Fundamentals of 3D mathematics [17] inverse square theorem

This theorem is used in a function of pushing mesh vertices with the mouse to simulate fingers pushing plasticine . Reprinted from Important theorems in mathematical physics ： Inverse square theorem

This is used to simulate that a mouse position close to the grid outside the grid sends a uniform force around , Let the mesh vertex produce correct displacement according to its relationship with the position .

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summary

Suppose we have a point light source . It will distribute energy equally in all directions . therefore , If you want to find all points in space with the same energy intensity , You must draw a sphere around the light point . The larger the radius of the sphere , Energy dispersive “ The surface of the ” The bigger it is .

The relationship between the radius and the surface area of the sphere is inverse square . This means that the strength will depend on 1/r^2. If you double the distance from the light source , The intensity will not just be halved , Instead, it will drop to a quarter of its value ; Three times the distance will reduce the intensity to one ninth , And so on .

Any point source , As long as it diffuses evenly in all directions without being limited by its scope , Will obey the inverse square law . A simple experiment can illustrate this point .

This comes from strict geometric derivation . At any given radius r The intensity at is the source intensity divided by the area of the sphere surface at that radius .

Since the origin of the inverse square law is a strict geometric property , So it applies to many different phenomena .

Gravitational source 、 The electric field 、 light 、 The sound 、 Point sources of electromagnetic or nuclear radiation follow the inverse square law . The following examples show the simulation effect of the inverse ratio theorem in some physical aspects with formulas , To show its meaning .
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gravity

As one of the fields obeying the general inverse square law , The gravitational field can be written as follows , Indicates the acceleration of gravity g Is an expression of the strength of the gravitational field .

The electric field

As one of the electric fields obeying the general inverse square law , The electric field of a point charge can be written as follows , The point charge Q Is the source of the electric field . Electricity in Coulomb's law follows the inverse square law .

radiation

As one of the fields obeying the general inverse square law , A point emitter can be described by the following relationship , Whether you're talking about Becquerel 、 Roentgen 、 Rudd or Remus . All exposures will decrease according to the inverse square law