Inverse Square Law in 1 dimensional space

[Izhaki]

Sorry for what could be a rather very stupid question. I'm not an expert in physics nor maths.

The inverse square law defines that strength is inversely proportional to the square of the distance from the source. This is justified by working out the surface area of a sphere, which involves a squared radius (distance).

Continuing this line of thought (based on the circumference of a circle), on a 2 dimensional space, I'd assume that we'll have the inverse proportional law. (/r or /d).

My question is, what law governs a 1 dimensional field? It appears to me all sensible that the rule will be the inverse square-root (/sqrt(d)). But, for the life of me, I can't work out why.

Thanks in advance,
Izhaki

 

[K^2]

No. It's a constant. The series is algebraic, not geometrical. So for any N-dimensional space, the law is thus.

$$1/R^{N-1}$$

For N=1, that comes out to just 1. No R-dependence.

 

[Izhaki]

Thanks,

/1 was my second option after /d, but as it implies that in a one dimensional space, a particle has the same force effect on a particle 3 places to its right, and a particle 5 places to its right. Which doesn't make much sense.

I take your answer as the correct one, but was wondering if you can point me to its source / proof? I don't need an explanation, just keywords to google.

Thanks again,
Izhaki

 

[Born2bwire]

You should think more about why the inverse square law comes about.

In three dimensions, let's say I have a isotropic point source. This means that it emits a spherical wavefront. In a lossless medium, the energy density of this wave front on the spherical shell must remain constant. However, the area of the shell grows as the wave propagates out. The surface area of the shell is proportional to r^2. Thus, we would expect that the energy density to drop off by 1/r^2 to keep the energy across the entire surface constant.

In two-dimensions, the point source (or a line source in 3D) now emits circular wavefronts that have a circumference that is proportional to r. Thus the energy must drop off by 1/r.

In one-dimension, the point source simply emits a wavefront that flows along a 1D line. The energy density does not decrease as the wavefront travels because there is no geometrical spreading of the wavefront. Hence, it is constant.

You could also look at the point source solutions for wave equations. In 3D, it's something like
$$\frac{e^{ikr}}{r}$$
In 2D:
$$H_\alpha^{1}(k\rho)$$
In 1D:
$$Acos(kx)+Bsin(kx)$$
The asymptotic behavior of the above is 1/r^2, 1/r and 1 for the power/energy (square of the amplitudes).

 

[Izhaki]

Brilliant,

The word 'wavefront' pretty much explained it all for me. On 3 dimensions the wavefront expands sphere-like, in 2D circle-like, and in 1D it doesn't expand at all (by way of loose visual analogy, it's always same-size two points getting away from the source).

Also, thanks for the wave equations reference.

If I'm taking your explanation a step further, would it be right to say that (strength-like) physical quantities are 'transmitted'. In other words, the Earth's gravity is not really a static force that pulls the moon to earth, but rather a dynamic traveling force (wavefront-like) that pulls the moon to earth? (If I'm not mistaken, gravity travels at the speed of light, which kind of supports this assumption.) By the way, I'm talking of Newtonian gravity, not Einstein.

Sorry for using everyday ambiguous language for what is really mathematical concepts...

 

[Born2bwire]

No, in Newtonian gravity there are no waves and the gravitational force is instantaneous. When we talk about the inverse square law it should be in the context of something like a wave equation. The fact that the gravitational force between bodies is similar in respect to the energy of a wave from a point source is coincidental here. The idea of waves and finite speed of gravitation is only something that comes about in terms of quantum field theory and general relativity.

 

[Izhaki]

OK,

So last (rather historical) question then:

Was the inverse square law an excepted, yet unexplained law of nature until General Relativity and Quantum field theory came about?

In other words, did any previous attempts to put this concept into a coherent proof failed? Newton used it, but could never explain it?

Thanks again for your reply.

 

[K^2]

Actually, even with instantaneous gravity/electric forces you can understand the inverse square law. Basically, it's just Gauss' Law applied to whatever dimensionality you happen to have. In N dimensional space, Gauss Surface will always be N-1 dimensional. And that gives you the correct power on the inverse law.

 

[Izhaki]

Thanks,

I'm not 100% fluent on Gauss' law, but definitely will look into it.

Thanks again,
Izhaki

 

[Izhaki]

Thanks,

I'm not 100% fluent on Gauss' law, but definitely will look into it.

Thanks again,
Izhaki