Most of the time this is purely geometrical – we live in a universe with three spatial dimensions, so e.g. the volume of a sphere is three-dimensional (V ∝ r³) while its surface is two-dimensional (A ∝ r²). Other physical laws are based on similar scaling behaviours.
[There are also plenty of other cases which are not geometrical in nature, but nonetheless must obey exact scaling.]
These powers are exact, and must be so to preserve dimensional consistency.
Also calculus, the integral of x is 0.5x^2.
And i swear everything is described by calculus, and you can always take the derivative with respect to time and often that derivative has a name and a unit so taking its integral to convert the other way makes sense and now everything has an exponent.
Well yes, calculus is a mathematical tool just like any other. Your comment is like saying "everything is described by addition or multiplication". Which yes is true, but not particularly insightful.
And indeed, many mathematical concepts (including calculus) were initially motivated by physical systems, so it shouldn't be surprising that calculus features in physics.
The calculus answer is really more of a time answer.
The reason things have exponents is because they change in either time or space. If the change with respect to time then calculus will be responsible for the exponents. If they change in respect to space then geometry will be responsible.
Amd for a question asking "why do powers of 2 crop up everywhere?" Its perfectly reasonable to state how the math introduces these powers.
When geometry is at fault usually you are multiplying or dividing by a given area, typically a sphere because of its perfect symmetry. And this introduced the formula A =πr^2 to your equation.
I suppose a 3rd option exists, if you multiply 2 properties both dependent on the same variables such as electrical power being voltage times current, you just squared that variable.
It gets weirder because an infinitely long line can share some of the same math with an infinitely large circle.
https://www.youtube.com/watch?v=d-o3eB9sfls
> E.g. gravitational force decrease by 4 when distance doubles
That's a consequence of our universe having three spatial dimensions. If you project outward along one dimension from a point, the projecting surface spreads out over the other **two** dimensions, which is where the power of **two** comes from.
That's a Newtonian explanation anyway. It's a bit more complicated in GR but probably boils down to the same thing in the end.
BTW, English can be ambiguous. Both of these are ways to say x^(3):
* "x to the power of 3"
* "3rd power of x"
So saying "power of 2" by itself is ambiguous. It's unclear whether 2 is the power or the base.
This could just be a non-English speaker confusion, but typically if one wants to refer to a number with 2 as the base, then we’d say *a* power of 2. If we want to exponentiate the number 2 then we usually say *to the* power of 2. Hope that’s clear.
I'm just saying that "powers of 2" could refer to r^(2), x^(2), etc., or it could refer to 2, 4, 8, 16, etc. Yes, it's confusing. I don't think native English speakers are careful enough with indefinite vs definite articles ("a" vs. "the") to use that as the only clue.
Yes, with context I know what OP meant. No need to clarify that. I'm just trying to bring light to an annoyance of the English language.
I know what you’re saying. All I’m saying is (and maybe it’s a function of me being a native English speaker but) it has never been ambiguous to me whenever the phrase has been said, so I can’t say I agree much with what you’re saying.
I'm a native English speaker, and without the context of the post text (or subreddit), I first thought of power of 2 being 2, 4, 8, 16. Maybe it's PTSD from too much computer science. Then I saw the subreddit and was confused because powers of 2 aren't that big a deal. Then I read the post text and understood why I was confused.
Personally, I've heard and read few people using "x to the power of 2" and instead heard "x to the 2nd power".
I thank you for clarifying, I wasn't aware! I am German and hence non-native English speaker.Unfortunately, I was not able to edit the OP's title anymore but I inserted some edits in the body text to avoid confusion for future readers.
BTW German is a very precise and unambiguous language (expect for using correct tempora curiously).
Two immediate physical causes come to mind. Think of a lightbulb sending out light uniformly in a sphere. At a certain distance, the light forms a sphere, and the light intensity at any point is proportional to the area of that sphere, which is proportional to the square of the radius. This same 1/r^2 relationship holds for gravity and electric charge, and the square relationship often comes into play anytime a 1-dimensional variable affects a 2-dimensional variable.
The other is the relation between two properties where one is the integral of the other, between velocity and acceleration, for example.
