DP IB Physics: SL

B. The particulate nature of matter

B.5 Current and circuits

DP IB Physics: SL

B. The particulate nature of matter

B.5 Current and circuits

Understandings
Students should understand:

That cells provide a source of emf

Chemical cells and solar cells as the energy source in circuits

That circuit diagrams represent the arrangement of components in a circuit

Direct current (dc) I as a flow of charge carriers as given by

[math]I = \frac{\Delta q}{\Delta t}[/math]

That the electric potential difference V is the work done per unit charge on moving a positive charge between two points along the path of the current as given by

[math]V = \frac{W}{q}[/math]

The properties of electrical conductors and insulators in terms of mobility of charge carriers

Electric resistance and its origin

Electrical resistance R as given by

[math]R = \frac{V}{I}[/math]

Resistivity as given by

[math]ρ = \frac{RA}{L}[/math]

Ohm’s law

the ohmic and non-ohmic behaviour of electrical conductors, including the heating effect of resistors

electrical power P dissipated by a resistor as given by

[math]P = IV = I^2 R = \frac{V^2}{R}[/math]

The combinations of resistors in series and parallel circuits

Series Circuits	Parallel circuits
[math]I = I_1 = I_2 = I_3 = Ω[/math]	[math]I = I_1 + I_2 + I_3 + Ω[/math]
[math]V = V_1 + V_2 + V_3 + Ω[/math]	[math]V = V_1 = V_2 = V_3 = Ω[/math]
[math]R = R_1 + R_2 + R_3 + Ω[/math]	[math]\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \Omega[/math]

that electric cells are characterized by their emf ε and internal resistance r as given by

[math]ϵ = I(R + r)[/math]

that resistors can have variable resistance.

a) Cells as a Source of EMF
Electromotive force (EMF), denoted as ε, is the energy per unit charge provided by a cell or any energy source in a circuit. It is not a force but rather a voltage that drives charge carriers through a circuit.
[math]ε = \frac{W}{q}[/math]
Where:
– ε = EMF (Volts, V)
– W = Energy supplied (Joules, J)
– q = Charge (Coulombs, C)
⇒ Cells Generate EMF:
Cells convert stored chemical energy into electrical energy, maintaining a potential difference across their terminals. When connected to a circuit, charges move due to this potential difference, creating an electric current.
Figure 1 Cells generate EMF
b) Types of Cells as Energy Sources
⇒ Chemical Cells
Chemical cells (batteries) use chemical reactions to generate EMF. They consist of:
1. Two electrodes – Usually different metals (e.g., Zinc and Copper)
2. Electrolyte – A solution or paste facilitating charge movement
⇒ Types of Chemical Cells
1. Primary Cells (Non-Rechargeable)
– Example: Alkaline batteries (used in remote controls, clocks)
– Irreversible chemical reaction → Once depleted, cannot be recharged
2. Secondary Cells (Rechargeable)
– Example: Lithium-ion batteries (used in smartphones, laptops)
– Reversible reaction → Can be recharged by applying an external voltage
Figure 2 Chemical cells as energy sources
⇒ Solar Cells (Photovoltaic Cells)
Solar cells convert light energy into electrical energy using the photovoltaic effect.
When light photons hit the solar cell, they excite electrons, allowing them to flow as an electric current.
Solar cells are widely used in solar panels, satellites, and renewable energy systems.
Figure 3 Photovoltaic cell

Feature	Chemical Cell	Solar Cell
Energy Source	Chemical reaction	Sunlight
Rechargeable?	Some are rechargeable	Continuous operation in daylight
Common Uses	Batteries, backup power	Solar panels, spacecraft

c) Circuit Diagrams and Components
A circuit diagram is a graphical representation of an electric circuit. It uses symbols to represent electrical components such as:
– Cell/Battery (Energy Source)
– Resistor (R)
– Light Bulb
– Switch
– Ammeter (Measures Current)
– Voltmeter (Measures Voltage)
The voltage provided by the battery creates an electric current that flows through the components.
d) Direct Current (DC) and Charge Flow
Electric current (I) is the rate of flow of charge (q) through a conductor. It is defined by:
[math]I = \frac{∆q}{∆t}[/math]
Where:
– I = Current (Amperes, A)
– Δq = Charge (Coulombs, C)
– Δt = Time (Seconds, s)
⇒ Direction of Current Flow
Conventional Current Flow: Flows from positive (+) to negative (-)
Electron Flow: Actual electrons move from negative (-) to positive (+) due to attraction towards the positive terminal.
⇒ Direct Current (DC) vs. Alternating Current (AC)

