# Is 1/4π built into G, the gravitational constant?

## Main Question or Discussion Point

We went over Coulomb's law today, which can be stated as

(1/ε0)(1/(4πr2))(|q1q2|)

This equation is very similar to Newton's law of gravitation, but it contains 1/4pir^2. This makes sense, because the electric force is being diluted over the surface of a sphere.

Is 1/(4πr2) built into G as well?

## Answers and Replies

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phinds
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2019 Award
Why would it be? Is gravity diluted over the surface of a sphere?

BobG
Homework Helper
We went over Coulomb's law today, which can be stated as

(1/ε0)(1/(4πr2))(|q1q2|)

This equation is very similar to Newton's law of gravitation, but it contains 1/4pir^2. This makes sense, because the electric force is being diluted over the surface of a sphere.

Is 1/(4πr2) built into G as well?
Yes. You have a certain amount of "stuff", whatever it is, that is being spread out over a larger surface area.

The inverse square law is ubiquitous. The reason for it is just a lot more clear in some equations.

collinsmark
Homework Helper
Gold Member
Yes, when comparing Newton's Universal Law of Gravity to Coulomb's law, the $4 \pi$ is built into the G in Newton's law. There's no need to include an additional $4 \pi$ when talking about Newton's equation.

On the other hand, as a consequence, You do need to include the $4 \pi$ when working with Gauss' Law of Gravitation. This is contrasted with Gauss' law of electrostatics, where there is no need to include the $4 \pi$.

In summary:
Gravitation: The $4 \pi$ is [needed] in Gauss' law, but not in the force equation.
Electrostatics: The $4 \pi$ is [needed within] part of the force equation, but not in the Gauss law.

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To add another perspective:

It depends on where do you start deriving the equation.

If you adding a constant to inverse square law of gravitational force, then there is no 4pi;
If you start by treating gravity as charges distributed in a sphere and apply Gauss's law, there is a 4pi.

collinsmark
Homework Helper
Gold Member
To add another perspective:

It depends on where do you start deriving the equation.

If you adding a constant to inverse square law of gravitational force, then there is no 4pi;
If you start by treating gravity as charges distributed in a sphere and apply Gauss's law, there is a 4pi.
Right, I agree it makes sense depending on how you approach it. My point though, is that for whatever reason (and there may be good reasons depending on how one approaches it) equations are different in this respect for gravity and electromagnetism. The placement of the $4 \pi$ is essentially swapped between conventional versions of gravity equations and conventional electrostatic equations.

Gravity:
Newton's Universal Gravitation:
$$\vec F = -G \frac{m_1 m_2}{r^2} \hat {a_r}$$
Gauss' law for gravity:
$$\oint \vec g \cdot \vec {dA} = -4 \pi Gm_{enc}$$

Electrostatics:
Coulomb's Law:
$$\vec F = \frac{1}{4 \pi \varepsilon_0}\frac{q_1 q_2}{r_2} \hat {a_r}$$
Gauss' Law
$$\oint \vec E \cdot \vec {dA} = \frac{q_{enc}}{\varepsilon_0}$$

Good point, didn't notice that before.

There is a historical reason of this swapping. For gravity, Newton's Law of Gravitation is well known before formulating gravitational Gauss's law, so the G is left as it is by convention.
For the Electrostatics, the reason to add a 4pi to Coulomb's law is to maintain numerical consistency with older unit of measurement.

Khashishi
The placement of the 4pi is just an arbitrary historical unit convention. There is no deeper meaning behind it.
There are two commonly used sets of metric units (and several less commonly used ones). In SI, the 4pi appears in Coulombs law. In Gaussian cgs units, 4pi doesn't appear there, but instead it shows up in the differential form of Coulomb's law.

Well this is out of context but it would be better if gravitational law is written in same form as Coulomb's law because the later is easier to understand .these fundamental mental forces of nature act very differently but can be described by almost same equations who doesn't like uniformity of equations?

phinds
Gold Member
2019 Award
Well this is out of context but it would be better if gravitational law is written in same form as Coulomb's law because the later is easier to understand .these fundamental mental forces of nature act very differently but can be described by almost same equations who doesn't like uniformity of equations?
There are numerous "conventions" in science that are present for historical reasons and would be better done some other way but they are what they are and the best one can do is accept it and move on since they are not going to change.

There are numerous "conventions" in science that are present for historical reasons and would be better done some other way but they are what they are and the best one can do is accept it and move on since they are not going to change.
There are numerous "conventions" in science that are present for historical reasons and would be better done some other way but they are what they are and the best one can do is accept it and move on since they are not going to change.
Yeah I guess we should respect the tradition :(

phinds