Electric current, charge, potential difference and electromotive force

Introduction to Electric Current, Charge, and Energy

Welcome to one of the most fundamental and exciting areas of physics: Electricity! Whether you are charging your phone, turning on a light switch, or powering an electric car, you are relying on the movement of invisible electric charges. In this chapter of your CCEA AS 1 Physics course, we will explore what electricity actually is at a microscopic level, how we measure it, and how energy is transferred around circuits.

Don't worry if electricity has felt abstract or confusing in the past. We will break every concept down into clear, bite-sized pieces with everyday analogies to help you build total confidence!

1. Electric Charge: The Foundation of Electricity

Before we can understand electric current, we must first understand electric charge (symbol \(Q\) or \(q\)). Charge is a fundamental property of matter, just like mass.

Key Facts About Charge

• Two Types of Charge: Charges can be either positive (\(+\)) or negative (\(-\)). Like charges repel one another, while opposite charges attract.
• The Unit of Charge: Electric charge is measured in Coulombs (\(\text{C}\)).
• The Elementary Charge: Charge is quantised, meaning it comes in discrete, indivisible packets. The smallest stable unit of free charge is the charge on a single proton or electron, called the elementary charge, \(e\):
\(e = 1.60 \times 10^{-19}\text{ C}\)

Quantisation of Charge Equation

Any measurable charge \(Q\) is simply a whole-number multiple of the elementary charge \(e\):

\(Q = \pm n e\)

Where:
• \(Q\) = total charge in Coulombs (\(\text{C}\))
• \(n\) = number of charge carriers (an integer: \(1, 2, 3, \dots\))
• \(e\) = elementary charge (\(1.60 \times 10^{-19}\text{ C}\))

Example: A single electron carries a charge of \(-1.60 \times 10^{-19}\text{ C}\), whereas a single proton carries \(+1.60 \times 10^{-19}\text{ C}\).

What are Charge Carriers?

Charge does not move on its own; it is carried by particles called charge carriers:
• In Metals (Conductors): The charge carriers are delocalised (free) electrons.
• In Electrolytes (Liquids/Solutions): The charge carriers are positive and negative ions.

Key Takeaway: Charge is measured in Coulombs (\(\text{C}\)) and is always a whole-number multiple of the elementary charge \(e = 1.60 \times 10^{-19}\text{ C}\).

2. Electric Current: Charge on the Move

Electric current (symbol \(I\)) is defined as the rate of flow of electric charge past a point in a circuit.

The Fundamental Current Equation

\(I = \frac{\Delta Q}{\Delta t}\)

Where:
• \(I\) = electric current in Amperes or Amps (\(\text{A}\))
• \(\Delta Q\) = charge flowing past a cross-section in Coulombs (\(\text{C}\))
• \(\Delta t\) = time taken in seconds (\(\text{s}\))

Defining the Ampere

From the formula, we can define the SI unit of current:
One Ampere (\(1\text{ A}\)) is the current flowing when a charge of one Coulomb (\(1\text{ C}\)) passes a given point in a circuit in one second (\(1\text{ s}\)).
Therefore: \(1\text{ A} = 1\text{ C s}^{-1}\).

Conventional Current vs. Electron Flow

This is a classic area where students can get caught out, but the history is very simple:

• Conventional Current: Defined historically by scientists before the electron was discovered. It is the direction a positive charge would move: from the positive (\(+\)) terminal to the negative (\(-\)) terminal. All standard circuit diagrams and formulas use conventional current.
• Electron Flow: The actual physical movement of electrons in a metal conductor: from the negative (\(-\)) terminal to the positive (\(+\)) terminal (since like charges repel, electrons are pushed away from the negative terminal).

Did you know? Even though electrons physically move toward the positive terminal, we still draw circuit arrows pointing from positive to negative. Don't worry—calculations work identically either way!

Key Takeaway: Current is the rate of flow of charge (\(I = \frac{\Delta Q}{\Delta t}\)), measured in Amperes (\(\text{A}\)). Conventional current flows \(+ \rightarrow -\), while electrons physically drift \(- \rightarrow +\).