Feature	Direct Current (DC)	Alternating Current (AC)
Charge Flow Direction	One direction only	Reverses periodically
Voltage Source	Batteries, solar cells	Power stations (e.g., mains electricity)
Common Uses	Electronics, vehicles	Home power supply

e) Electric Potential Difference (V)
Electric potential difference, denoted as V, is defined as the amount of work W required to move a unit positive charge q from one point to another in an electric field. It is mathematically expressed as:
[math]V = \frac{W}{q}[/math]
Where:
– V = Electric potential difference (volts, V)
– W = Work done in moving the charge (joules, J)
– q = Charge moved (coulombs, C)
⇒ Explanation:
When a charge moves through an electric circuit, energy is transferred. This energy is provided by a power source (like a battery or generator).
The work done per unit charge is what we measure as voltage.
If a charge moves from a higher potential to a lower potential, it releases energy, which can be used to power electrical devices.
Figure 4 Potential difference
f) Conductors and Insulators in Terms of Charge Carrier Mobility
⇒ Conductors:
Electrical conductors are materials that allow electric charges (typically electrons) to move freely through them. This movement of charge carriers (electrons or ions) enables the conduction of electric current. Examples include metals like copper, silver, and aluminum.
⇒ Properties of Conductors:
1. High Charge Carrier Mobility – Free electrons in metals move easily due to weak bonding between them and the metal atoms.
2. Low Electrical Resistance – Conductors offer very little resistance to the flow of electric current.
3. Good Thermal Conductivity – Conductors also transfer heat effectively due to the free movement of electrons.
4. Obeys Ohm’s Law – For most metals, the current flowing is proportional to the voltage applied.
⇒ Insulators:
Electrical insulators are materials that do not allow free movement of charge carriers, meaning they resist the flow of electric current. Examples include rubber, glass, and plastic.
Properties of Insulators:
1. Low Charge Carrier Mobility – Electrons in insulators are tightly bound to atoms, preventing free movement.
2. High Electrical Resistance – Since there are few or no free electrons, insulators oppose the flow of current.
3. Used for Electrical Protection – Insulators are used to coat wires and components to prevent accidental electric shocks.
Figure 5 Difference between conductor, semiconductor, and insulator
h) Electrical Resistance and Its Origin
Electrical resistance (R) is the opposition that a material offers to the flow of electric current. It is a measure of how difficult it is for electrons to move through a conductor.
It is mathematically given by Ohm’s Law:
[math]R = \frac{V}{I}[/math]
Where:
– R = Resistance (ohms, Ω)
– V = Voltage across the material (volts, V)
– I = Current flowing through the material (amperes, A)
⇒ Origin of Electrical Resistance:
Resistance arises due to collisions between moving electrons and atoms within the conductor.
As electrons move through a conductor, they experience resistance from vibrating atoms (lattice structure), impurities, and defects in the material.
The more frequent these collisions, the greater the resistance of the material.
Figure 6 Electrical resistances
⇒ Factors Affecting Resistance:
1. Material of the Conductor – Different materials have different numbers of free electrons; copper has lower resistance than iron.
2. Length of the Conductor (L) – Longer conductors have more resistance.
3. Cross-sectional Area (A) – Wider conductors have lower resistance.
4. Temperature – Resistance increases with temperature for most conductors because atoms vibrate more, increasing collisions.
i) Resistivity (ρ)
Electrical resistivity is an intrinsic property of a material that quantifies how strongly it resists the flow of electric current. It is given by the formula:
[math]ρ = \frac{RA}{L}[/math]
Where:
ρ = Resistivity (ohm-meter, Ωm)
R = Electrical resistance (Ω)
A = Cross-sectional area of the conductor (m²)
L = Length of the conductor (m)
Figure 7 Resistivity
⇒ Explanation:
Resistivity is a property specific to each material, independent of its shape or size.
Conductors like copper and silver have low resistivity, allowing easy flow of electric current.
Insulators like rubber and glass have high resistivity, preventing the flow of current.
⇒ Factors Affecting Resistivity:
1. Material Type – Different materials have different inherent resistivities.
2. Temperature – In metals, resistivity increases with temperature due to increased atomic vibrations.
3. Impurities – More impurities in a material increase resistivity by disrupting electron flow.
j) Ohm’s Law