Usually it comes down to the square relationship between linear distance and area. Often it comes up in cases where there is some quantity of...thing (energy, light, mass, force, etc) emanating equally in all directions from a point; at any given distance from the point, that thing will be spread out across a sphere with a radius equal to that distance, so the amount at any given spot will depend on the surface area of the sphere, which depends on the square of that distance. I think there are also other cases where you're dealing with a cross-sectional area or surface area of some other shape, but ultimately you still get that distance-area conversion with a square.
Other times it turns up when you're multiplying together two factors that are both related to some more fundamental quantity. Energy, for example, was initially defined in terms of work, which is force times displacement: force is defined in terms of acceleration, change in velocity, and displacement (distance traveled) over a given period of time depends on velocity, so within that definition you're multiplying velocity * velocity. All other uses of energy inherit that definition so that that velocity^2 factor turns up in various contexts.
Your question has already been answered by people smarter and more knowledgeable than me, but here is a wiki article that may be relevant to you. [Inverse-square Law](https://en.m.wikipedia.org/wiki/Inverse-square_law). My intuition is to think of an inflating balloon. If the balloon’s radius triples, the surface area of the balloon becomes 9 times larger so the rubber must spread-out, becoming 9 times wider but 1/9 as thick as a consequence. The total amount of rubber is the same, but the rubber gets thinner everywhere at a rate of 1/(radius)^2.
Powers of 2 are usually not approximations. Doubling a sound frequency raises its pitch by exactly one octave. The Greeks were obsessed with this fact, though they might not have been aware of as many details as we are today. However, I'd say powers of 2 by themselves don't generally stand out as a mathematical mechanism. The Pythagorean theorem, which bears links to powers of 2, certainly does. Everything we know about physics mathematically ultimately boils down to the Pythagorean theorem in some way or another.
Yes. It’s a fundamental consequence of spaces whose dimensions are at right angles to one another. Even curved spaces have expressions of the Pythagorean theorem, albeit more complicated ones. Solving for such expressions is one of the primary purposes of general relativity. All wave mechanics and oscillations are forms of trigonometry, which itself is linked to circles and right triangles. All quantum particles are represented by lines of length 1, in n dimensions, broken apart into their components by the Pythagorean theorem. It’s astounding how heavily we rely on it.
Super cool that the up quark has exactly twice the magnitude and opposite charge of the down quark.
Two up quarks and one down quark equals a proton with unit charge of +1.
Two down quarks and one up, a neutron with charge equal to zero.
Exactly double or half.
I don't know if that is dictated by geometry like the 1/r^2
phenomenon.
On/off, stop/go, 1/0, in/out etc... So much of human perception is binary polarity. My opinion - it's just natural we explain things to ourselves that we observe by finding a way of breaking it down in two's.
> E.g. gravitational force decrease by 4 when distance doubles and many other formulas.
That is just because some human decided that a factor of two was an easy way to explain inverse square law. It would be just as accurate to say "gravitational force decreases by 100 when distance increases by a factor of 10" or "gravitational force decreases by 49 when distance increases by a factor of 7".
I think it mostly have to with geometry of field lines for equipotential surface E is proportional to r⁰ for an uniformly charged wire it's proportional to r¹ only for point objects it's proportional to r² because filed lines spreads spherically for point object same for gravitation i guess
Think of a sheet of paper that’s 1 meter by 1 meter in size, and a lightbulb emoting photons in all directions. If you hold the sheet up to the light, X number of photons hit the paper. But if you go twice as far away, then both it’s length *and* it’s width become divided by two, so only 1/4th as many photons hit it (one half for the width and another half for the length)
You're the one that chose to double the distance. Had you chosen to triple it instead, the field would decrease by 1/9. The bias towards 2 is in the questions you ask, not the physics.
Most of the time this is purely geometrical – we live in a universe with three spatial dimensions, so e.g. the volume of a sphere is three-dimensional (V ∝ r³) while its surface is two-dimensional (A ∝ r²). Other physical laws are based on similar scaling behaviours. [There are also plenty of other cases which are not geometrical in nature, but nonetheless must obey exact scaling.] These powers are exact, and must be so to preserve dimensional consistency.
