A Few Facts On Coulomb’s Law

A few words on the modern interpretation of Coulomb’s Law in terms of Fields Theory.

At the base of all the electromagnetism theory lays Coulomb’s Law, which describes why and how charges move in a medium, whether the medium is a conductor or the void.

Electrical and Electronics engineers are therefore supposed to be very familiar with this law. Let’s spend some time to provide a few important facts of the law. For that, we have to go back to the concept of ‘field’.

In modern physics, we usually tend to identify the effect of an entity over another entity with the name of field. Mathematically speaking, a field is a 3-dimensional matrix that assigns a specific value, usually a vector, to each point in the observed space.

Each time we put in that space an object capable of reacting with the field, we end up with a force applied to that object that is the product of the field in that point and the measurable value of that object.

So, in the case of charges, a single charge creates a field in the surrounding space. When we put in that space a second charge, the new charge will be subject to a force that is the product of the charge itself and the value of the field in that point in space.

Because charges can repel or attract, depending on their positive or negative sign, the field itself is directional, and therefore represented with vectors..

Here is a picture representing a field generated by a positive charge:

And here is a picture representing a field generated by a negative charge:

If the charge creating the field is Q, then the field can be represented by the following equation:

where E is the value of the field, actually the module of the field vector, and r is the distance from the charge of a point in space where the field is calculated. The constant ε0 is called “dielectric constant” and, in this formula, it assumes that the charge is located in the void or in thin air. An adjustment to the constant needs to be made when the medium is different.

We call this field an Electrostatic field, hence the E, because the charge that generates it does not move, it is static.

The force on a charge q put in the field can be calculated as:

And, since E is actually a vector, F is therefore also a vector.

If we want to represent this in actual vector format, we can write it as:

where

is the position vector of the charge q with respect to the charge Q. This is what is called Coulomb’s Law.

You can see that if the charges are both positive or both negative, the vector F is oriented such that the charge q moves away from the charge Q and, vice versa, when the charges have opposite sign, F is oriented toward Q.

That is exactly what we expected: charges of the same sign repel each other and charges of opposite sign attract each other.

As a further example, here is the graphical representation of an electrostatic field generated by two charges of opposite sign:

… and finally an electrostatic field generated by two charges of the same sign:

Conductors, Insulators, and Semiconductors

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Everybody knows that an electric wire, usually made of copper, is a conductor. And everybody knows that all metals are conductors.

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Everybody also knows that plastic is a good electrical insulator, as well as other materials such as glass and rubber.

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But how about semiconductors? What are they? And how do we really distinguish among conductors, semiconductors and insulators?

To answer all these question we need to look deeper inside the materials.We know that matter is made of atoms, and atoms are made of protons, neutrons and electrons. Protons and neutrons reside at the center of the atom structure, called the nucleus. Electrons are allocated all around the nucleus, at a long distance from it, relatively to the scale of the nucleus itself.

Electrons have a certain amount of energy, that is always an integral value of a certain amount that is called quantum of energy. Depending on the amount of energy they posses, they are locate closer or farther away from the nucleus. The more energy, the farther they are.

Based on quantum mechanics, which we are not going to talk in details in this context, electrons occupy bands of energy. The farther bands in the atom are the so called Valence Band and Conduction Band.

The valence band contains all those electrons that allow the atoms to stick together and forming molecules by bonding with other atoms of the same or a different substance.

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For certain materials, rather than having molecules, the atoms form what is called a crystalline lattice, which is the case we are more interested in this context. It is worth noting that a new theory, highly based on quantum mechanics, is also distinguishing between actual crystalline lattices and material networks. However, for all the scope and purpose of this context, we will make a simplification and name both of them as crystalline lattices.

lattice(This picture, courtesy of Wikipedia)

In certain conditions, electrons in the valence band can jump to a higher level of energy, thus moving in what is called the conduction band. In the conduction band, electrons are no more stuck to their own atoms, but can start moving freely in the lattice that makes up the material. When that happens, we can control their movement by applying an electric field by the means, for example, of a battery. The voltage of the battery, applied to the two ends of the same block of material (for example a wire), produces the electric field inside the material and forces the electrons to move toward the positive electrode of the battery while, in the mean time, the negative electrode of the battery provides electrons to the material, to replace those that have entered the positive electrode of the battery.

