What characterizes an electric current? We know that in a good electrical conductor there
is a crystal lattice formed by the atoms that make it up. We also know that the only elements of
atoms that have the ability to move within the conductor are the electrons. And for there
to be a flow of electrons within the conductor we must have an electric field
oriented within it. It is this electric field, interacting with the electron charge, that
gives sufficient electrical force for the electrons to move within the conductor. Thus we can
understand electric current as a movement of electric charges. Let's look at
this situation in more detail.
We have already seen in chapter Electric Field a eq. 71-02 which relates electric force,
electric charge and electric field. For the sake of didactic let's repeat here.
eq. 71-02
We must understand that within a conductor, when an electric field acts, an electric force arises that causes the
electrons to accelerate, gaining kinetic energy. In the displacement of electrons there are several collisions
with metal ions. In these collisions the electrons transfer a significant portion of kinetic energy, contributing to
the increase of the thermal energy of the metal. In turn, this transfer of energy causes the metal to rise in temperature.
After the collision, the electron acquires a new velocity and direction in its path. And so, the cycle begins again.
This causes the electron to acquire a
average velocity other than zero. All this due to the electric field and which gives rise to
the so-called drift speed. Based on the principles of mechanics, we know that the speed of an object is
proportional to its acceleration and travel time. Thus, we can write:
eq. 73-01
Where the barred variable represents the average value of the magnitude, e is the electron charge and m is
the electron mass. In eq. 73-01 we assume that the initial electron velocity is
null , which is plausible when E = 0 . We are calling the variable △t—
as mean time between collisions , which depends on the metal
temperature, but does not depend on the intensity of the electric field, E . Then we can write for the drift speed the equation:
eq. 73-02
It should be noted that each conductive material has a charge density , also known as
electron density is symbolized by the variable ne. This variable represents
the number of carriers per volume unit and is a characteristic of each material. The Table 73-01
relates some of the major metals to the available conduction electron density. Note that the values are
very close to each other.
Considering a piece of electrical wire of length L and using the principles of mechanics , we can write that L = Vd △t. We can approximate a piece of electrical wire of a certain length by a cylinder of area A and length L.
So the volume of this cylinder is given by V = A L . On the other hand we can write
L = Vd△t, where △t is the time interval that electrons travel
L. Thus, we can relate the total number of electrons, Ne, contained in volume
V, according to eq. 73-03.
eq. 73-03
And by definition, electric current is the rate of amount of charge that passes through a section of conductor
per unit of time and its unit of measure is coulomb / second or ampère . In addition, we define the
conventional sense of electric current as that which flows from the highest potential to the lowest potential.
To determine the total charge of all electrons passing through the wire, we use the fact that Q = e Ne,
where e is the charge of the electron and its value in coulomb is worth e = 1.6 x 10- 19C.
Then we can write that:
eq. 73-04
Making the simplifications possible, we arrive at the electric current equation:
eq. 73-05
Electricity Sense Convention
It is conventional that the direction of the electric current has the same direction as the
electric field , that is, it is as if the electric current is the result of a positive
charge movement. For a macroscopic theory there is no distinction whether what moves inside a
conductor is negative or positive charges. Thus, once a convention is established the results
will be correct.
Another interesting quantity to study is the so-called current density, J , defined as the relation between the
electric current in ampère that runs through a wire and the section area wire cross-section in square meter.
eq. 73-06
The eq. 73-06 can also be expressed in easily measurable quantities if we consider a piece of wire of length
L and cross-sectional area. A. We know that the law of Ohm is given by V = R I, where
V is the potential difference between the ends of the wire. But we also know that V = E L. And J = I / A. So we can
relate all these equations and write:
eq. 73-07
Working algebraically at eq. 73-07, we get the following expression:
eq. 73-08
Where L is the wire length in m , A is the cross section of the wire in m2
and R is the electrical resistance of the wire in Ω . Note that the expression in parentheses
will have as a unit of measure Ω-1m-1, which is exactly the unit of
measurement for conductivity, which we will study in the next item. We should note that this justifies the eq. 73-11.
As we saw in the previous item, the current density is given by eq. 73-06 , where its direct relation with the
electron drift velocity is evident, Vd. Putting this equation together with eq. 73-02, we can write that:
eq. 73-09
Note that eq. 73-09 shows that for a given electric field intensity, the higher current density if the conducting material has a large electron density, ne, or a long collision time. Thus, the higher the value of these variables, the better the conductive material. Then we can define
conductivity, σ, from a material like:
eq. 73-10
Comparing eq. 73-08 with eq. 73-11 it is possible to relate the quantities involved with the definition of conductivity. From this definition we can rewrite eq. 73-06 as follows:
eq. 73-11
From this equation we can get some important information, such as:
Every electrical current is caused by an electric field that exerts forces on the charge carriers.
The density of electric current and, consequently, electric current, depend linearly on the intensity of the electric field.
The density of electric current also depends on the electrical conductivity of the material.
It is noteworthy that the conductivity is affected by the temperature of the material, the crystalline structure and the impurities contained on it.
Another value of great practical utility is the so-called resistivity, ρ, of the material, which is defined as the inverse of conductivity.. Then:
eq. 73-12
The electrical resistivity of a material represents the difficulty that electrons have in moving within a conductor due to the application of an electric field over the material.
The more known and most commonly used unit of measurement for the resistivity é o Ω m. Since conductivity is the inverse of resistivity, so its unit of measurement is the Ω-1 m-1.
Considering that the electric current is a consequence of a movement of electrons within a
conductor due to the electric field between its ends, we can say that:
"The rate at which electrons exit from one end of the conductor is exactly equal to
rate at which electrons enter the other end of the conductor."
This means that electric current cannot be consumed or created within a
conductor. So, what is used by the circuit is the kinetic energy that the electrons have, being the same
dissipated in collisions with the conductor wire mesh ions, generating as a consequence, an increase in
conductor temperature. So, we conclude that:
Principle of Conservation of Electric Current
"The current is the same everywhere in a conductor carrying an electric current."