Ohm’s Law is a fundamental principle in electrical circuits that defines the relationship between voltage (V), current (I), and resistance (R). It states that the current passing through a conductor is directly proportional to the voltage applied across it and inversely proportional to its resistance. Mathematically, it is expressed as:
[math]V = IR[/math]
Where:
V = Voltage (Volts, V)
I = Current (Amperes, A)
R = Resistance (Ohms, Ω)
Explanation:
When a voltage is applied across a conductor, it pushes free electrons through the material, creating an electric current.
The amount of current depends on the material’s resistance. A higher resistance means less current for the same voltage.
⇒ Example:
If a 10V battery is connected across a 5Ω resistor, the current flowing through it is:
[math]I = \frac{V}{R} \\
I = \frac{10}{5} \\
I = 2 \text{ A}[/math]
⇒ Graphical Representation of Ohm’s Law:
For materials that obey Ohm’s Law (ohmic conductors), a graph of voltage (V) versus current (I) is a straight line with a constant slope, where the slope represents the resistance R.
k) Ohmic and Non-Ohmic Behavior of Electrical Conductors
1. Ohmic Conductors:
These materials obey Ohm’s Law, meaning their resistance remains constant regardless of voltage or current.
The V−I graph is a straight line passing through the origin.
Examples: Metals like copper, silver, and aluminum at constant temperature.
2. Non-Ohmic Conductors:
These materials do not obey Ohm’s Law; their resistance changes with voltage or current.
The V−I graph is nonlinear (curved).
⇒ Examples:
Semiconductors (e.g., diodes, transistors) – Their resistance changes based on applied voltage.
Filament lamps – The resistance increases as temperature rises due to heating effects.
Electrolytes and gases – Their conductivity varies based on ionization levels.
⇒ Graphical Comparison:
– Ohmic Conductor: A straight-line V−I
– Non-Ohmic Conductor: A curved V−I graph (e.g., diode’s graph is exponential).
Figure 8 Ohmic and non-ohmic conductor’s graphs
⇒ Heating Effect of Resistors
When current flows through a resistor, electrical energy is converted into heat. This effect is known as Joule Heating and is the basis for devices like electric heaters, irons, and filament bulbs.
⇒ Cause of Heating:
– As electrons move through a resistor, they collide with the atoms of the material.
– These collisions transfer energy to the atoms, increasing their vibrational motion.
– This energy is released as heat, which can warm up the resistor.
⇒ Applications of Joule Heating:
– Electric heaters, kettles, and toasters (where heat is useful).
– Incandescent light bulbs (heat is a byproduct).
– Overheating in electrical circuits (undesirable; requires cooling mechanisms like heat sinks).
l) Electrical Power Dissipated by a Resistor
The power (P) dissipated in a resistor is the rate at which electrical energy is converted into heat. It is given by:
[math]P = IV[/math]
Using Ohm’s Law (V=IR), we can derive two additional formulas:
[math]P = I^2 R \\
P = \frac{V^2}{R}[/math]
Where:
– P = Power (Watts, W)
– I = Current (Amperes, A)
– V = Voltage (Volts, V)
– R = Resistance (Ohms, Ω)
⇒ Explanation of Formulas:
[math]P = IV[/math]: Directly relates power to voltage and current.
[math]P = I^2R[/math]: Shows that power is proportional to the square of the current and resistance (used when current is known).
[math]P = \frac{V^2}{R}[/math]: Shows that power is proportional to the square of the voltage and inversely proportional to resistance (used when voltage is known).
⇒ Example Calculations:
1. Given: I=3A, R=5Ω
[math]P = I^2 R \\
P = (3)^2 (5) \\
P = 45 \text{ W}[/math]
2. Given: V=10V, R=5Ω
[math]P = \frac{V^2}{R} \\
P = \frac{(10)^2}{5} \\
P = 20 \text{ W}[/math]
Power Dissipation in Circuits:
In a series circuit, total resistance increases, and power dissipation is shared across resistors.
In a parallel circuit, each branch dissipates power independently, and total power consumption increases.
m) Combinations of Resistors in Series and Parallel Circuits
Resistors can be connected in two basic configurations: series and parallel. These configurations affect the overall resistance, voltage distribution, and current flow in different ways.
⇒ Series Circuits
In a series circuit, resistors are connected end-to-end, meaning the same current flows through each resistor.
Figure 9 Series circuit
⇒ Current in a Series Circuit
Since there is only one path for current flow, the current remains the same across all resistors:
[math]I = I_1 = I_2 = I_3 = ⋯[/math]
Where:
– I is the total current in the circuit.
– [math]I_1, I_2, I_3[/math] are the currents through individual resistors, which are equal in a series connection.
⇒ Voltage in a Series Circuit