Don’t think that electrons move very fast when they do that. Electrons, in fact, move very slowly, but it is the huge amount of them that help creating a measurable current.

Then you would ask: but when I turn the switch on, the light comes out of a lamp instantly. If the electrons move slowly, shouldn’t a lamp emit light only after a while?

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Well, yes and no. In fact, the lamp does not light up immediately. It takes a certain amount of time to do that. But that time is so small that for us the event happens instantly.

Also, when you turn the switch on, the electrons closest to the positive electrode move into it almost immediately, not because they are fast, but because they are so close to it. At the same time, new electrons are fed to the material from the negative electrode. So, at the end, all the electrons in the wire start moving simultaneously inside of it, causing the current to start flowing immediately.

But I am digressing. Let’s go back to our primary subject.

We have talked about electrons in the valence band and the possibility they have to jump to the conduction band and, thus, helping creating a current if we apply a voltage.

But how much energy do the electrons need to jump to the conduction band?

Here it comes the definition of conductors, semiconductors and insulators.

conductors_semiconductors_insulators

In the materials called conductors, the level of energy that the valence electrons need to jump to the conduction band is basically ‘0’. In fact, valence band and conduction band overlap each other and, therefore, some or all the electrons in the valence band are also, already, in the conduction band. This happens mostly with metals, like copper, iron, aluminum, and so forth. Depending on the particular metal, the conduction and valence bands are more or less overlapped. Those material where there is more overlap are those that conduct electricity better. Those material where there is less overlap, are those that are worst to conduct current, although still conductors.

In the materials called insulators, the gap between the valence band and the conduction band is so high that electrons cannot jump from the valence band to the conduction band, and so they cannot generate an electrical current.
Of course, if we apply a voltage high enough, we can still provide them the energy to make the jump. However, in that case, because of the very high voltage, electrons jump from one band to the other in a disruptive way, causing the material to break. Once that happened, the insulator loose its property and it is no good anymore as such.

Finally, in the materials called semiconductors, the valence band and the conduction band are separated, but close together. It is relatively easy for an electron in the valence band to jump to the conduction band if we only heat a little bit the semiconductor, maybe just with our bare hands. The heat provides enough energy to the electrons to jump to the conduction band. However, not many electrons will do so, unless we keep heating the semiconductor. So, at the end, although capable of conducting some current, semiconductors are not good in doing so. Thus the name of their category.

In this article, we have talked about energy bands in materials, and how materials behave based on the position of the two highest energy band levels.

We have said that conductors are those were valence and conduction bands are partially overlapping.

Conversely, insulators are those that have a high gap in between the valence and the conduction bands.

And finally, semiconductors are those somewhat in between. For them, the valence and conduction bands are separated with a gap, but that gap is small enough to allow, under certain conditions, for the electrons to jump from the valence to the conduction band. That’s why they perform poorly both as conductors and insulators. However, we will see later on how semiconductors can be of great advantage for us, as long as they are treated in a certain way. They are those that allow us to build all the wonders of modern electronics.

Let’s Talk About Voltage

What is voltage? Where the name comes from? How it makes electric current flow?

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Voltage is a very common word in the context of electrical and electronics engineering. However, common as it is, it is a concept that is often not fully understood.

How many people know that voltage is a concept related with potential energy? The same kind of energy that is so often used in mechanical physics!

When we talked about electrical current in the previous post, we said that current moves from a higher electrical potential energy to a lower one. We also defined the current as the flow of positive charges moving from the positive electrical potential energy to the negative one.

charges_in_wire

In order for the charges to move, there has to be a force that puts them in motion in a specific direction. Such force will have to do some sort of work on the charges and the work done by the force can be translated in a change in energy of the charges.

We define Electromotive Force the ratio between the energy change of the charges and the amount of charges. This can be represented with the following formula:

emf

Note that, despite the name, the Electromotive Force is not a force in the mechanical sense. Instead it is a potential energy per unit of charge, which we also call electric potential, or just potential.

