Coulomb's law explanation, formula and units, exercises, experiments

The Coulomb law it is the physical law that governs 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.

These small spheres can be considered as point loads, 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..

With this in mind, Charles Coulomb found the following:

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

-This force is always directed along the line connecting the charges..

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

Article index

• 1 Formula and units of Coulomb's law
• 2 How to apply Coulomb's law
• 3 Solved exercises
  • 3.1 - Exercise 1
  • 3.2 - Exercise 2
• 4 Experiments
• 5 References

Formula and units of Coulomb's law

Thanks to these observations, Coulomb concluded that the magnitude of the force F between two point charges what₁ Y what_two, separated a distance r, is given mathematically as:

Since the force is a vector magnitude, to express it completely a unit vector is defined r 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_and 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 N.m^two / C^two, to get newtons as a result. The value of the constant was determined experimentally as:

k_and = 8.89 x 10 ⁹ N.m^two / C^two ≈ 9 x 10 ⁹ N.m^two / C^two

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 F₁ Y F_two they are the same, the direction is the same, but the senses 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:

F_net = F_i1 + F_i2 + F_i3 + F_i4 +… = ∑ F_ij

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

- You should always be consistent with the units. The most frequent thing is to work with the electrostatic constant in SI units, then it is necessary 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.

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:

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

Let's call F₁ the force exerted by charge q on charge -q, are directed along the x-axis and point in the negative direction, therefore:

Analogously, it is calculated F_two:

Note that the magnitude of F_two is half of that of F₁, although the load is double. To find the net force, finally they are added vectorially F₁ Y F_two:

F_net = (-k + k / 2). (q^two / d^two) (x) N = - (k / 2). (Q^two / d^two) (x) N

- 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 statement situation is described in figure 5a.

We choose one of the spheres and on it we draw the isolated body diagram, which includes three forces: weight W, tension in the rope 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^-two/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. Importantly, the tension T is slanted 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^two / d^two, clears Q:

Q = 2 × 10^-eleven 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 scale, is suspended by a thread. The experiment consists of touching the discharged elderberry spheres with another metallic sphere charged with Q charge.

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 that 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 electrostatic repulsion force.

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 S.A. by C.V.
5. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 2.