Modern Physics

Punctual load: properties and Coulomb’s law

A point charge , in the context of electromagnetism, is that electrical charge of dimensions so small that it can be considered a point. For example, the electrically charged elementary particles, the proton and the electron, are so small that their dimensions can be omitted in numerous applications. Considering that a charge is punctual, it makes the work of calculating its interactions and understanding the electrical properties of matter much easier.

Elementary particles are not the only ones that can be point charges. The same happens with the ionized molecules, the charged spheres that Charles A. Coulomb (1736-1806) used in his experiments and even the Earth itself. All can be considered point charges, as long as we see them at distances much greater than the object’s size.

Since all bodies are made of elementary particles, electrical charge is an inherent property of matter, as is mass. You cannot have a massless and also uncharged electron.


As far as we know today, there are two types of electrical charge: positive and negative. Electrons have a negative-type charge, while protons have a positive charge.

Charges of the same sign repel each other, while those of the opposite sign attract. This is valid for any type of electrical charge, punctual or distributed on an object of measurable dimensions.

Furthermore, careful experiments have proven that the charge on the proton and that on the electron have exactly the same magnitude.

Another very important point to consider is that the electrical charge is quantized. So far, no isolated electrical charge of magnitude smaller than the electronic charge has been found. All are multiples of that.

Finally, the electrical charge is conserved. In other words, electrical charge is not created or destroyed, but can be transferred from one object to another. This way, if the system is isolated, the total load will remain constant.

Electric charge units

The unit of electrical charge in the International System of Units (SI) is the Coulomb, abbreviated with a capital C, in honor of Charles A. Coulomb (1736-1806), who discovered the law that bears his name and describes the interaction between two charges punctual. We’ll talk about it later.

The electric charge of the electron, the smallest possible that can be isolated in nature, has a magnitude of:

and  = 1.6 x 10-16 C

Coulomb is a fairly large unit, so submultiples are often used:

-1 milli C = 1 mC = 1 x 10 -3 C

-1 micro C = 1 μC = 1 x 10 -6 C

-1 nano C = 1 nC = 1 x 10 -9 C

And, as we mentioned earlier, the sign of  is negative. The proton charge has exactly the same magnitude, but with a positive sign.

Signals are a matter of convention, that is, there are two types of electricity and must be distinguished; therefore, one is assigned a (-) sign and the other (+) sign. Benjamin Franklin made this designation and also enunciated the principle of cargo conservation.

In Franklin’s day, the internal structure of the atom was still unknown, but Franklin had observed that a bar of glass rubbed with silk was electrically charged, calling this type of electricity positive.

Any object that was attracted by this electricity had a negative sign. After the electron was discovered, it was observed that the charged glass rod attracted them and that is how the charge on the electron was negative.

Coulomb’s law for punctual charges

In the late 18th century, Coulomb, an engineer in the French army, spent a great deal of time studying the properties of materials, the forces acting on beams, and the force of friction.

But he is best remembered for the law that bears his name and that describes the interaction between electrical charges at two points.

Leave two electrical charges 1 and 2 . Coulomb determined that the force between them, by attraction or repulsion, was directly proportional to the product of both charges and inversely proportional to the square of the distance between them.


∝ q 1 . q 2 / r 2

In this equation, F represents the magnitude of the force and r is the distance separating the loads. Equality requires a proportionality constant, which is called the electrostatic constant and is denoted as and .


F = k. q 1 . q 2 / r 2

In addition, Coulomb found that the force was directed along the line that joined the loads. Therefore, if r is the unit vector in this line, Coulomb’s law as a vector is:

Punctual load: properties and Coulomb's law 2

Coulomb Police

Coulomb used a device called a torsion scale for his experiments. Through it, the value of the electrostatic constant can be established in:

e = 8.99 x 10 9 N m 2 / C 2 ≈ 9.0 x 10 9 N m 2 / C 2

Next, we’ll look at an app. Three point loads are taken q A , Q B Q C which are at the positions shown in Figure 2. Calculate the net force on Q B .

Figure 2. The force on the negative charge is calculated by Coulomb’s law. Source: F. Zapata.

The charge q A attracts the charge q B , because they are of opposite signs. The same can be said about q C . The isolated body diagram is in figure 2 on the right, where you can see that both forces are directed along the vertical axis or along the y axis and have opposite directions.

The net force on charge q B is:

R = AB + CB (Overlapping principle)

It remains only to replace the numerical values, taking care to write all units in the International System (SI).

AB = 9.0 x 10 9 x 1 x 10 -9 x 2 x 10 -9 / (2 x 10 -2 ) 2 N (+ y) = 0.000045 (+ y) N

CB = 9.0 x 10 9 x 2 x 10 -9 x 2 x 10 -9 / (1 x 10 -2 ) 2 N ( -y ) = 0.00036 ( -y ) N

R = AB + CB = 0.000045 (+ y) + 0.00036 (- y ) N = 0.000315 (- y) N

gravity and electricity

These two forces have identical mathematical form. Obviously, they differ in the value of the proportionality constant and in gravity it works with masses, whereas electricity does with charges.

Related:   Pressure Gradient: What It Is and How It Is Calculated

But the important thing is that both depend on the inverse squared distance.

There is a unique type of mass and it is considered positive; therefore, gravitational force is always attractive, while charges can be positive or negative. Therefore, electrical forces can be attractive or repulsive depending on the case.

And we have this detail that derives from the above: all objects in free fall have the same acceleration while they are close to the Earth’s surface.

But if we release a proton and an electron close to a charged plane, for example, the electron will have a much greater acceleration than the proton. Furthermore, the accelerations will have opposite directions.

Finally, the electrical charge is quantized as indicated. This means that we can find charges 2.3 or 4 times that of the electron – or that of the proton – but never 1.5 times that charge. Masses, on the other hand, are not multiples of a single mass.

In the world of subatomic particles, the electrical force exceeds the gravitational force in magnitude. However, on macroscopic scales, the force of gravity prevails. Where At the level of planets, solar system, galaxy and much more.

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