The unit for the emf is called Volt, and is the ratio between one unit of energy, or 1 Joule, and one unit of charge, or 1 Coulomb:

volt

And that’s where the name voltage comes from: it is a name derived from the measurement unit of the electric potential. Although the correct name is electric potential, to make things easier in the day to day talk, we just call it voltage.

So, again, what is voltage? The voltage is the difference of electrical potential energy that allows a unit of charge to move and produce a current.

Now, do you think it is correct to say that when there is voltage we have a current? Do the two things always go together? Well, the answer is no. You can actually have one without the other.

An example of voltage without a current is the battery. If you don’t connect the battery to a circuit, there is no current involved!

batteries

And what about the current? This example is definitively less intuitive. So, let’s just say for now that if we force a current in a ring of a special material called superconductor and then we remove the cause that generated the current, the current keeps flowing in the superconductor, even in the absence of a voltage. Maybe we can explore this concept a little more in a future post, if I see there is an interest for it. Translation: let me know what you think! Comments are always welcome!

superconductor_ring

Let’s now go back to the battery. The battery provides a voltage between its positive and negative poles. With that, if we connect the battery to an electric load, current starts to flow immediately and the higher the voltage the greater the amount of current (more on that in a future post).

current_with battery

Like the current, there are two forms of voltage:

  1. The DC voltage

  2. The AC voltage

The DC voltage is the one typically provided by batteries. It is a voltage of a fixed value and of fixed polarity. The plus and minus on the electrodes of the battery never change. One electrode will always be the positive one, and the other electrode will be the negative one.

dc

When we connect a battery to an electric circuit, we have a flow of DC current.

The AC voltage is the one created for example in the power plants and provided at the wall outlets in your house.

power_plant_scaled

outlet_scaled

The voltage at the outlet is not constant as the one in the batteries. Instead, it changes continuously following the shape of a sine wave. Because of that, the polarity at each electrode of the outlet changes over time from positive to negative and vice versa, following the shape of the sine wave.

sine_wave

When we connect a device to the electric outlet, the current that will flow through that device will be an AC current as well.

The sinusoidal shape of the AC voltage depends on the way the electricity is generated. In the power plants there are devices called alternators, a much bigger version of those that you can find inside your car to recharge the battery, or on a bike, to provide electricity to turn on the lights at night.

Depending on the power plant, a different kind of energy is used to put in motion the alternator. It could be fossil fuel or nuclear energy that heat a reservoir of water and create the steam that makes the alternator rotate.

big_alternator_scaled

Or it could be the rotation of a propeller-like device that is put in motion by the wind.

propeller

Whatever the source is of the mechanical energy, the alternator converts that energy in electrical energy. But, since the rotation translates into a sine wave when described on a Cartesian reference system, the resulting electrical energy acquires that shape too.

DC and AC are both necessary to us to power devices that make our lives easier. Some devices need to be powered with DC, others need the AC.

DC voltage is necessary, for example, to power electronic devices: your smartphone or radio or computer, for example.

AC voltage is used to transmit the electrical energy from the places where it is created to the places where it is used. When AC reaches our home, it can be used as is to power electric motors like those in the refrigerator or the washing machine, or other household appliance. Or it can be converted to DC to power other devices, like your TV set.

Summarizing:

  1. To generate a current, we need to provide some energy to the charges in the conductor.

  2. The potential energy per unit of charge is called emf, or electromotive force. That measures the capability of the generator (a battery, for example) of generating a current.

  3. Most used synonyms of the emf are: voltage, difference of potential, and potential. Somebody also uses the name “tension”.

  4. The measurement unit of the potential is the Volt, which is the potential energy of 1 Joule per unit of charge, or 1 Coulomb.

  5. The Volt is a differential measure, not an absolute one. You always measure Volts with respect to a point that we arbitrarily define as 0V, or ground.

  6. Emf can be DC or AC and, correspondingly, it can generate a DC or an AC current when connected to an electrical circuit.

  7. DC voltage is normally generated through chemical reactions in a battery, or can be obtained from the AC through a process called “rectification”.

  8. AC voltage is normally generated with alternators like those used in power plants.

If you want to know more on this topic, I suggest you to watch the companion video of this article, which I posted on my YouTube channel:

https://www.youtube.com/watch?v=vPX-B4xAdtk

Thank you for reading this article and, as always,

Happy Experiments!