Conduction of electricity - Physics Online Tuition

AS UNIT 2 Electricity and light 2.1 Conduction of electricity Learners should be able to demonstrate and apply their knowledge and understanding of:
a)	The fact that the unit of charge is the coulomb (C), and that an electron’s charge, e, is a very small fraction of a coulomb
b)	The fact that charge can flow through certain materials, called conductors
c)	Electric current being the rate of flow of charge
d)	The use of the equation [math]I = \frac{∆Q}{∆t}[/math]
e)	Current being measured in amperes (A), where [math]A = C s^{-1}[/math]
f)	The mechanism of conduction in metals as the drift of free electrons
g)	The derivation and use of the equation [math]I = nAve[/math] for free electrons

(a) The Unit of Charge and the Electron’s Charge
⇒ The Coulomb (C) as the Unit of Charge
The coulomb (C) is the standard SI unit of electric charge.
It is defined as the amount of charge that flows through a conductor carrying a steady current of 1 ampere in 1 second:
[math]1C = 1A . 1s[/math]
– Here, ampere (A) is the unit of electric current, and second (s) is the unit of time.
A charge of 1 coulomb represents a significant quantity of electric charge.
⇒ Charge of an Electron (e)
The elementary charge, denoted as e, is the magnitude of the electric charge carried by a single electron or proton:
[math]e = 1.602 × 10^{-19} C[/math]
Electrons have a charge of −e, while protons have a charge of +e.
Figure 1 Electric Charges
Since e is a very small fraction of a coulomb, a single electron or proton carries an almost negligible amount of charge.
⇒ Number of Electrons in 1 Coulomb
Because the charge of one electron is so small, it takes a very large number of electrons to make up 1 coulomb of charge:
[math]\begin{gather}
\text{Number of electrons} = \frac{1 \text{ C}}{1.602 \times 10^{-19} \text{ C/electron}} \\
\text{Number of electrons} \approx 6.242 \times 10^{18} \text{ electrons}
\end{gather}[/math]
This illustrates that the coulomb represents a macroscopic (large-scale) quantity of charge.
⇒ Comparison of Coulomb and Elementary Charge
The charge of everyday objects is typically measured in coulombs because the elementary charge (e) is too small to describe macroscopic amounts of charge.
For instance:
– A 1 ampere current flowing for 1 second moves approximately [math]6.242 × 10^{18}[/math]
(b) Charge Flow Through Conductors
⇒ Conductors
Conductors are materials that allow electric charge to flow through them easily.
This property arises from the structure of the material, where certain electrons (called free electrons) are not tightly bound to their atoms and can move freely.
Figure 2 Flow of electric Charges
⇒ How Charge Flows in Conductors
1. Free Electrons in Conductors:
Metals such as copper, silver, and aluminum are excellent conductors because their outermost electrons (valence electrons) are loosely held by their nuclei.
These free electrons form an “electron cloud” or “sea of electrons” that can move through the material when a potential difference (voltage) is applied.
2. Electric Field and Current:
When a conductor is connected to a voltage source (e.g., a battery), an electric field is established inside the material.
This field exerts a force on the free electrons, causing them to drift in the direction opposite to the electric field. This motion of electrons constitutes an electric current.
3. Charge Carriers:
In metals, electrons are the primary charge carriers.
In electrolytes (e.g., saltwater), ions (positive and negative) carry the charge.
⇒ Examples of Conductors
Metals: Copper, silver, gold, and aluminum are common electrical conductors.
Example: Copper is widely used in electrical wiring due to its low resistance and high conductivity.
Graphite: A non-metal that conducts electricity because of the mobility of its delocalized electrons.
⇒ Ionic Solutions (Electrolytes):
Salt dissolved in water allows ions to move freely, enabling the flow of charge.
⇒ Insulators
Insulators are materials in which electrons are tightly bound to their atoms and cannot move freely. Examples include rubber, glass, and plastic.
