Unizor - Physics4Teens - Electromagnetism - Electric Current - Speed of ...

Notes to a video lecture on http://www.unizor.com

Speed of Electrons

Electric current is a movement of electrons. We know from experience that, when we turn on the switch, the lights in the room are lit practically immediately. Does it mean that electrons from one terminal of a switch go to the light fixture and back to another terminal of a switch that fast?
No.

Let's calculate the real speed of electrons, first, theoretically and then in some practical case.

Assume, the amperage of the electric current going through a wire, that is the number of coulombs of electric charge going through a wire per second, is I, and the wire has cross-section area A.
Assume further that we know all the physical characteristics of a material our wire is made of, which will be introduced as needed.

Based on this information, our plan is to determine the number of electrons going through the wire per unit of time and, knowing the density of electrons per linear unit of length in the wire, determine the linear speed of these electrons.

Obviously, to determine the linear density of electrons, we will need physical characteristics of a wire.

The number of electrons going through a wire per unit of time is easily determined from the amperage I. Since I represents the number of coulombs of electric charge going through a wire per second, we just have to divide this by the charge of a single electron in coulombs e=−1.60217646·10⁻¹⁹C.
So, the number of electrons going through a wire per second is
N_e = I / e.

Now we will determine the linear density of electrons in the wire.

First of all, we have to know how many active electrons in an atom of material our wire is made of participate in the transfer of electric charge, because not all electrons of each atom are freely moving in the electric field, but only those on the outer orbit. Let's assume, this number is n_e.
Using this number, we convert the number of electrons participating in the transfer of electric charge N_e into the number of atoms N_atoms in that part of a wire occupied by all electrons transferring the given charge per second I.
N_atoms = N_e / n_e = I / (n_e·e).

Next, from the number of atoms we will find their mass and, using the density of wire material, the volume.
Dividing the volume by a cross-section of a wire, we will get the length of a segment of wire occupied by those electrons transferring charge per second, which is the speed of electrons or drift.

Knowing the number of atoms, to get to their mass, we will use the Avogadro number
N_A=6.02214076·10²³
that represents the number of particles in one mole of a substance. One mole of material our wire is made of is the number of grams equal to its atomic mass m_a, known for any material used for a wire.
If N_A atoms have mass of m_a gram, N_atoms have total mass
M_atoms = m_a · N_atoms / N_A =
= m_a·I / (n_e·e·N_A).

Knowing the total mass M_atoms of all atoms that contain all electrons traveling through a wire per second, we calculate the volume V_atoms by using the density ρ of a material the wire is made of.
V_atoms = M_atoms / ρ =
= m_a·I / (n_e·e·N_A·ρ)

Dividing the volume V_atoms by the cross-section area of a wire, we will get the length of the wire L occupied by electrons traveling through it in one second
L = V_atoms / A =
= m_a·I / (n_e·e·N_A·ρ·A)
where
m_a - atomic mass of wire's material (assuming it's one atom molecules, like copper)
I - electric current - amperage in the wire
n_e - number of active electrons in each atom of wire's material that participate in the transfer of electric charge
e - electric charge of one electron
N_A - number of atoms in 1 mol of conducting material - Avogadro Number
ρ - density of wire's material
A - cross-section area of a wire

The above formula represents the length of a wire occupied by all active electrons traveling through it during one second, which is the speed of movement of electrons making up an electric current, called drift.

Let get to practical examples.

Assume, the voltage or the difference of electric potential E between two ends of a copper wire is maintained at 110V (standard voltage for apartments in the USA).
This wire connects a lamp that consumes 120W of electric power P (or wattage).
Let the cross-section area of a wire be 3 mm².

First of all, let's calculate the amount of electricity moving through the wire per unit of time - amperage I.
As we know, the amperage I, multiplied by voltage E, is the electric power P (wattage).
Therefore,
E = 110V
P = 120W
I = P/E = 60W/110V ≅ 1.09A

The atomic number of copper is Z=29. It means, the atom of copper has 29 protons in the nucleus and 29 electrons orbiting a nucleus. These 29 electrons are in four orbits: 2+8+18+1.
The outer orbit has only one electron that participates in the movement of electric charge, so the number of active electrons in an atom of copper is
n_e=1.

The nucleus of an atom of copper has 29 protons and 34 or 36 neutrons, its atomic mass is
m_a = 63.546g/mol

The electric charge of one electron is
e = −1.60217646·10⁻¹⁹C.

The Avogadro number is
N_A=6.02214076·10²³

Density of copper is
ρ = 8.96 g/cm³ = 0.00896 g/mm³

Cross-section area of a wire is
A = 3 mm²

Using the formula above with values listed, we obtain
L = m_a·I / (n_e·e·N_A·ρ·A)
we get
L ≅ 0.0267 mm/sec
This is the speed of electrons traveling along a copper wire in this case.
Pretty slow!