3. Microscopic Model of Current: Drift Velocity

When an electrical circuit is switched on, the light bulb lights up almost instantaneously. However, individual electrons inside the wire are actually moving surprisingly slowly! Let's understand why.

Random Thermal Motion vs. Net Drift

Inside a metal, free electrons are constantly zooming around at very high thermal speeds (around \(10^5\text{ m s}^{-1}\) to \(10^6\text{ m s}^{-1}\)) in random directions, constantly colliding with vibrating metal lattice ions. Because this motion is random, there is no net movement in any direction.

When an electric field is applied across the wire (by connecting a battery), a small net force acts on the electrons. They begin to slowly drift in one overall direction. This average forward velocity is called the drift velocity (symbol \(v\)), and it is typically less than \(1\text{ mm s}^{-1}\)!

The Transport Equation: \(I = n A v e\)

We can relate the macroscopic electric current \(I\) to the microscopic motion of charge carriers using the transport equation:

\(I = n A v e\)

Where:
• \(I\) = electric current (\(\text{A}\))
• \(n\) = number density of charge carriers: the number of free electrons per unit volume (\(\text{m}^{-3}\))
• \(A\) = cross-sectional area of the conductor (\(\text{m}^2\))
• \(v\) = mean drift velocity of the charge carriers (\(\text{m s}^{-1}\))
• \(e\) = elementary charge (\(1.60 \times 10^{-19}\text{ C}\))

Step-by-Step Derivation of \(I = n A v e\)

1. Consider a cylinder of wire with cross-sectional area \(A\) and length \(L\).
2. The volume of this section of wire is: \(\text{Volume} = A \times L\).
3. If there are \(n\) free electrons per unit volume, the total number of free electrons in this section is: \(N = n \times \text{Volume} = n A L\).
4. The total charge \(\Delta Q\) in this section is: \(\Delta Q = N \times e = n A L e\).
5. All of these electrons will leave the cylinder in a time \(\Delta t = \frac{L}{v}\), where \(v\) is the drift velocity.
6. Using the definition of current \(I = \frac{\Delta Q}{\Delta t}\):
\(I = \frac{n A L e}{\frac{L}{v}} = n A v e\).

Understanding Number Density (\(n\))

The value of \(n\) determines how well a material conducts electricity:
• Conductors (e.g., Copper): Very high \(n\) (roughly \(10^{28}\text{ to } 10^{29}\text{ m}^{-3}\)). Because \(n\) is huge, a large current can flow with a very small drift velocity \(v\).
• Semiconductors (e.g., Silicon): Moderate \(n\) (roughly \(10^{15}\text{ to } 10^{19}\text{ m}^{-3}\)). To carry the same current, electrons must move faster, or \(n\) can be increased by raising the temperature or doping.
• Insulators (e.g., Plastic, Glass): Extremely low \(n \approx 0\). There are virtually no free electrons to carry a current.

Key Takeaway: The transport equation is \(I = nAve\). While electrical signals travel near the speed of light, the actual drift velocity \(v\) of electrons is very slow (fractions of a millimetre per second).

4. Potential Difference and Electromotive Force

To make charges move around a circuit, energy must be supplied to them and transferred from them. This brings us to two of the most important concepts in circuit theory: Electromotive Force (e.m.f.) and Potential Difference (p.d.).

Electromotive Force (e.m.f., symbol \(\mathcal{E}\) or \(E\))

Definition: Electromotive force is the energy transferred from chemical (or other) energy forms into electrical energy per unit charge passing through the source.

\(\mathcal{E} = \frac{W}{Q}\)

• Source: Provided by energy sources such as batteries, power supplies, solar cells, and dynamos.
• Everyday Analogy: Think of the battery as a ski lift that takes skiers (charges) from the bottom of the mountain and gives them gravitational potential energy (electrical energy) to reach the top.

