Article status: Draft
Time Estimate for Reading: 30 min
Learning Objectives: Concept of "Field"
Effort Required: Medium
Pedagogy Model: Evolution, Formula Analysis, Inter-disciplinary
Prior Physics Concepts: displacement, velocity, acceleration, mass, power, force
Prior Math Tools: Secondary school level Arithmetic, geometry and algebra
This article helps in understanding the concept of "field" as used in physics.
Who said only mathematics can have constants like pi and e? Physics can also have constants. The one dimensional gravitational constant G, the first of the physical constants (1640 AD). What seems continuous to our senses seems to have a discrete value. This paves the way for further exploration and leads to Avogadro's constant (a fixed number of atoms or molecules matter can contain), Boltzmann constant (relating energy of particle and temperature), velocity of light (maximum limit for electromagnetic radiation) and the all important Planck's constant (the minimum quantity of energy). There are more...
Being the first physical constant, it gets close to heart. So, let us explore it a little.
In the process of understanding how Newton would have arrived at this formula involving gravitational constant 'G' a couple of questions comes to the mind.
1. Why product of the masses. it could have been an addition as well?
2. Why square of distance in the denominator?
3. How did newton eliminate other parameters like magnetism, electricity, atmospheric pressure, friction and air drag from appearing in this formula?
Newton must be right. so, we take this for granted as it is a law and move on to static electricity. A similar formula pops up.
Force due to gravity and force due to electric charges. just replacing mass with charge and G with 1/Eo. But there is a 4*Pi downstairs.
'Eo', (the number of field lines or permittivity) of electric charges depends on the medium.
Compare this with G. G is a constant through out the universe. It seems, gravity can to travel through everything. Whether we hide inside a metal cage (faraday's cage), enter into a vacuum chamber, heat it, change the color. Gravity is all prevailing. Initially it was thought that light is made of photons, photons do not have mass and cannot be attracted. But with Einsteins prediction of space-time curvature, light bends around massive objects like sun. Philosophically speaking we may even say guilt and grudge pulls us down and may be gravity is the reason.
Having discussed about Eo, we need to understand the 4*pi*r^2 downstairs. For that we need to introduce and understand the concept called field. Let us start with equation for the electric field,
Getting to understand the concept of 'Field' was not so easy. Isaac Asimov, the famous fiction author, came to my help. I learned to see through the formulas from his books on science.
Seeing through the formula, these are the observations. A 'q' has been removed from the right hand side of the equation. 'F' has been replaced by 'E'. So, mathematically 'q' should have gone under 'F'. Scientists and physicists naturally are tempted to give a name for concepts involving more than one variable. so. F/Q is given a name "Field". it is similar to naming 'mv' as momentum and 'ma' as force.
Why should we remove the Q in the first instance? It becomes simple from the mathematical perspective. Instead of 2 variables, we just have 1 variable and is easy to solve. More importantly, it becomes very easy when we consider the effects of multiple objects (more of this later).
It is ok for mathematics. But, how do we intuitively understand the concept of 'Field'. The clue is in 4*Pi*r^2 downstairs. This incidentally is the surface area of a sphere. let us imagine that, we place a charge in the center of a sphere. what happens when a charge is divided by surface area. we shall take help of an analogy.
Sphere is three dimensional. but let us begin with 2 dimensions and later extend it to three dimensions. Consider a pizza placed in the center of table of radius 'r'. let us imagine people of unit dimensions (1) occupying the circumference of the table. The single pizza will be divided equally among all the persons. Keeping the pizza size same, if we increasing the radius of the table, the same pizza will be shared by more people (circumference 2*pi*r increases with r). Lets us extend this to a spherical table and assume that people (imagine house flies for a better scale) can sit around such a spherical table. Each one will get a very little share. This is the idea of a field.
So, a force field can be imagined to get shared or rather get diluted as we move away from the center. it can be calculated by dividing the force (considered to be concentrated at the center) by surface area of the sphere enclosed.