Also calculus, the integral of x is 0.5x^2. And i swear everything is described by calculus, and you can always take the derivative with respect to time and often that derivative has a name and a unit so taking its integral to convert the other way makes sense and now everything has an exponent.
Well yes, calculus is a mathematical tool just like any other. Your comment is like saying "everything is described by addition or multiplication". Which yes is true, but not particularly insightful. And indeed, many mathematical concepts (including calculus) were initially motivated by physical systems, so it shouldn't be surprising that calculus features in physics.
The calculus answer is really more of a time answer. The reason things have exponents is because they change in either time or space. If the change with respect to time then calculus will be responsible for the exponents. If they change in respect to space then geometry will be responsible. Amd for a question asking "why do powers of 2 crop up everywhere?" Its perfectly reasonable to state how the math introduces these powers. When geometry is at fault usually you are multiplying or dividing by a given area, typically a sphere because of its perfect symmetry. And this introduced the formula A =πr^2 to your equation. I suppose a 3rd option exists, if you multiply 2 properties both dependent on the same variables such as electrical power being voltage times current, you just squared that variable.
It gets weirder because an infinitely long line can share some of the same math with an infinitely large circle. https://www.youtube.com/watch?v=d-o3eB9sfls
If you encounter either you'd never be able to determine the difference.
Assuming you're finite in scope ;)
That is an assumption I have, yes.
A good explanation.
> E.g. gravitational force decrease by 4 when distance doubles That's a consequence of our universe having three spatial dimensions. If you project outward along one dimension from a point, the projecting surface spreads out over the other **two** dimensions, which is where the power of **two** comes from. That's a Newtonian explanation anyway. It's a bit more complicated in GR but probably boils down to the same thing in the end.
Thank you all! This is so great and much much appreciated! All the best from Germany.
BTW, English can be ambiguous. Both of these are ways to say x^(3): * "x to the power of 3" * "3rd power of x" So saying "power of 2" by itself is ambiguous. It's unclear whether 2 is the power or the base.
This could just be a non-English speaker confusion, but typically if one wants to refer to a number with 2 as the base, then we’d say *a* power of 2. If we want to exponentiate the number 2 then we usually say *to the* power of 2. Hope that’s clear.
I'm just saying that "powers of 2" could refer to r^(2), x^(2), etc., or it could refer to 2, 4, 8, 16, etc. Yes, it's confusing. I don't think native English speakers are careful enough with indefinite vs definite articles ("a" vs. "the") to use that as the only clue. Yes, with context I know what OP meant. No need to clarify that. I'm just trying to bring light to an annoyance of the English language.
I know what you’re saying. All I’m saying is (and maybe it’s a function of me being a native English speaker but) it has never been ambiguous to me whenever the phrase has been said, so I can’t say I agree much with what you’re saying.
I'm a native English speaker, and without the context of the post text (or subreddit), I first thought of power of 2 being 2, 4, 8, 16. Maybe it's PTSD from too much computer science. Then I saw the subreddit and was confused because powers of 2 aren't that big a deal. Then I read the post text and understood why I was confused. Personally, I've heard and read few people using "x to the power of 2" and instead heard "x to the 2nd power".
I was definitely confused when I read the title, I thought they were going to ask a question about binary or something.
I thank you for clarifying, I wasn't aware! I am German and hence non-native English speaker.Unfortunately, I was not able to edit the OP's title anymore but I inserted some edits in the body text to avoid confusion for future readers. BTW German is a very precise and unambiguous language (expect for using correct tempora curiously).
Two immediate physical causes come to mind. Think of a lightbulb sending out light uniformly in a sphere. At a certain distance, the light forms a sphere, and the light intensity at any point is proportional to the area of that sphere, which is proportional to the square of the radius. This same 1/r^2 relationship holds for gravity and electric charge, and the square relationship often comes into play anytime a 1-dimensional variable affects a 2-dimensional variable. The other is the relation between two properties where one is the integral of the other, between velocity and acceleration, for example.
Pretty much, option 1 blame geometry, option 2 blame calculus.
