COULOMB'S LAW: EXPLANATION, FORMULA AND UNITS, EXERCISES, EXPERIMENTS - PHYSICAL - 2024

The Coulomb law is the physical law governing the interaction between electrically charged objects. It was enunciated by the French scientist Charles Augustin de Coulomb (1736-1806), thanks to the results of his experiments using the torsion balance.

In 1785, Coulomb experimented innumerable times with small electrically charged spheres, for example moving two spheres closer or farther apart, varying the magnitude of their charge and also their sign. Always carefully observing and recording each answer.

Figure 1. Scheme showing the interaction between point electric charges using Coulomb's law.

These small spheres can be considered as point charges, that is, objects whose dimensions are insignificant. And they fulfill, as has been known since the time of the ancient Greeks, that charges of the same sign repel and those of a different sign attract.

Figure 2. The military engineer Charles Coulomb (1736-1806) is considered the most important physicist in France. Source: Wikipedia Commons.

With this in mind, Charles Coulomb found the following:

-The force of attraction or repulsion between two point charges is directly proportional to the product of the magnitude of the charges.

-Said force is always directed along the line that joins the charges.

-Finally, the magnitude of the force is inversely proportional to the square of the distance that separates the charges.

Formula and units of Coulomb's law

Thanks to these observations, Coulomb concluded that the magnitude of the force F between two point charges q ₁ and q ₂, separated by a distance r, is mathematically given as:

As the force is a vector magnitude, to express it completely a unit vector r is defined in the direction of the line joining the charges (a unit vector has magnitude equal to 1).

Additionally, the constant of proportionality necessary to transform the previous expression into an equality is called k _e or simply k: the electrostatic constant or Coulomb's constant.

Finally, Coulomb's law is established for point charges, given by:

Force, as always in the International System of Units, comes in newton (N). Regarding the charges, the unit is named coulomb (C) in honor of Charles Coulomb and finally the distance r comes in meters (m).

Looking closely at the above equation, it is clear that the electrostatic constant must have units of Nm ² / C ², to get newtons as a result. The value of the constant was determined experimentally as:

k _e = 8.89 x 10 ⁹ Nm ² / C ² ≈ 9 x 10 ⁹ Nm ² / C ²

Figure 1 illustrates the interaction between two electric charges: when they are of the same sign they repel, otherwise they attract.

Note that Coulomb's law conforms to Newton's third law or law of action and reaction, therefore the magnitudes of F ₁ and F ₂ are equal, the direction is the same, but the directions are opposite.

How to apply Coulomb's law

To solve problems of interactions between electric charges, the following must be taken into account:

- The equation applies exclusively in the case of point charges, that is, electrically charged objects but of very small dimensions. If the loaded objects have measurable dimensions, it is necessary to divide them into very small loads and then add the contributions of each of these loads, for which an integral calculation is required.

- The electric force is a vector quantity. If there are more than two interacting charges, the net force on the charge q _i is given by the superposition principle:

_Net F = F _i1 + F _i2 + F _i3 + F _i4 +… = ∑ F _ij

Where the subscript j is 1, 2, 3, 4… and represents each of the remaining charges.

- You must always be consistent with the units. The most common is to work with the electrostatic constant in SI units, so you have to make sure that the charges are in coulombs and the distances in meters.

- Finally, the equation applies when the charges are in static equilibrium.

Solved exercises

- Exercise 1

In the following figure there are two point charges + q and + 2q. A third point charge –q is placed at P. It is asked to find the electric force on this charge due to the presence of the others.

Figure 3. Diagram for the resolved exercise 1. Source: Giambattista, A. Physics.

Solution

The first thing is to establish a suitable reference system, which in this case is the horizontal axis or x axis. The origin of such a system can be anywhere, but for convenience it will be placed at P, as shown in figure 4a:

Figure 4. Scheme for the resolved exercise 1. Source: Giambattista, A. Physics.

A diagram of the forces on –q is also shown, taking into account that it is attracted by the other two (figure 4b).

Let us call F ₁ the force exerted by the charge q on the charge –q, they are directed along the x-axis and point in the negative direction, therefore:

Analogously, F ₂ is calculated:

Note that the magnitude of F ₂ is half that of F ₁, although the charge is double. To find the net force, finally F ₁ and F ₂ are added vectorially:

- Exercise 2

Two polystyrene balls of equal mass m = 9.0 x 10 ^-8 kg have the same positive charge Q and are suspended by a silk thread of length L = 0.98 m. The spheres are separated by a distance of d = 2 cm. Calculate the value of Q.

Solution

The situation of the statement is described in figure 5a.

Figure 5. Schemes for the resolution of exercise 2. Source: Giambattista, A. Physics / F. Zapata.

We choose one of the spheres and on it we draw the isolated body diagram, which includes three forces: weight W, tension in the string T and electrostatic repulsion F, as it appears in figure 5b. And now the steps:

Step 1

The value of θ / 2 is calculated with the triangle in figure 5c:

θ / 2 = arcsen (1 x 10 ^-2 /0.98) = 0.585º

Step 2

Next we must apply Newton's second law and set it equal to 0, since the charges are in static equilibrium. It is important to note that the tension T is inclined and has two components:

∑F _x = -T. sin θ + F = 0

∑F _y = T.cos θ - W = 0

Step 3

We solve for the magnitude of the stress from the last equation:

T = W / cos θ = mg / cos θ

Step 4

This value is substituted into the first equation to find the magnitude of F:

F = T sin θ = mg (sin θ / cos θ) = mg. tg θ

Step 5

Since F = k Q ² / d ², we solve for Q:

Q = 2 × 10 ^-11 C.

Experiments

Checking Coulomb's law is easy using a torsion balance similar to the one Coulomb used in his laboratory.

There are two small elderberry spheres, one of which, the one in the center of the balance, is suspended by a thread. The experiment consists of touching the discharged elderberry spheres with another metallic sphere charged with Q charge.

Figure 6. Coulomb's torsion balance.

Immediately the charge is distributed equally between the two elderberry spheres, but then, as they are charges of the same sign, they repel each other. A force acts on the suspended sphere which causes the twisting of the thread from which it hangs and immediately moves away from the fixed sphere.

Then we see that it oscillates a few times until it reaches equilibrium. Then the torsion of the rod or thread that holds it is balanced by the force of electrostatic repulsion.

If originally the spheres were at 0º, now the moving sphere will have rotated an angle θ. Surrounding the scale, there is a tape graduated in degrees to measure this angle. By previously determining the torsion constant, then the repulsive force and the value of the charge acquired by the elderberry spheres is easily calculated.

References

1. Figueroa, D. 2005. Series: Physics for Sciences and Engineering. Volume 5. Electrostatics. Edited by Douglas Figueroa (USB).
2. Giambattista, A. 2010. Physics. Second Edition. McGraw Hill.
3. Giancoli, D. 2006. Physics: Principles with Applications. 6th. Ed Prentice Hall.
4. Resnick, R. 1999. Physics. Vol. 2. 3rd Ed. In Spanish. Compañía Editorial Continental SA de CV
5. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 2.