Now we know where Gauss's Law comes from. The flux is the integral product of 'field and surface area'. Interesting.
We have added field and flux to our vocabulary. we just have an intuitive understanding. A mathematical treatment of the same should await a treatment with calculus.
Having understood, the reason for 4*Pi*r^2 being in the downstairs of the formula for electrostatic force, we may apply the same concept to the formula for gravitational force. We need to multiply and divide the equation by 4Pi
If you have followed our discussion on the 3-D 'G' and 1-D 'G', we can intuitively understand the need for h, h-bar and h-bar/2.
h is a three dimensional constant used in wave equations (3-dimensional;). Dividing h-bar by 2 will be h/4*pi. this incidentally can be visualized as a 1-D Planck's constant. Since sigma x, the standard deviation of position (1-D) and sigma p, the standard deviation of momentum (1-D), h/4*pi can be considered to be a 1-D Planck's constant. The r^2 term disappears as we consider unit radius.
You may refer to this article for more information on self-organizing systems. Science gets amazing.
This article might seem to be more like a random walk, transgressing various fields and seemingly unrelated areas. At the end of it, it is nice to know that, Newton was dealing with one dimensional motion or one dimensional forces (though circular and elliptic orbits are 2 dimensional, the force that makes it happen is one dimensional) and he should have been happy with one dimensional constant.
Note to readers:
If you come across any formulas that involves 'G' please do send me a note.
References:
http://mathworld.wolfram.com/FeigenbaumConstant.html
https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
https://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity
https://en.wikipedia.org/wiki/Planck_constant
https://en.wikipedia.org/wiki/Self-organization
Time Estimate for Reading: 30 min
Learning Objectives: Concept of "Field"
Effort Required: Medium
Pedagogy Model: Evolution, Formula Analysis, Inter-disciplinary
Prior Physics Concepts: displacement, velocity, acceleration, mass, power, force
Prior Math Tools: Secondary school level Arithmetic, geometry and algebra
This article helps in understanding the concept of "field" as used in physics.
Who said only mathematics can have constants like pi and e? Physics can also have constants. The one dimensional gravitational constant G, the first of the physical constants (1640 AD). What seems continuous to our senses seems to have a discrete value. This paves the way for further exploration and leads to Avogadro's constant (a fixed number of atoms or molecules matter can contain), Boltzmann constant (relating energy of particle and temperature), velocity of light (maximum limit for electromagnetic radiation) and the all important Planck's constant (the minimum quantity of energy). There are more...
Being the first physical constant, it gets close to heart. So, let us explore it a little.
1. Why product of the masses. it could have been an addition as well?
2. Why square of distance in the denominator?
3. How did newton eliminate other parameters like magnetism, electricity, atmospheric pressure, friction and air drag from appearing in this formula?
Newton must be right. so, we take this for granted as it is a law and move on to static electricity. A similar formula pops up.
Force due to gravity and force due to electric charges. just replacing mass with charge and G with 1/Eo. But there is a 4*Pi downstairs.
'Eo', (the number of field lines or permittivity) of electric charges depends on the medium.
Compare this with G. G is a constant through out the universe. It seems, gravity can to travel through everything. Whether we hide inside a metal cage (faraday's cage), enter into a vacuum chamber, heat it, change the color. Gravity is all prevailing. Initially it was thought that light is made of photons, photons do not have mass and cannot be attracted. But with Einsteins prediction of space-time curvature, light bends around massive objects like sun. Philosophically speaking we may even say guilt and grudge pulls us down and may be gravity is the reason.
Having discussed about Eo, we need to understand the 4*pi*r^2 downstairs. For that we need to introduce and understand the concept called field. Let us start with equation for the electric field,
Seeing through the formula, these are the observations. A 'q' has been removed from the right hand side of the equation. 'F' has been replaced by 'E'. So, mathematically 'q' should have gone under 'F'. Scientists and physicists naturally are tempted to give a name for concepts involving more than one variable. so. F/Q is given a name "Field". it is similar to naming 'mv' as momentum and 'ma' as force.
