POINT CHARGE: PROPERTIES AND COULOMB'S LAW - PHYSICAL - 2024

A point charge, in the context of electromagnetism, is that electrical charge of such small dimensions that it can be considered a point. For example, elementary particles that have an electric charge, the proton and the electron, are so small that their dimensions can be omitted in many applications. Considering that a charge is point-oriented makes the job 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. Also can be 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 size of the object.

Figure 1. Point charges of the same sign repel each other, while those of the opposite sign attract. Source: Wikimedia Commons.

Since all bodies are made of elementary particles, electric charge is an inherent property of matter, just like mass. You cannot have an electron without mass, and also not without charge.

Properties

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

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

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

Another very important point to consider is that the electrical charge is quantized. To date, no isolated electric charges of a magnitude less than the charge of the electron have been found. They are all multiples of this.

Finally, the electric charge is conserved. In other words, electric charge is neither created nor destroyed, but it can be transferred from one object to another. In this way, if the system is isolated, the total load remains constant.

Units of electrical charge

The unit for electric 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 point charges. We will talk about it later.

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

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

And as we mentioned before, the sign of e ^- is negative. The charge on the proton has exactly the same magnitude, but with a positive sign.

The signs are a matter of convention, that is, there are two types of electricity and it is necessary to distinguish them, therefore one is assigned a sign (-) and the other sign (+). Benjamin Franklin made this designation, and also enunciated the principle of conservation of charge.

By Franklin's time, the internal structure of the atom was still unknown, but Franklin had observed that a rod of glass rubbed with silk became electrically charged, calling this kind of electricity positive.

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

Coulomb's law for point charges

At the end of the 18th century, Coulomb, a French army engineer, spent a lot 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 two point electric charges.

Let be two electric charges q ₁ and q ₂. Coulomb determined that the force between them, either attraction or repulsion, was directly proportional to the product of both charges, and inversely proportional to the square of the distance between them.

Mathematically:

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

Thus:

Furthermore Coulomb found that the force was directed along the line connecting the charges. So, if r is the unit vector along said line, Coulomb's law as a vector is:

Application of Coulomb's law

Coulomb used a device called a torsion balance for his experiments. Through it it was possible to establish the value of the electrostatic constant in:

Next we will see an application. Three point loads are taken q _A, q _B q _C that are in the positions shown in Figure 2. calculate the net force on q _B.

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

The charge q _A attracts the charge q _B, because they have opposite signs. The same can be said about q _C. The isolated body diagram is in figure 2 on the right, in which it is observed that both forces are directed along the vertical axis or y axis, and have opposite directions.

The net force on charge q _B is:

F _R = F _AB + F _CB (Principle of superposition)

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

F _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

F _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

F _R = F _AB + F _CB = 0.000045 (+ y) + 0.00036 (- y) N = 0.000315 (- y) N

Gravity and electricity

These two forces have the same mathematical form. Of course, they differ in the value of the constant of proportionality and in that gravity works with masses, while electricity works with charges.

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

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

And we have this detail that is derived from the above: all objects in free fall have the same acceleration, as long as they are close to the surface of the Earth.

But if we release a proton and an electron near 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 electric charge is quantized, just like said. That 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 this charge. The masses, on the other hand, are not multiples of some single mass.

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

References

1. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 5. Electrostatics. Edited by Douglas Figueroa (USB).
2. Giancoli, D. 2006. Physics: Principles with Applications. 6th. Ed Prentice Hall.
3. Kirkpatrick, L. 2007. Physics: A Look at the World. 6th abridged edition. Cengage Learning.
4. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson.
5. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. V 2.