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In electromagnetics, Coulomb’s law is a very classic formula. Almost all textbooks give this formula, but none of them give a detailed corresponding theoretical derivation. In order for beginners of physics to better understand the physical meaning of this formula, we explored the source, the physical model and mechanism of this formula. Based on the principle that the interaction between two different fields can generate energy density, which is equal to the pressure, we analyzed the distribution of the electric field energy density as well as the corresponding pressure on the charged surface. Through the rigorous mathematical derivation, we give the theoretical derivation of this formula.

Coulomb’s law is a mathematical description of the electric force between charged objects. According to Coulomb, the electric force for charges at rest has the following properties: If two electrically charged bodies are placed nearby each other there will be an attraction or a repulsion force acting on them depending upon the nature of the charge of the bodies. The formula for the force acting between two electrically charged bodies was first developed by Charles-Augustinde in 1785. The electric force is proportional to the product of the charge of the charged bodies and inversely proportional to the square of the distance between the center of the charged bodies.

Coulomb’s law is summarized from a large number of experiments. Then, almost all the classic electronic books have directly given the formula of Coulomb’s law, and hardly given the theoretical derivation of the formula [

As we all know, there is an electric field around an electric charge, which is a characteristic of electric charge. This electric field has energy and mass. Here we give a classical model of electric field generated by a point charge for the following discussion, as shown in

E = q 4 π ε 0 r 3 r (1)

Here E is the electrostatic field intensity, ε 0 is the dielectric constant in vacuum. q is the electric quantity of a charge. r is the position vector from a point to the charge.

For two point charges with a certain distance, the distribution of electric field is shown in

G = 1 2 D ⋅ E = 1 2 ( D 1 + D 2 ) ⋅ ( E 1 + E 2 ) (2)

Here G is defined as the energy density, E 1 , E 2 is the electrostatic field intensity generated by the first-point charge and the second point charge, respectively (seeing

G = 1 2 D 1 ⋅ E 1 + 1 2 D 1 ⋅ E 2 + 1 2 D 2 ⋅ E 1 + 1 2 D 2 ⋅ E 2 (3)

Using D = ε 0 E , Equation (2) can be changed to

G = 1 2 ε 0 E 1 2 + ε 0 E 1 ⋅ E 2 + 1 2 ε 0 E 2 2 (4)

Now, we are more interested in the second term of the Formula (4), which represents the energy density generated by the interaction of E 1 with E 2 . Equation (4) can be written as

G = 1 2 ε 0 E 1 2 + ε 0 E 1 E 2 cos θ + 1 2 ε 0 E 2 2 (5)

Here θ is the angle between E 1 and E 2 . The classical physics tells that the energy density is equal to the pressure difference. So, Equation (5) can be written as

P = 1 2 ε 0 B 1 2 + ε 0 E 1 E 2 cos θ + 1 2 ε 0 E 2 2 (6)

Here P is defined as the pressure.

Because the size of the charge is too small, the distance between two point charges is much greater than the radius of charge. Thus, the electric field generated by the first charge around the second charge can be regarded as a uniform electric field. For the following research, we enlarge the size of the second charge, as shown in _{A}, P_{B}), respectively,

{ P A = G A = 1 2 ε 0 E 1 2 + ε 0 E 1 E 2 cos θ 1 + 1 2 ε 0 E 2 2 P B = G B = 1 2 ε 0 E 1 2 + ε 0 E 1 E 2 cos θ 2 + 1 2 ε 0 E 2 2 (7)

Here θ 1 and θ 2 are the angle between E 1 vector and E 2 vector at the A and B-point of the second charge, respectively.

The first and third terms in Equation (7) indicate that the corresponding energy density is constant at the different position of the second charge surface, which suggests that the corresponding pressure of the two terms is unchanged on this surface. Thus, the total force generated from them on the whole surface is zero. Only the second term in Equation (7) shows that the corresponding energy density is different at the different positions of the second charge surface, which can produce a pressure difference. That is, the total force generated from the second terms in Equation (7) is not zero, which is the source of the electrostatic force. Therefore, we only consider the calculation of the second term in Equation (7). Here, we set

P A = ε 0 E 1 E 2 cos θ (8)

For the charge surface, the force can be written as

d F A o = P A d S = ( ε 0 E 1 E 2 cos θ ) ( R d θ ) ( 2 π R sin θ ) e A O (9)

Here R is the radio of the second charge, e A o is the unit vector from A-point to O-point along the radius of chare (seeing

Seeing from Equation (8), there is a difference in energy density at the tangent direction of the charged surface, which also leads to the force through the pressure difference at the direction (seeing

d F O C = d F A O + d F A C (10)

d F O C = d F A O cos θ = 1 cos θ [ q 1 4 π ε 0 r 1 2 q 2 4 π ε 0 R 2 cos θ ε 0 2 π R sin θ R d θ ] (11)

Equation (11) can be simplified to be

d F O C = q 1 q 2 4 π ε 0 r 1 2 1 2 sin θ d θ (12)

Now, through the integration of the whole surface of the charge, we get the formula of the electrostatic attraction force

F O C = ∫ 0 π q 1 q 2 4 π ε 0 r 1 2 1 2 sin θ d θ = q 1 q 2 4 π ε 0 r 1 2 (13)

Here F O C is the force acting between two charges. Equation (13) is the complete expression of Coulomb’s law. This above derivation method proves that the force acting between two objects in space essentially comes from the interaction of electronic fields. The above derivation process also suggests that all forces in space are derived from the difference in energy density produced by the interaction of one field with other fields. There are different fields in the universe, such as electric field, magnetic field, gravitational field, etc. Their interaction will produce energy density. For a certain object, the difference in energy density around it will cause a pressure difference, which can produce a force.

For any formula in physics, there should be many different ways to derive it. Many textbooks only give some conclusive formulas, and did not provide the physical and mathematical explanations of these formulas, which is not conducive to the understanding of students and teachers, and it is not conducive to improving students’ interest in physics. The derivation process in this paper can reveal the corresponding physical principles of the formula, which is very meaningful for physicists to deeply understand the physical model and mechanism corresponding to the formula.

The interaction of the different electric field results in a different energy density difference on the surface of the charge, which causes attraction or repulsion between different charges. Using the difference of energy density, we set a model to derive the formula of Coulomb’s law. So far, this derivation method is not reported in other papers. Physics is a discipline that explains the origin of this world and the laws of matter movement. It aims to tell people the truth of this objective world. Therefore, our in-depth thinking on all physical laws can further unlock the truth of this world, which is the value of this paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Zhang, H.W. and Feng, S.M. (2020) One Method to Derivate Coulomb’s Law between Two Charges. Journal of Applied Mathematics and Physics, 8, 2880-2885. https://doi.org/10.4236/jamp.2020.812213