Exploring Inverse Square Laws in Newton's and Coulomb's Laws of Force

In summary, the laws of universal gravitation and Coulomb's law are inverse square laws because of the mathematical structure of space-time and the fundamental principles of homogeneity and isotropy. As the distance increases, the force weakens because it is acting over a larger area, resulting in a decrease in the number of lines per square meter. These laws are described by local field equations and result in radial forces that decrease with the square of the distance.
  • #1
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Why are Newtons law of universal gravitation,

[itex]F=G\frac{m_{1}m_{2}}{r^{2}}[/itex]

and Coulombs law,

[itex] F = K_{e}\frac{q_{1}q_{2}}{r^{2}}[/itex]

inverse square laws? I understand why they are inverse because the force decreases with distance but why is the distance, r, squared?

Thanks
AL
 
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  • #2
In gravitation and electricity the force can be represented by lines radiating from a point in 3 dimensions. If you can accept that the strength of the force is something to do with the number of lines passing through a square metre then as distance increases the number of lines per square metre decreases and the force weakens. The surface through which all the lines pass is a sphere (if a point source) and the surface area is proportional to r^2 therefore the number of lines per square metre decreases as 1/r^2.
This simplified explanation assumes that the lines are not absorbed or stopped in any way (true for gravity but less true for electrical lines)
hope this helps
 
  • #3
This is a quite deep question about the physical laws. The answer is the mathematical structure of space-time. The laws you are looking at are all in the approximation of Newtonian mechanics or the special theory of relativity. In both space-time models, there always exists (by assumption!) at least one frame of reference, where the special principle of relativity, i.e., the principle of inertia is valid, i.e., a force-free particle always moves in straight lines or stays at rest. In addition, for any inertial observer, i.e., an observer who is at rest relatively to such an inertial frame, space is described by a Euclidean space.
Particularly space is symmetric under arbitrary rotations around any point (isotropy of space, i.e., no direction in space is special). It is also homogeneous, i.e., it's invariant under translations: There's no special place in space. This means that on a fundamental level all laws of nature must be described by equations that are consistent with homogeneity and isotropy of space.

Another very successful concept of physics is the description of forces by local fiel equations, and if you work out the very equations that obey the above mentioned symmetry principles you come to quite simple general forms of such equations. One such equation is Laplace's or Poisson's equation,

[tex]\Delta \Phi(\vec{x})=-\rho(\vec{x}).[/tex]

It's fundamental solution reads

[tex]\Phi(\vec{x})=\int \mathrm{d}^3 \vec{x}' \frac{\rho(\vec{x}')}{4 \pi |\vec{x}-\vec{x}'|}.[/tex]

Particularly, if the source term on the right-hand side is taken as a pointlike unstructured object at the origin of the coordinate system, you get

[tex]\Phi(\vec{x})=\frac{Q}{4 \pi |\vec{x}|}.[/tex]

The corresponding vector field, describing the forces is given by the gradient, and this leads to radial forces which decrease with the square of the distance,

[tex]\vec{F}=-\vec{\nabla} \Phi=\frac{Q}{4 \pi} \frac{\vec{x}}{|\vec{x}|^3}.[/tex]
 
  • #4
Thanks technician and vanhees71.

I think I understand it now, maybe not the mathematical reason as vanhees presented.

Would it be fair to say that in laymans terms as the distance increases the force is acting over a larger area so therefore its weaker because there is less force acting upon a given area?
 
  • #5


Both Newton's law of universal gravitation and Coulomb's law are considered inverse square laws because the force between two objects decreases with the square of the distance between them. This means that as the distance between two objects increases, the force between them decreases exponentially. The reason for this is due to the concept of a spherical surface. As an object moves away from another object, the force is spread out over a larger and larger surface area. Since the surface area of a sphere increases with the square of the radius, the force decreases with the square of the distance.

Additionally, both laws also involve the concept of an inverse square law of flux. This means that the amount of force passing through a given area decreases with the square of the distance from the source of the force. This is due to the fact that the force is spread out over a larger and larger area as the distance increases.

Overall, the inverse square law is a fundamental principle in physics and is observed in many different phenomena, including gravity, electromagnetism, and light. It is a mathematical representation of how forces and energies behave in relation to distance and is an important concept in understanding the behavior of the universe.
 

1. What are inverse square laws?

Inverse square laws state that the strength of a physical force is inversely proportional to the square of the distance between two objects. This means that as the distance between two objects increases, the force between them decreases.

2. How do Newton's and Coulomb's laws of force demonstrate inverse square laws?

Both Newton's law of universal gravitation and Coulomb's law of electrostatic force follow the inverse square law. This means that the force between two objects due to gravity or electric charge decreases as the distance between them increases.

3. Why is it important to understand inverse square laws?

Understanding inverse square laws helps us understand the nature of physical forces and how they behave. It also allows us to make accurate predictions and calculations in fields such as physics, engineering, and astronomy.

4. Are there any exceptions to inverse square laws?

There are some cases where inverse square laws do not apply, such as in the case of electromagnetic radiation, where the intensity is inversely proportional to the distance rather than the square of the distance.

5. How can inverse square laws be applied in real-life situations?

Inverse square laws have practical applications in fields such as astronomy, where they can be used to calculate the gravitational force between celestial bodies, and in engineering, where they can be used to design systems for transmitting and receiving radio waves.

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