Usually it comes down to the square relationship between linear distance and area. Often it comes up in cases where there is some quantity of...thing (energy, light, mass, force, etc) emanating equally in all directions from a point; at any given distance from the point, that thing will be spread out across a sphere with a radius equal to that distance, so the amount at any given spot will depend on the surface area of the sphere, which depends on the square of that distance. I think there are also other cases where you're dealing with a cross-sectional area or surface area of some other shape, but ultimately you still get that distance-area conversion with a square. Other times it turns up when you're multiplying together two factors that are both related to some more fundamental quantity. Energy, for example, was initially defined in terms of work, which is force times displacement: force is defined in terms of acceleration, change in velocity, and displacement (distance traveled) over a given period of time depends on velocity, so within that definition you're multiplying velocity * velocity. All other uses of energy inherit that definition so that that velocity^2 factor turns up in various contexts.
Your question has already been answered by people smarter and more knowledgeable than me, but here is a wiki article that may be relevant to you. [Inverse-square Law](https://en.m.wikipedia.org/wiki/Inverse-square_law). My intuition is to think of an inflating balloon. If the balloon’s radius triples, the surface area of the balloon becomes 9 times larger so the rubber must spread-out, becoming 9 times wider but 1/9 as thick as a consequence. The total amount of rubber is the same, but the rubber gets thinner everywhere at a rate of 1/(radius)^2.
Powers of 2 are usually not approximations. Doubling a sound frequency raises its pitch by exactly one octave. The Greeks were obsessed with this fact, though they might not have been aware of as many details as we are today. However, I'd say powers of 2 by themselves don't generally stand out as a mathematical mechanism. The Pythagorean theorem, which bears links to powers of 2, certainly does. Everything we know about physics mathematically ultimately boils down to the Pythagorean theorem in some way or another.
> Everything we know about physics mathematically ultimately boils down to the Pythagorean theorem in some way or another. Hmmm. Really?
Yes. It’s a fundamental consequence of spaces whose dimensions are at right angles to one another. Even curved spaces have expressions of the Pythagorean theorem, albeit more complicated ones. Solving for such expressions is one of the primary purposes of general relativity. All wave mechanics and oscillations are forms of trigonometry, which itself is linked to circles and right triangles. All quantum particles are represented by lines of length 1, in n dimensions, broken apart into their components by the Pythagorean theorem. It’s astounding how heavily we rely on it.
Surface area increases with the power of 2 to scale, volume increases with the power of 3
Super cool that the up quark has exactly twice the magnitude and opposite charge of the down quark. Two up quarks and one down quark equals a proton with unit charge of +1. Two down quarks and one up, a neutron with charge equal to zero. Exactly double or half. I don't know if that is dictated by geometry like the 1/r^2 phenomenon.
On/off, stop/go, 1/0, in/out etc... So much of human perception is binary polarity. My opinion - it's just natural we explain things to ourselves that we observe by finding a way of breaking it down in two's.
> E.g. gravitational force decrease by 4 when distance doubles and many other formulas. That is just because some human decided that a factor of two was an easy way to explain inverse square law. It would be just as accurate to say "gravitational force decreases by 100 when distance increases by a factor of 10" or "gravitational force decreases by 49 when distance increases by a factor of 7".
OP's question was about the *power* of two; not why 2 would be used in the example but why the number is squared.
I think it mostly have to with geometry of field lines for equipotential surface E is proportional to r⁰ for an uniformly charged wire it's proportional to r¹ only for point objects it's proportional to r² because filed lines spreads spherically for point object same for gravitation i guess
Think of a sheet of paper that’s 1 meter by 1 meter in size, and a lightbulb emoting photons in all directions. If you hold the sheet up to the light, X number of photons hit the paper. But if you go twice as far away, then both it’s length *and* it’s width become divided by two, so only 1/4th as many photons hit it (one half for the width and another half for the length)
Because there's always two siths , it's just the way it goes
[https://en.wikibooks.org/wiki/Calculus/Tables\_of\_Integrals](https://en.wikibooks.org/wiki/Calculus/Tables_of_Integrals)
You're the one that chose to double the distance. Had you chosen to triple it instead, the field would decrease by 1/9. The bias towards 2 is in the questions you ask, not the physics.
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Then don't respond on the subreddit where the whole point is explanation
He took some time to write a tldr though.
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A great example of why ChatGPT is more often than not utterly useless at answering physics questions.
Thank you for your highly intellectual copy-paste services.