In insulators, there are no free charge carriers to support the flow of electric charge.
c) Electric Current as the Rate of Flow of Charge
⇒ Definition of Electric Current
Electric current (I) is defined as the rate at which electric charge (Q) flows through a conductor or a circuit.
Mathematically, it is expressed as:
[math]\begin{gather}
I = \frac{\Delta Q}{\Delta t} \\
I = \frac{5 \text{ C}}{10 \text{ s}} \\
I = 0.5 \text{ A}
\end{gather}[/math]
Where:
– I: Electric current (measured in amperes, A).
– ΔQ: Total electric charge that flows through a point (measured in coulombs, C),
– Δt: Time during which the charge flows (measured in seconds, s).
⇒ Units of Electric Current
The ampere (A) is the SI unit of electric current.
It is defined as:
[math]1A = 1 C/s[/math]
This means a current of 1 ampere corresponds to 1 coulomb of charge flowing past a given point in 1 second.
⇒ Understanding Current
Positive and Negative Charges:
– By convention, the direction of current is the flow of positive charge. In most cases, however, current in a metal is due to the movement of electrons, which flow in the direction opposite to the conventional current.
Steady Current:
– If the charge flow is constant over time, the current is said to be steady, and the equation simplifies to:
[math]I = \frac{Q}{t}[/math]
d) Application of the Equation [math]I = \frac{\Delta Q}{\Delta t}[/math]
Example 1: Calculating Current
– If 5 C of charge flows through a circuit in 10 s, the current is:
[math]\begin{gather}
I = \frac{\Delta Q}{\Delta t} \\
I = \frac{5 \text{ C}}{10 \text{ s}} \\
I = 0.5 \text{ A}
\end{gather}[/math]
– This means a current of [math][/math] is flowing through the circuit.
Example 2: Determining Charge
– Rearranging the equation:
[math]\Delta Q = I \cdot \Delta t[/math]
– If a current of 2 A flows for 4 s, the total charge transferred is:
[math]\begin{gather}
\Delta Q = I \cdot \Delta t \\
\Delta Q = (2)(4) \\
\Delta Q = 8 \text{ C}
\end{gather}[/math]
Example 3: Real-Life Scenario
– In a smartphone battery, suppose 2 A flows while charging for 3000 s (50 minutes).
– The total charge delivered is:
[math]\begin{gather}
\Delta Q = I \cdot \Delta t \\
\Delta Q = (1.2)(3000) \\
\Delta Q = 3600 \text{ C}
\end{gather}[/math]
1. Physical Meaning:
Current measures how quickly charge is flowing in a circuit.
A higher current means more charge is passing through a point in the circuit per unit time.
2. Direction of Current:
Conventional current flows from the positive terminal to the negative terminal of a power source.
In reality, electrons (negative charges) flow in the opposite direction.
3. Steady and Time-Varying Current:
For steady currents, the equation [math]I = \frac{\Delta Q}{\Delta t}[/math] is used directly.
For time-varying currents, more advanced calculus-based analysis is required.

e) Current Measured in Amperes (A)
⇒ Definition of Ampere
Electric current (I) is the rate of flow of electric charge (Q) over time (t):
[math]I = \frac{∆Q}{∆t}[/math]
The SI unit of electric current is the ampere (A), defined as:
[math]1A = 1C/s[/math]
This means a current of 1 ampere represents the flow of 1 coulomb of charge through a point in a conductor every second.
⇒ Characteristics of Current Measurement
Conventional Current:
– By convention, current is the flow of positive charge. However, in most metallic conductors, current results from the movement of free electrons, which flow in the opposite direction to the conventional current.
Practical Units:
– Larger currents are often measured in amperes (A).
– Smaller currents can be expressed in:
Milliamperes (mA): [math]1 mA = 10^{-3} A[/math]
Microamperes (μA): [math]1 μA = 10^{-6} A[/math]
f) Mechanism of Conduction in Metals
⇒ Structure of Metals:
Metals consist of a lattice of positive metal ions surrounded by a “sea” of free electrons.
These free electrons are loosely bound to their atoms and can move freely within the metal.
