Welcome to Resistance and Resistivity!
Have you ever wondered why a light bulb gets warm when it is switched on, or why super-long extension cables need to be thick? The answers lie in the concepts of resistance and resistivity. In this chapter, we will explore how electrical conductors resist the flow of electric charge, how temperature alters this behaviour, and how some materials can even conduct electricity with zero resistance at all.
Don't worry if physics equations sometimes look intimidating at first! We will break everything down step-by-step using clear analogies, straightforward maths, and practical tips to help you ace your CCEA AS 1 exam.
1. Understanding Electrical Resistance
When an electric current flows through a conductor (such as a copper wire), free electrons drift through a lattice of fixed positive metal ions. As these electrons move, they collide with the vibrating ions. These collisions oppose the flow of charge — this opposition is what we call electrical resistance.
Definition and Formula
The resistance \(R\) of a component is defined as the ratio of the potential difference \(V\) across it to the current \(I\) flowing through it:
\(R = \frac{V}{I}\)
Where:
• \(R\) is the resistance measured in ohms (\(\Omega\)).
• \(V\) is the potential difference (p.d.) measured in volts (\(\text{V}\)).
• \(I\) is the electric current measured in amperes (\(\text{A}\)).
From the definition, \(1\ \Omega = 1\ \text{V A}^{-1}\). This means a component has a resistance of \(1\ \Omega\) if a potential difference of \(1\ \text{V}\) causes a current of \(1\ \text{A}\) to flow through it.
Water Analogy: Imagine water flowing through a garden pipe. The water pressure represents potential difference (\(V\)), the volume of water flowing per second is the current (\(I\)), and any pebbles or narrowing in the pipe represent resistance (\(R\)). More pebbles mean less water flows for the same pressure!
Key Takeaway: Resistance is the opposition to current flow. Always define resistance using the formula \(R = \frac{V}{I}\) (the ratio of p.d. to current), not just as "slowing down current".
2. Ohm's Law
Many conductors obey a fundamental relationship discovered by Georg Ohm.
Ohm's Law Statement: The current through a metallic conductor is directly proportional to the potential difference across it, provided physical conditions (especially temperature) remain constant.
Mathematically, we write this as:
\(I \propto V\quad\text{or}\quad V \propto I\quad (\text{at constant temperature})\)
Common Exam Pitfall to Avoid:
Warning: Do not write "\(V = IR\) is Ohm's Law". The equation \(R = \frac{V}{I}\) is simply the definition of resistance (which applies to all components). Ohm's Law is specifically the proportionality between \(I\) and \(V\) when temperature is constant.
Key Takeaway: If a conductor obeys Ohm's Law, it is called an Ohmic conductor. Its resistance remains constant as voltage changes, provided temperature does not change.
3. Current-Voltage (\(I\)-\(V\)) Characteristics
An \(I\)-\(V\) characteristic graph shows how the current through a component changes as the potential difference across it is varied.
A. Metallic Conductor at Constant Temperature (Ohmic Conductor)
• Graph shape: A straight line passing directly through the origin \((0,0)\).
• Explanation: Resistance is constant because the gradient (\(\frac{I}{V} = \frac{1}{R}\)) is constant. Both positive and negative voltages give identical linear behavior.
B. Filament Lamp
• Graph shape: An 'S'-shaped curve passing through the origin, which curves and flattens out at higher voltages.
• Explanation: As voltage increases, current increases. The increased collisions between electrons and ions transfer energy to the lattice, causing the filament to heat up. Higher temperature causes the positive metal ions to vibrate with greater amplitude, leading to more frequent collisions with electrons. Therefore, resistance increases as current and temperature rise.
C. Semiconductor Diode
• Forward Bias (positive voltage): Very little current flows until a threshold voltage is reached (typically around \(0.6\ \text{V}\) to \(0.7\ \text{V}\)). Beyond this threshold, current increases rapidly for very small increases in voltage, meaning resistance drops to a very low value.
• Reverse Bias (negative voltage): The diode has extremely high resistance, allowing virtually zero current to flow.
D. Negative Temperature Coefficient (NTC) Thermistor
• Graph shape: A curve that gets steeper as voltage increases.
• Explanation: As current flows, the thermistor warms up. In semiconductors, higher temperatures release many more charge carriers (free electrons). This large increase in charge carriers outweighs the effect of increased lattice vibrations, causing the overall resistance to decrease as temperature increases.
Quick Review Box:
• Ohmic wire: Constant \(R\) (straight line).
• Filament lamp: \(T \uparrow \implies R \uparrow\) (gradient decreases).
• NTC Thermistor: \(T \uparrow \implies R \downarrow\) (gradient increases).
• Diode: Conducts in one direction only past threshold voltage.
4. Resistivity (\(\rho\))
Resistance depends on the size and shape of an object, but resistivity is a property of the material itself, regardless of its dimensions.
Factors Affecting the Resistance of a Wire:
1. Length (\(L\)): Longer wire \(\implies\) more collisions \(\implies R \propto L\).
2. Cross-Sectional Area (\(A\)): Thicker wire \(\implies\) more paths for electrons \(\implies R \propto \frac{1}{A}\).
3. Material: Different materials have different atomic structures, described by resistivity (\(\rho\)).
The Resistivity Equation
Combining these factors gives:
\(R = \frac{\rho L}{A}\)
Rearranging for resistivity \(\rho\) (the Greek letter 'rho'):
\(\rho = \frac{R A}{L}\)
Where:
• \(\rho\) is the resistivity of the material in ohm-metres (\(\Omega\ \text{m}\)).
• \(R\) is the resistance in ohms (\(\Omega\)).