Potential Difference (p.d., symbol \(V\))

Definition: Potential difference is the energy transferred from electrical energy to other forms of energy (such as heat, light, or sound) per unit charge passing between two points.

\(V = \frac{W}{Q}\)

• Location: Measured across circuit components that transfer electrical energy, such as resistors, lamps, and motors.
• Everyday Analogy: Think of components as the ski slopes and obstacles where skiers spend their stored energy coming back down.

The Unit of Potential Difference and e.m.f.: The Volt (\(\text{V}\))

Both e.m.f. and p.d. are measured in Volts (\(\text{V}\)).
One Volt (\(1\text{ V}\)) is defined as one Joule of energy transferred per Coulomb of charge:
\(1\text{ V} = 1\text{ J C}^{-1}\).

Comparison: e.m.f. vs. Potential Difference

• e.m.f. (\(\mathcal{E}\)): Energy converted to electrical energy from other forms per unit charge. Associated with energy suppliers (batteries, generators).
• p.d. (\(V\)): Energy converted from electrical energy to other forms per unit charge. Associated with energy users (resistors, bulbs, heaters).

Key Takeaway: Both e.m.f. and p.d. are defined by \(W/Q\) and measured in Volts (\(\text{J C}^{-1}\)). e.m.f. puts energy in to the circuit; p.d. takes energy out of the circuit.

5. Electrical Energy and Electrical Power

Now that we have linked charge, current, and potential difference, we can calculate the electrical work done and power dissipated in circuits.

Electrical Energy / Work Done (\(W\))

From the definition of potential difference, \(W = Q V\).
Since charge is related to current by \(Q = I t\), we can substitute this in to get:

\(W = I V t\)

Where:
• \(W\) = energy transferred in Joules (\(\text{J}\))
• \(I\) = current in Amperes (\(\text{A}\))
• \(V\) = potential difference in Volts (\(\text{V}\))
• \(t\) = time in seconds (\(\text{s}\))

Electrical Power (\(P\))

Power is defined as the rate of energy transfer: \(P = \frac{W}{t}\).
Substituting \(W = I V t\) into the power equation gives the fundamental power formula:

\(P = I V\)

Where:
• \(P\) = power in Watts (\(\text{W}\)), where \(1\text{ W} = 1\text{ J s}^{-1}\)
• \(I\) = current in Amperes (\(\text{A}\))
• \(V\) = potential difference in Volts (\(\text{V}\))

Memory Trick: Remember "PIV" (\(P = I \times V\)) to easily recall the relationship between Power, Current, and Potential Difference!

Key Takeaway: Energy transferred is \(W = I V t\), and electrical power is \(P = I V\), measured in Watts (\(\text{W}\)).

6. Summary and Quick Review

Formula Summary Table

• Total Charge: \(Q = n e\)
• Electric Current: \(I = \frac{\Delta Q}{\Delta t}\)
• Microscopic Current (Drift Velocity): \(I = n A v e\)
• Potential Difference & e.m.f.: \(V = \frac{W}{Q}\) and \(\mathcal{E} = \frac{W}{Q}\)
• Electrical Work / Energy: \(W = I V t = Q V\)
• Electrical Power: \(P = I V\)

Common Mistakes to Avoid in Exams

1. Confusing \(n\) with total number of electrons: In \(I = nAve\), \(n\) is the number density (free electrons per unit volume, unit \(\text{m}^{-3}\)), not the total number of electrons.
2. Mixing up e.m.f. and p.d. definitions: Always specify the direction of energy conversion. e.m.f. is other forms \(\rightarrow\) electrical; p.d. is electrical \(\rightarrow\) other forms.
3. Forgetting to convert units: Cross-sectional areas are often given in \(\text{mm}^2\). Remember: \(1\text{ mm}^2 = 1 \times 10^{-6}\text{ m}^2\). Time must always be in seconds (\(\text{s}\)), not minutes or hours.
4. Stating drift velocity is near the speed of light: The electric field/signal propagates extremely fast, but the charge carriers themselves move very slowly (typically \(10^{-4}\text{ m s}^{-1}\)).