Figure 3 Metallic bonding
⇒ Drift of Free Electrons
1. Random Motion:
In the absence of an electric field, free electrons move randomly within the metal.
There is no net flow of charge because the random motions cancel out.

Figure 4 Drift Velocity
2. Electric Field:
When a potential difference (voltage) is applied across the metal, an electric field is created.
The electric field exerts a force on the free electrons, causing them to drift in the direction opposite to the field (since electrons are negatively charged).
This drift motion of electrons constitutes the electric current in the metal.
Figure 5 Electric Field
⇒ Drift Velocity (v):
The drift velocity is the average velocity of electrons as they move under the influence of the electric field.
It is typically very small (on the order of [math]10^{-4}[/math]m/s) because electrons collide frequently with the metal ions, slowing their progress.
g) Derivation and Use of the Equation [math]I = nAve[/math]
The equation [math]I = nAve[/math] provides a relationship between the macroscopic current (I) and the microscopic properties of the conductor.
⇒ Parameters in the Equation
– I: Electric current (A),
– n: Number density of free electrons (electrons per unit volume, measured in [math]m^{-3}[/math]),
– A: Cross-sectional area of the conductor ( [math]m^2[/math]),
– v: Drift velocity of the electrons (m/s),
– e: Charge of an electron ( [math]1.602 × 10^{-19} C[/math]).
⇒ Derivation of [math]I = nAve[/math]
1. Electric Current:
Current is defined as the total charge passing through the cross-sectional area of the conductor per second:
[math]I = \frac{∆Q}{∆t}[/math]
2. Charge Flow:
The total charge (Q) flowing through the conductor in time Δt is the product of:
– The number of electrons (N),
– The charge of each electron (e):
[math]Q = N . e[/math]
3. Number of Electrons:
The total number of electrons passing through the cross-sectional area is:
[math]N = n ⋅ A ⋅ v ⋅ Δt[/math]
Where:
– n: Number density of free electrons,
– A: Cross-sectional area of the conductor,
– v: Drift velocity,
– Δt: Time period.
4. Substituting for Q:
[math]Q = n ⋅ A ⋅ v ⋅ Δt ⋅ e.[/math]
5. Current (I):
Dividing by Δt to find the current:
[math]I = \frac{∆Q}{∆t} = n ⋅ A ⋅ v ⋅ Δt ⋅ e.[/math]
⇒ Physical Meaning of the Equation
[math]I = nAve[/math] relates the macroscopic current (I) to the microscopic properties of the conductor:
– n: Determines how many free electrons are available to carry charge.
– A: The larger the cross-sectional area, the more electrons can pass through at once.
– v: The drift velocity of electrons is directly proportional to the current.
– e: The charge of each electron defines the contribution of each electron to the total current.
⇒ Practical Applications of [math]I = nAve[/math]
1. Calculating Drift Velocity:
If the current, number density, and conductor area are known, the drift velocity can be calculated:
[math]v = \frac{I}{nAe}[/math]
2. Analyzing Conductors:
The equation can be used to compare the conductivity of different materials by considering their number density (n).
3. Understanding Current Flow:
Provides insight into the microscopic processes that generate a macroscopic current in conductors.
⇒ Example Calculation
A copper wire has a cross-sectional area of [math]A = 1 mm^2 = 1 × 10^{-6} m^2[/math], carries a current of [math]I = 3A[/math], and has a number density [math]n = 8.5 × 10^{28} m^{-3}[/math]. The drift velocity (v) can be calculated as:
[math]v = \frac{I}{n A e}[/math]
Substituting:
[math]\begin{gather}
v = \frac{I}{n A e} \\
v = \frac{3}{(8.5 \times 10^{28}) (1 \times 10^{-6}) (1.602 \times 10^{-19})} \\
v \approx 2.2 \times 10^{-4} \text{ m/s}
\end{gather}[/math]
The drift velocity is very small, which highlights that even small velocities can result in large currents due to the high number density of electrons in metals.