3.2 Resistance
Resistance is an electrical property that opposes the flow of electric current through a conductor. Conductors are classified into ohmic conductors, which obey Ohm’s Law, and non-ohmic conductors, which do not. The resistance of a conductor is influenced by physical factors such as length, cross-sectional area, material resistivity, and temperature. Understanding resistance allows for the analysis of series and parallel combination circuits, as well as the practical selection of materials for heating elements, household wiring, and high-efficiency superconducting devices.
Learning Objectives
- Differentiate between ohmic conductors and non-ohmic conductors using their current-potential difference (\(V\)-\(I\)) graphs.
- Solve numerical problems involving current, potential difference, and effective resistance in series, parallel, and combination circuits.
- Identify and describe the factors that affect the resistance of a wire (length, cross-sectional area, resistivity, and temperature).
- Define resistivity of a conductor (\(\rho\)) and use the formula \(R = \frac{\rho l}{A}\) to solve related problems.
- Compare the resistivity and characteristics of non-conductors, semiconductors, conductors, and superconductors.
- Describe practical applications of materials with different resistivities in daily appliances and technology.
Summary of Factors Affecting Wire Resistance and Material Resistivity Classification
Link the physical parameters \(l\), \(A\), and \(\rho\) to the formula \(R = \frac{\rho l}{A}\), and distinguish between ohmic and non-ohmic \(V\)-\(I\) characteristic graphs.
Short Explanation
Ohmic Conductor
An ohmic conductor obeys Ohm’s Law. Its graph of \(V\) against \(I\) is a straight line passing through the origin. Example: constantan wire at constant temperature.
Non-Ohmic Conductor
A non-ohmic conductor does not obey Ohm’s Law. Its graph of \(V\) against \(I\) is a curve with an increasing gradient. Example: filament bulb.
Series Circuit
Current is the same through each resistor: \(I = I_1 = I_2 = I_3\). Total potential difference: \(V = V_1 + V_2 + V_3\). Effective resistance: \(R = R_1 + R_2 + R_3\).
Parallel Circuit
Total current: \(I = I_1 + I_2 + I_3\). Potential difference is the same across each branch: \(V = V_1 = V_2 = V_3\). Effective resistance: \(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\).
Resistance of a wire
\(R = \dfrac{\rho l}{A}\)Where \(R\) is resistance in \(\Omega\), \(\rho\) is resistivity in \(\Omega\text{ m}\), \(l\) is length in \(\text{m}\), and \(A\) is cross-sectional area in \(\text{m}^2\).
| Category | Definition / Characteristic | Resistivity Level | Examples | Applications |
|---|---|---|---|---|
| Non-conductor (Insulator) | Material that does not conduct electricity; good insulator | Highest resistivity | Plastic, wood | Plastic cover of fuse box, cable casing |
| Semiconductor | Conducts electricity better than an insulator but not as well as a conductor | Between non-conductor and conductor | Silicon, germanium | Silicon chips, microelectronics |
| Conductor | Material that conducts electricity readily | Low resistivity | Copper, iron, carbon, nichrome | Connecting wires (copper), heating coils (nichrome) |
| Superconductor | Conducts electricity without any resistance at or below its critical temperature | Zero resistivity at critical temperature (\(T_c\)) | Caesium (\(T_c \le 1.5\text{ K}\)) | Superconducting coils in MRI machines |
Try to Answer First
Answer in your mind, then press “Check Answer”.
What happens to the resistance of a filament bulb as the current flowing through it increases? Explain why.
Check Answer
Two identical constantan wires, each of length \(50\text{ cm}\) and resistance \(4\ \Omega\), are connected in parallel. What is the effective resistance of the combination?
Check Answer
A wire of length \(2.0\text{ m}\) with a cross-sectional area of \(1.0 \times 10^{-6}\text{ m}^2\) has a resistance of \(3.4\ \Omega\). What is the resistivity of the wire material?
Check Answer
Common Mistakes
- Using wire diameter (\(d\)) directly into \(A = \pi r^2\) without dividing by 2 first.
- Forgetting to convert lengths from centimetres (\(\text{cm}\)) to metres (\(\text{m}\)) and cross-sectional areas from \(\text{mm}^2\) to \(\text{m}^2\) (\(1\text{ mm}^2 = 1 \times 10^{-6}\text{ m}^2\)).
- Correctly calculating \(\frac{1}{R_p}\) (e.g. \(\frac{4}{12}\)) but forgetting to invert the fraction to get \(R_p\) (\(R_p = \frac{12}{4} = 3\ \Omega\)).
- Treating resistance (\(R\), measured in \(\Omega\)) and resistivity (\(\rho\), measured in \(\Omega\text{ m}\)) as the same quantity.
Concept Misunderstandings
Standard Wire Gauge (s.w.g.) increases as wire thickness increases.
A higher s.w.g. number actually denotes a thinner wire with a smaller diameter and cross-sectional area.
Tungsten is inherently non-ohmic under all conditions.
Tungsten wire is an ohmic conductor when its temperature is kept constant. It only displays non-ohmic behavior in a bulb because its temperature rises significantly during operation.
Superconductors reduce resistance to a tiny value close to zero.
At or below the critical temperature (\(T_c\)), the electrical resistivity of a superconductor is strictly zero, meaning current flows without any loss of energy as heat.
Summary
- Ohmic conductors have constant resistance and a straight \(V\)-\(I\) graph through the origin; non-ohmic conductors (like filament bulbs) have increasing resistance as temperature rises, giving a curved \(V\)-\(I\) graph.
- Circuit analysis: Series circuits share current (\(I\) is constant); parallel circuits share potential difference (\(V\) is constant).
- Factors affecting wire resistance: \(R \propto l\), \(R \propto \frac{1}{A}\), and \(R \propto \rho\), yielding the relationship \(R = \frac{\rho l}{A}\).
- Resistivity (\(\rho\)) is an intrinsic material property measured in \(\Omega\text{ m}\).
- Material classification: Non-conductors have the highest resistivity, semiconductors are intermediate, conductors have low resistivity, and superconductors have zero resistivity at or below critical temperature \(T_c\).
Short Activity
Objective: Test your understanding of resistance, resistivity, and material classification.
A. Objective Quiz
1 Which of the following statements correctly describes a tungsten filament bulb during operation?
2 What happens to the resistance of a constantan wire if its length is doubled and its cross-sectional area is halved at constant temperature?
B. Fill in the Blanks
3 The SI unit for the resistivity of a conductor is the .
4 The temperature at which the resistivity of a superconductor becomes zero is called the temperature.
C. Matching / Drag and Drop
Match each electrical material or component to its correct characteristic or function based on its resistivity. If using a phone, tap the answer first, then tap the matching box.