• \(A\) is the cross-sectional area in square metres (\(\text{m}^2\)).
• \(L\) is the length of the conductor in metres (\(\text{m}\)).
Memory Aid for Units: Think of the formula units: \(\frac{\Omega \times \text{m}^2}{\text{m}} = \Omega\ \text{m}\). Do not confuse \(\Omega\ \text{m}\) with \(\Omega\ \text{m}^{-1}\)!
Key Takeaway: Resistivity allows us to compare materials directly. Good conductors (like copper) have very low resistivity (\(\approx 10^{-8}\ \Omega\ \text{m}\)), while insulators have very high resistivity (\(> 10^{10}\ \Omega\ \text{m}\)).
5. Practical: Determining the Resistivity of a Metal Wire
Measuring the resistivity of a wire is a core experimental technique in A-Level Physics. Here is the standard laboratory procedure:
Apparatus
• Test wire (e.g., constantan or nichrome) clamped to a metre rule.
• Power supply, ammeter (in series), voltmeter (in parallel across the measured length), switch, and variable resistor.
• Micrometer screw gauge.
Step-by-Step Procedure
1. Measure Diameter: Use a micrometer screw gauge to measure the diameter \(d\) of the wire at several different positions along its length, taking measurements at different orientations. Calculate the mean diameter \(d\).
2. Calculate Area: Determine the cross-sectional area using the formula \(A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}\).
3. Measure Resistance for Various Lengths: Connect flying leads to measure a chosen length \(L\) of wire. Record current \(I\) and voltage \(V\), and calculate \(R = \frac{V}{I}\).
4. Repeat: Change the length \(L\) (e.g., from \(0.20\ \text{m}\) to \(1.00\ \text{m}\) in steps of \(0.10\ \text{m}\)) and record new values of \(V\) and \(I\).
Graphical Analysis
Using \(R = \frac{\rho}{A} L\), comparing this to the equation of a straight line \(y = mx + c\):
• Plot \(R\) on the \(y\)-axis against \(L\) on the \(x\)-axis.
• A straight line through the origin should be obtained with a gradient equal to \(\frac{\rho}{A}\).
• Calculate resistivity: \(\rho = \text{gradient} \times A\).
Experimental Precautions to Reduce Errors
• Keep currents small: Use a variable resistor or disconnect between readings to prevent wire heating, which would change the resistance.
• Avoid parallax error: View the metre rule directly from above when measuring length.
• Zero error: Check and correct for zero error on the micrometer before measuring the diameter.
6. Temperature Dependence: Metals vs. Semiconductors
Why do materials react differently to heating? It all comes down to two competing microscopic processes: lattice vibrations and charge carrier density.
Metallic Conductors (Positive Temperature Coefficient)
• In metals, there is already a huge, fixed number of free conduction electrons.
• When temperature rises, the metal ions vibrate faster and with larger amplitude.
• This causes more frequent collisions with moving electrons, reducing their drift velocity.
• Result: Resistance increases as temperature increases.
NTC Thermistors / Semiconductors (Negative Temperature Coefficient)
• In semiconductors, few free electrons are available at room temperature.
• Heating provides thermal energy to break covalent bonds, releasing many more electrons into the conduction band.
• The massive increase in the number density of charge carriers (\(n\)) far outweighs the small increase in collisions from vibrating ions.
• Result: Resistance decreases as temperature increases.
Did You Know? NTC thermistors are widely used in digital thermometers, car engine temperature sensors, and fire alarms because of their sharp, predictable response to temperature changes.
7. Superconductivity
What if you could transmit electricity without losing any energy at all as heat?
What is Superconductivity?
Superconductivity is a property of certain materials that have zero electrical resistance and zero resistivity below a specific temperature called the critical temperature (\(T_c\)) (also known as the transition temperature).
Key Characteristics
• Above \(T_c\), the material behaves like a normal conductor with standard resistance.
• At and below \(T_c\), resistance drops sharply to exactly zero.
• Once a current is initiated in a closed superconducting loop, it will flow indefinitely without needing a power supply!
Applications of Superconductors
1. High-Field Electromagnets: Used in MRI scanners in hospitals and particle accelerators (such as the Large Hadron Collider at CERN) to produce extremely strong magnetic fields without overheating.
2. Maglev Trains: Magnetic levitation allows high-speed trains to float above the tracks, eliminating mechanical friction.
3. Power Transmission: Lossless electrical power cables (currently limited by cooling costs).
Limitations
• Most traditional superconductors require cooling with liquid helium close to absolute zero (below \(10\ \text{K}\)).
• "High-temperature" superconductors still require cooling with liquid nitrogen (\(\approx 77\ \text{K}\) or \(-196^\circ\text{C}\)), which is expensive and technically challenging to maintain over long distances.
Key Takeaway: Below the critical temperature \(T_c\), superconductors exhibit zero resistance, allowing huge currents to flow with zero energy dissipation as heat.
Summary Checklist for Revision
Make sure you can confidently do the following before your exam:
• State the definition of resistance: \(R = \frac{V}{I}\).
• State Ohm's Law and identify when it applies.
• Sketch and explain the \(I\)-\(V\) graphs for an Ohmic resistor, filament lamp, diode, and NTC thermistor.
• Calculate resistivity using \(\rho = \frac{RA}{L}\) and convert units accurately (especially \(\text{mm}\) to \(\text{m}\) and \(\text{mm}^2\) to \(\text{m}^2\)).
• Describe the experimental method and graphical analysis to determine the resistivity of a wire.
• Explain the difference in temperature response between metals and semiconductors using microscopic ideas.
• Define superconductivity and critical temperature (\(T_c\)), and state real-world applications.