The total voltage supplied by the source is divided across each resistor:
[math]V = V_1 + V_2 + V_3 …[/math]
Where:
– V is the total voltage from the source.
– [math]V_1, V_2, V_3[/math] are the voltages across individual resistors.
Each resistor drops a portion of the total voltage according to Ohm’s Law (V=IR).
⇒ Resistance in a Series Circuit
The total resistance in a series circuit is the sum of the individual resistances:
[math]R_{total} = R_1 + R_2 + R_3 …[/math]
Where:
– [math]R_{total}[/math] is the equivalent resistance of the circuit.
– [math]R_1, R_2, R_3[/math] are the individual resistances.
Figure 10 Resistance in series circuit
⇒ Example:
If three resistors of 2Ω, 4Ω, and 6Ω are connected in series, the total resistance is:
[math]R_\text{total} = R_1 + R_2 + R_3 \\
R_\text{total} = 2 + 4 + 6 \\
R_\text{total} = 12 \, \Omega[/math]
⇒ Characteristics of Series Circuits:
Same current flows through all resistors.
Voltage divides among resistors.
Total resistance increases as more resistors are added.
⇒ Parallel Circuits
In a parallel circuit, resistors are connected across the same two points, meaning each resistor has the same voltage across it.
Current in a Parallel Circuit
The total current in the circuit is the sum of the currents through each resistor:
[math]I = I_1 + I_2 + I_3 + …[/math]
Where:
– I is the total current supplied by the source.
– [math]I_1, I_2, I_3[/math] are the individual branch currents.
Each branch has a different current based on its resistance.
Voltage in a Parallel Circuit
Since all resistors are connected across the same two points, they all have the same voltage:
[math]V = V_1 = V_2 = V_3 = ⋯[/math]
Where:
– V is the total voltage.
- [math]V_1, V_2, V_3[/math] are the voltages across each resistor, which are equal.
⇒ Resistance in a Parallel Circuit
The total resistance in a parallel circuit is found using the reciprocal formula:
[math]\frac{1}{R_\text{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots[/math]
Where:
– [math]R_{\text{total}}[/math] is the equivalent resistance.
– [math]R_1, R_2, R_3[/math] are the individual resistances.
Figure 11 Resistance in parallel circuit
⇒ Example:
If three resistors of 2Ω, 4Ω, and 6Ω are connected in parallel, the total resistance is:
[math]\frac{1}{R_\text{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \\
\frac{1}{R_\text{total}} = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} \\
\frac{1}{R_\text{total}} = \frac{11}{12} \\
R_\text{total} = \frac{12}{11} \\
R_\text{total} \approx 1.09 \, \Omega[/math]
⇒ Characteristics of Parallel Circuits:
– Voltage remains the same across all resistors.
– Current divides among resistors.
– Total resistance decreases as more resistors are added.
n) Electric Cells, EMF, and Internal Resistance
An electric cell is a source of electrical energy that generates voltage through chemical reactions.
⇒ Electromotive Force (EMF, ϵ)
EMF is the total energy supplied per unit charge by the cell. It is measured in volts (V) and represents the ideal voltage output of the battery.
However, real batteries have internal resistance (r), which causes a voltage drop inside the battery itself.
⇒ Internal Resistance and Terminal Voltage
The actual voltage across the external circuit (called terminal voltage, V) is given by:
[math]V = ϵ – Ir[/math]
Where:
– V = Terminal voltage (volts)
– ϵ = EMF of the battery (volts)
– I = Current in the circuit (amperes)
– r = Internal resistance of the battery (Ω)
Using Ohm’s Law, the total voltage can also be expressed as:
[math]ϵ = I(R + r)[/math]
Where:
– R = External resistance of the circuit.
Figure 12 Terminal voltage
⇒ Effects of Internal Resistance:
When no current flows (I=0), terminal voltage equals EMF (V=ϵ).
When current flows, internal resistance reduces terminal voltage.
High internal resistance reduces battery efficiency.
⇒ Example Calculation:
A battery has ϵ=12V and r=1Ω. If a 5Ω resistor is connected, the current is:
[math]I = \frac{\mathcal{E}}{R + r} \\
I = \frac{12}{5 + 1} \\
I = 2 \text{ A}[/math]
The terminal voltage is:
[math]V = \mathcal{E} – I r \\
V = 12 – (2 \times 1) \\
V = 12 – 2 \\
V = 10 \text{ V}[/math]
o) Variable Resistors (Rheostats and Potentiometers)
Resistors can have fixed or variable resistance.
1. Rheostat
A rheostat is a variable resistor used to control current in a circuit.
– Used in dimmer switches, fan regulators, and volume controls.
– Works by adjusting the length of the resistance wire in the circuit.
Figure 13 Rheostat
2. Potentiometer
A potentiometer is a three-terminal variable resistor used to adjust voltage levels.
– Used in audio equipment, sensor circuits, and tuning devices.
– Works by varying the position of a sliding contact along a resistive strip.
Figure 14 Potentiometer