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I very well understand the proportionality relation that was used to derive these laws like $F$ is proportional to product of masses and inversely to radius squared and hence its proportional to the product of these two and = some $k_e$ times product masses times $1/r^2.$ And the same was used in Coulombs Law. But how exactly these $k_e$s are defined is what am not getting. I read somewhere that if the unit of charge is to be defined using Coulomb's the choice of $k_e$ is free to us. What does this mean? Is this related to that fact why we take $k=1$ for $F=k \,ma$ relationship?

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Units of physical quantities are almost always arbitrarily chosen before one formulated such a law. Then one always needs constants in order to well fit the proportionality expressed by the law with the already existing units. In many cases one starts to reformulate units when the law already exist, in order to simplify its expression of the law and its applications. It happened for example with those who took the speed of light c = 1, so the speed-addition became just (u + v) /(1-uv). Or with people who defined the eV (electron-volt) as a more convenient unit than the Joule.

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Coulomb's law of electrostatic force states:

Two charges exert a force on each other that is:

1. Directly proportional to the product of the magnitude of the charges

2. Inversely proportional to the square of the distance between the charges

The $k$ in Coulomb's law is the constant of proportionality,known as Coulomb's constant which is $k=1/4\pi\epsilon_0$ where $\epsilon_0$, is the permittivity of free space. The value of Coulomb's constant is

$$k=8.99\times 10^9 \cdot ~\mathrm{\frac{Nm^2}{C}}=8.99\times 10^9 \cdot ~\mathrm{\frac{F}{m}}\text{. (to 3 significant figures)}$$

Newton's law of universal gravitation states:

For any two bodies, there is an attraction between them that is:

1. Directly proportional to the product of their masses

2. Inversely proportional to the square of the distance between the bodies

The G in the universal gravitational law is the constant of proportionality, known as Newton's gravitational constant. The value of the constant is

$$G= 6.67\times 10^{-11} ~\mathrm{Nm^2 kg^{-2}}\text{. (to 3 significant figures)}$$

I hope that this clears things up. Both $k$ and $G$ are constants of proportionality, but they differ in value, and well, units.

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  • $\begingroup$ +1 A warm welcome to physics SE! :o) Learn how to use MathJax by looking at the changes at your post by clicking at the edit button at the left bottom corner. $\endgroup$ – Stefan Bischof Mar 13 '16 at 20:00
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I bet you know that the k in your question $k=1/4\cdot\pi\epsilon$ where epsilon is the permittivity of medium hence $k$ is related to the polarising power of the medium. It will be more clear if you read HC Verma's book on this topic.

The constant $k$ of Newton's 2nd law is very different from the previous $k$ as it is simply a proportionality constant and its value (1) supports the fact that force is exactly equal to mass multiplied by acceleration.

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Yes your guess right. It is very much similar as taking k=1 in Newton's first law,i.e.,

F=kma

When Coulomb discovered his law there was no unit of charge defined at the time. Coulomb arrived at the law by performing simple experiments. He took a pair of turbo electric materials( materials that charge up while rubbing) and charges them by rubbing them. So both the bodies were equally charged but with opposite polarity (one positively and the negatively). He then calculated the force acting between them, when seperated by a certain distance, with the help of a torsion balance, a device which measures very feeble forces.

He halved the charges on the bodies by making them touch any other conductor so that charge shot distributed equally on both the bodies and thus got halved, then he again measured the force between them seperated by the same distance as previous. Next he halved or doubled the distance and again measured the force. Thus, Coulomb did not measured the charges on the bodies but he was able to arrive at the law.

But,after arriving at the equation, he could use it to define the unit of charge( since it was not previously defined) just like Newton used his first law to define force(read my answer https://physics.stackexchange.com/a/530555/243233 )

Now you can see that Newton took k as 1 but Coulomb didn't. Why? In fact Coulomb actually did it. You can see that in CGS system of units it's value is 1. But in SI system it's value is 9×10^9 .Why SI system adopted this value only.

The fact is that SI system didn't adopt the value, instead they calculated it. Just as Newton used one of his laws to define force(Newton's first law) and then used the definition to find the value of the constant G in his law of Gravitation, SI system used the formula of current I=Q/t to define charge and then used that definition to find the value of k in Coulomb's law.( If you have problem over this para then read the answer in the link first).

Now I hope that you understand why it can be used as the definition of charge (in fact, it is used in CGS system).

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