared in the formula. You can theoretically add mass at will, the force will never become infinite. But if you reduce distance to zero, then the mathematical result is indeed infinite.

 

And why squared?

 

Thanks for pinch-hitting swansont. I would have used the example of Faraday's lines of force concept or an analogy to intensity of light.

Intensity of light is radiation.

Does gravitation act as a radiation?

[imatfaal]

I must be the only one to find all this mind blowing.

Nope - but scientists have to get some work done sometimes, so they have to put the amazement on hold and do the hard graft based on what they believe to be true rather than ponder why it is true (i don't think any of them ignore the wonder but they must put it aside). For amateurs like me, I realised I could really try and get to grips with the science or I could lose myself in a mixture of pop-science and wonder; therefore I try to be rigorous at least some of the time.

What is in the distance that makes a force to drop? (MigL will say ; 'geometry", I guess)

Not all forces do drop - it is thought that the strong force that holds quarks together to make hadrons does not diminish over distance; such that eventually the energy required to pull them apart is less than that to create new particles - thus you never get a lone quark. completely crazy - but the models work! they make valid and repeatable predictions upon which much of our knowledge rests.

 

Something that an object cannot do because as you said, adding objects elsewhere do not change the force.

Something more than adding mass to the object (or to the Earth), because mass is not squared in the formula. You can theoretically add mass at will, the force will never become infinite. But if you reduce distance to zero, then the mathematical result is indeed infinite.

When you get very close to themass of an object you might find that newtonian mechanics breaks down you need gr - but at zero distance it becomes paradoxical; forces act between objects over a distance - if there is no distance and there are two objects in one piece of space can you really have a force at all?

And why squared?

Observationally it works. Not all forces obey inverse square - but gravity does, it fits the experimental data. There are various heuristics about why this is so - but there is no basic " first why" no grundnorm.

 

Intensity of light is radiation.

Does gravitation act as a radiation?

You can work out how much radiation the sun emits in total by studying the solar radiation of one square metre of space a known distance from the sun. to an extent the same thing applies to em interaction and gravity - you can think of these force as messenger particles going off in every direction, or of lines of force spreading out from a source; each of these two analogies will have the density of the lines of force or the intensity of the particles fall off with the inverse square law (surface area of sphere is 4 pi r^2). But in the end these analogies are not real physics - the real physics is in the modelling and experimentation not in the philosophizing of why things really happen.

[swansont]

For any sphere you draw around a light source, like the sun (ignoring its satellites), all of the light passes through it. If you make the radius twice as large, the surface area increases by a factor of 4, so it is an inverse-square relation. Yes, it's a geometric argument.

 

This works for flux lines as well, as long as there are no additional sinks or sources. Any field that emanates from a point source and behaves in a similar fashion is going to also be inverse-square. Photons are the force carriers for the electromagnetic interaction, so Electric fields behave this way. Magnetic fields don't have monopoles, so the "no additional sink or source" rule is violated and it is not inverse-square. Gravitons are modeled as the carriers of gravity in the quantum picture, and they behave similarly to photons.

 

The weak and residual string forces don't, because their force carriers are massive and have a finite lifetime. The argument that the number of particles passing through any surface is constant fails.

[michel123456]

Nope - but scientists have to get some work done sometimes, so they have to put the amazement on hold and do the hard graft based on what they believe to be true rather than ponder why it is true (i don't think any of them ignore the wonder but they must put it aside). For amateurs like me, I realised I could really try and get to grips with the science or I could lose myself in a mixture of pop-science and wonder; therefore I try to be rigorous at least some of the time.

Not all forces do drop - it is thought that the strong force that holds quarks together to make hadrons does not diminish over distance; such that eventually the energy required to pull them apart is less than that to create new particles - thus you never get a lone quark. completely crazy - but the models work! they make valid and repeatable predictions upon which much of our knowledge rests.

 

 

When you get very close to themass of an object you might find that newtonian mechanics breaks down you need gr - but at zero distance it becomes paradoxical; forces act between objects over a distance - if there is no distance and there are two objects in one piece of space can you really have a force at all?

Observationally it works. Not all forces obey inverse square - but gravity does, it fits the experimental data. There are various heuristics about why this is so - but there is no basic " first why" no grundnorm.

 

 

You can work out how much radiation the sun emits in total by studying the solar radiation of one square metre of space a known distance from the sun. to an extent the same thing applies to em interaction and gravity - you can think of these force as messenger particles going off in every direction, or of lines of force spreading out from a source; each of these two analogies will have the density of the lines of force or the intensity of the particles fall off with the inverse square law (surface area of sphere is 4 pi r^2). But in the end these analogies are not real physics - the real physics is in the modelling and experimentation not in the philosophizing of why things really happen.

agree on almost all but not the "completely crazy".

 

Quite the contrary. The strong force acts like a rope.

When you hold your dog with a rope, the distance to your dog (the length of the rope) has no importance. And as much your dog pulls on the rope the more resistance exists. If you replace the rope with a rubber band, you'll see that as much the distance increases as much the force increases too.

So it is not "completely crazy", it is similar to a macroscopic experience on how material objects behave.

 

The "completely crazy" stuff is about force reducing with distance (squared). A situation in which the rubber band becomes more and more liquid as long as it extends.

 

IMHO people are not enough surprised.

[Strange]

The "completely crazy" stuff is about force reducing with distance (squared). A situation in which the rubber band becomes more and more liquid as long as it extends.

 

IMHO people are not enough surprised.

 

I think the reason we are not too surprised by this is that we are familiar with it (qualitatively) from things like the way the intensity of light or heat, or strength of a magnet, falls of with distance. Then, learning about Newton's laws of gravity at a young age. It ends up feeling quite intuitive.

 

For me, at least, it was more of a shock to find that other forces didn't follow the same law.

 

The "rope" analogy for the strong force is limited (well, it is an analogy) in that if you break the rope, your dog runs free. If you provide enough energy to "break" the strong force, the energy is converted into more quarks ... all still bound by the strong force.

[swansont]

The thing that the 1/r² forces have in common is they are action-at-a-distance forces. A rope is a contact force — there is a continuous object connecting you to the dog. The non-contact forces have an exchange particle, and that means you can apply the idea of the density of the exchange particles, or lines of force.