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January 24th, 2019, 12:18 PM   #1
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Why are things squared in formulas about light, space etc.

As a beginning math student I've wondered: why is the square prominent in math formulas about the natural world, instead of the circle? Circular light beams, elliptical solar systems and galaxies, Inverse Square Law, seem to me better defined in terms of the circle than squares, which don't appear in the natural world. Thanks in advance for explanation.
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January 24th, 2019, 12:38 PM   #2
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the square in inverse square law simply means that the field strength varies as $\dfrac{1}{r^{\Large 2}}$ as opposed to say $\dfrac 1 r$ or $\dfrac{1}{r^{3.5}}$

where $r$ is the distance between the source and observer

It has nothing to do with the polygon that is a square.

In 3D space this does generate spheres of equal field strength so your intuition regarding circles is somewhat correct.
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January 24th, 2019, 01:54 PM   #3
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Originally Posted by romsek View Post
It has nothing to do with the polygon that is a square.
I rather think that it (often) does.



This one is also interesting and slightly related.

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Last edited by skipjack; January 24th, 2019 at 09:20 PM.
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January 24th, 2019, 02:31 PM   #4
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Quote:
Originally Posted by paulm View Post
As a beginning math student I've wondered: why is the square prominent in math formulas about the natural world, instead of the circle? Circular light beams, elliptical solar systems and galaxies, Inverse Square Law, seem to me better defined in terms of the circle than squares, which don't appear in the natural world. Thanks in advance for explanation.
What is a "circular light beam?"

Actually, it all depends on what part of Physics you are looking at. For example, there are crystals that have symmetries of triangles, squares, and hexagons (speaking in only 2D). Magnetic fields, though also of an inverse square nature, have dipolar fields, not spherical ones.

And, of course, the ellipses in orbital motion are derived from an inverse square law, but Newtonian gravity in the Solar System is better approximated by a law that can be modeled as an inverse square plus inverse cubed formula in certain cases.

The inverse square laws, though common, are far from the only ones useful. As always, Nature is full of surprises.

-Dan

Last edited by skipjack; January 24th, 2019 at 09:23 PM.
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January 24th, 2019, 06:22 PM   #5
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Quote:
Originally Posted by romsek View Post
the square in inverse square law simply means that the field strength varies as $\dfrac{1}{r^{\Large 2}}$ as opposed to say $\dfrac 1 r$ or $\dfrac{1}{r^{3.5}}$

where $r$ is the distance between the source and observer

It has nothing to do with the polygon that is a square.

In 3D space this does generate spheres of equal field strength so your intuition regarding circles is somewhat correct.
Thanks, I now see distinction between polygon square and squaring distance (inverse intensity) or increasing velocity (gravity).

Those 3blue1brown videos are inspiring to learn math.
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January 26th, 2019, 09:22 AM   #6
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After understanding that squaring time, distance (gravity) or speed (light) has to do with intensity, not a geometric square, I'm still confused why these things are squared and not cubed or some value * pi. Like E = mc^2 for example: why is speed of light squared and not cubed, or multiplied * e or pi to some power? Same question goes for Inverse Square Law and 9.8 m/sec^2. Why is square all over the place?
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January 26th, 2019, 09:27 AM   #7
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That was covered in the first 3Brown1Blue video above. The effects of gravity and light from a point source are shared over a surface a distance $r$ from the source. This surface has an area which is proportional to $r^2$ (because it's an area) and therefore the intensity of the effect is proportional to $\frac1{r^2}$. This is required so that the total strength of the effect over the area isproportional to $r^2 \cdot \frac1{r^2} = 1$, the strength of the source, not the distance from it.
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January 26th, 2019, 09:48 AM   #8
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Why not the surface area of a circle or some multiple of pi?
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January 26th, 2019, 09:55 AM   #9
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Why not the surface area of a circle or some multiple of pi?
$\pi r^2.$

You are still confusing the meaning of square computationally and spatially.

Notice r squared in the formula for a circle's area.

You are creating a problem in your own mind because a word has more than one meaning and you keep trying to pretend it has one meaning.
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Last edited by JeffM1; January 26th, 2019 at 10:02 AM.
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January 26th, 2019, 10:33 AM   #10
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$\pi r^2.$

You are still confusing the meaning of square computationally and spatially.

Notice r squared in the formula for a circle's area.

You are creating a problem in your own mind because a word has more than one meaning and you keep trying to pretend it has one meaning.
Good point, I'll try to work it out in my head a while. Thanks.
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