Why should we remove the Q in the first instance? It becomes simple from the mathematical perspective. Instead of 2 variables, we just have 1 variable and is easy to solve. More importantly, it becomes very easy when we consider the effects of multiple objects (more of this later).
It is ok for mathematics. But, how do we intuitively understand the concept of 'Field'. The clue is in 4*Pi*r^2 downstairs. This incidentally is the surface area of a sphere. let us imagine that, we place a charge in the center of a sphere. what happens when a charge is divided by surface area. we shall take help of an analogy.
Sphere is three dimensional. but let us begin with 2 dimensions and later extend it to three dimensions. Consider a pizza placed in the center of table of radius 'r'. let us imagine people of unit dimensions (1) occupying the circumference of the table. The single pizza will be divided equally among all the persons. Keeping the pizza size same, if we increasing the radius of the table, the same pizza will be shared by more people (circumference 2*pi*r increases with r). Lets us extend this to a spherical table and assume that people (imagine house flies for a better scale) can sit around such a spherical table. Each one will get a very little share. This is the idea of a field.
So, a force field can be imagined to get shared or rather get diluted as we move away from the center. it can be calculated by dividing the force (considered to be concentrated at the center) by surface area of the sphere enclosed.
Now we know where Gauss's Law comes from. The flux is the integral product of 'field and surface area'. Interesting.
We have added field and flux to our vocabulary. we just have an intuitive understanding. A mathematical treatment of the same should await a treatment with calculus.
Having understood, the reason for 4*Pi*r^2 being in the downstairs of the formula for electrostatic force, we may apply the same concept to the formula for gravitational force. We need to multiply and divide the equation by 4Pi
The gravitational constant now will have a value of 8.385E-10 instead of 6.674E-11. With all these, we have converted the 1 dimensional gravitational constant into a three dimensional Gravitational constant.
Why do we need a three dimensional gravitational constant?
It makes understanding simple. Let us consider 2 examples.
1. Gauss law of gravity
With 3 dimensional G, we can do away with the 4*pi here.
2. Intuitive understanding of Planck's constant and Heisenberg's uncertainty principle.
Planck's Constant,
E = hf ; E is the packet of energy, h the Planck's Constant and f the frequency of radiation.
Heisenberg's uncertainty principle
Notice a strike over the h on the right hand side of the equation. it is called h-bar
h is a three dimensional constant used in wave equations (3-dimensional;). Dividing h-bar by 2 will be h/4*pi. this incidentally can be visualized as a 1-D Planck's constant. Since sigma x, the standard deviation of position (1-D) and sigma p, the standard deviation of momentum (1-D), h/4*pi can be considered to be a 1-D Planck's constant. The r^2 term disappears as we consider unit radius.
On a similar note, dividing h by 2*Pi makes h a 2-D Planck's constant and is used in angular momentum calculations.
Would there be any implications on Eintein's General Theory of Relativity?
If 'G' in that equation is multiplied by 4*Pi, the 8*pi in the numerator of the equation gets replaced by '2'. Does it lead to some other interpretations?
There seems to be order in Chaos too. The Feigenbaum constant for bifurcation rate in chaos (and also to fractals). What seems to be chaos at first, may lead to order in the long run and result into self organizing systems.
You may refer to this article for more information on self-organizing systems. Science gets amazing.
This article might seem to be more like a random walk, transgressing various fields and seemingly unrelated areas. At the end of it, it is nice to know that, Newton was dealing with one dimensional motion or one dimensional forces (though circular and elliptic orbits are 2 dimensional, the force that makes it happen is one dimensional) and he should have been happy with one dimensional constant.
Note to readers:
If you come across any formulas that involves 'G' please do send me a note.
References:
http://mathworld.wolfram.com/FeigenbaumConstant.html
https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
https://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity
https://en.wikipedia.org/wiki/Planck_constant
https://en.wikipedia.org/wiki/Self-organization
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