Welcome to Resistance and Resistivity
Welcome to one of the most practical and exciting areas of AS Physics! Every electrical device you use daily — from your smartphone to an electric kettle — relies entirely on controlling how electric current flows. In this chapter, we explore why materials oppose electric current, how different components behave when connected to a circuit, how we calculate a material's intrinsic resistivity, and the phenomenon of superconductivity.
Don't worry if electricity formulas have felt confusing in the past. We will break every concept down step-by-step with clear analogies, memory tricks, and worked examples.
1. Understanding Electrical Resistance
What is Resistance?
When an electric current flows through a conductor, free conduction electrons move through a metal lattice. As these electrons drift, they collide with the vibrating positive metal ions in the lattice. These collisions oppose the flow of charge and transfer kinetic energy from the electrons to the lattice as thermal energy (heat).
Resistance is defined as the ratio of the potential difference (\(V\)) across a component 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, measured in volts (\(\text{V}\))
• \(I\) is the current, measured in amperes (\(\text{A}\))
From this equation, we define the unit: \(1\text{ ohm } (1\text{ }\Omega)\) is the resistance of a conductor when a potential difference of \(1\text{ V}\) produces a current of \(1\text{ A}\) through it (\(1\text{ }\Omega = 1\text{ V A}^{-1}\)).
Everyday Analogy: The Crowded Corridor
Imagine students (conduction electrons) trying to run down a school hallway. The hallway is packed with teachers standing in place (positive lattice ions). As the students try to move forward, they bump into the teachers. The more collisions that occur, the harder it is to move forward — that opposition is resistance!
Key Takeaway
Resistance measures how much a component opposes the flow of electric current. It is calculated by dividing voltage across the component by the current passing through it (\(R = \frac{V}{I}\)).
2. Ohm's Law and \(I\text{--}V\) Characteristics
Ohm's Law
Ohm's Law states that: For a metallic conductor at constant temperature, the current flowing through it is directly proportional to the potential difference across it.
\(I \propto V\) (provided temperature and other physical conditions remain constant)
Comparing Component Characteristics
A graph of current (\(I\)) on the y-axis against potential difference (\(V\)) on the x-axis is called an \(I\text{--}V\) characteristic graph.
A. Metallic Conductor at Constant Temperature (Ohmic Conductor)
• Graph shape: A straight line passing directly through the origin \((0,0)\).
• Explanation: Because the gradient (\(\frac{I}{V}\)) is constant, the resistance (\(R = \frac{V}{I} = \frac{1}{\text{gradient}}\)) remains constant.
• Important note: A standard fixed resistor behaves as an ohmic conductor when kept at a steady temperature.
B. Filament Lamp (Non-Ohmic Conductor)
• Graph shape: An 'S'-shaped curve passing through the origin that curves so that the gradient decreases at higher voltages (in both positive and negative directions).
• Explanation step-by-step:
1. As the potential difference \(V\) increases, the current \(I\) increases.
2. The increased rate of electron collisions transfers more energy to the lattice, increasing the filament wire's temperature.
3. The positive ions vibrate with greater amplitude.
4. This increases the frequency of collisions with moving electrons, causing resistance to increase.
5. Since \(R\) increases, the gradient of the \(I\text{--}V\) graph (\(\frac{1}{R}\)) becomes flatter/less steep.
C. Semiconductor Diode
• Graph shape: In the forward direction (forward bias), virtually no current flows until a threshold voltage is reached (typically around \(0.6\text{ V}\) to \(0.7\text{ V}\) for silicon). Above this voltage, current rises sharply. In the reverse direction (reverse bias), the current is practically zero.
• Explanation: A diode is designed to allow current to flow in only one direction. It has an extremely high resistance in reverse bias and very low resistance above the threshold voltage in forward bias.
D. Negative Temperature Coefficient (NTC) Thermistor
• Graph shape: A curve where the gradient increases as voltage (and hence current and temperature) increases.
• Explanation: In a semiconductor thermistor, increasing the temperature provides energy to release more free charge carriers (electrons) into the conduction band. The dramatic increase in charge carrier density outweighs the effect of lattice vibrations, causing the overall resistance to decrease as temperature increases.
Common Mistake to Avoid
When finding resistance from an \(I\text{--}V\) curve at a particular point, do not calculate the tangent gradient! Resistance is always the total value of voltage divided by current at that specific point: \(R = \frac{V}{I}\).
Key Takeaway
Ohmic conductors have constant resistance (\(I \propto V\)). Filament lamps increase in resistance as they heat up. NTC thermistors decrease in resistance as temperature rises. Diodes conduct current in one direction only.
3. Resistivity
What is Resistivity?
Resistance is a property of an object (it depends on its size and shape), whereas resistivity (\(\rho\)) is an intrinsic property of the material itself, regardless of its dimensions.
Factors Affecting the Resistance of a Wire
1. Length (\(L\)): Resistance is directly proportional to length (\(R \propto L\)). Doubling the length doubles the number of collisions electrons must make.
2. Cross-Sectional Area (\(A\)): Resistance is inversely proportional to cross-sectional area (\(R \propto \frac{1}{A}\)). A wider wire provides more parallel paths for electrons, reducing resistance.
3. Material: Different materials have different internal structures and free electron densities, accounted for by the resistivity (\(\rho\)).
The Resistivity Equation
\(R = \frac{\rho L}{A}\)
Rearranging for resistivity:
\(\rho = \frac{R A}{L}\)
Where:
• \(\rho\) is resistivity, measured in ohm-metres (\(\Omega\text{ m}\))
• \(R\) is resistance (\(\Omega\))
• \(A\) is cross-sectional area (\(\text{m}^2\))
• \(L\) is length (\(\text{m}\))
Calculating Cross-Sectional Area
Wires are circular in cross-section. Given the diameter \(d\) or radius \(r\):
\(A = \pi r^2 = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}\)
Did You Know?
Good conductors like copper have very low resistivities (around \(1.7 \times 10^{-8}\text{ }\Omega\text{ m}\)), while good insulators like rubber have resistivities higher than \(10^{13}\text{ }\Omega\text{ m}\)!
Key Takeaway
Resistivity (\(\rho = \frac{R A}{L}\)) is a material property measured in \(\Omega\text{ m}\). Long, thin wires have high resistance; short, thick wires have low resistance.
4. Required Practical: Determining the Resistivity of a Metal Wire
Apparatus
• Power supply (low voltage DC)
• Ammeter and Voltmeter (or digital multimeters)
• Metre rule (to measure length \(L\))
• Micrometer screw gauge (to measure diameter \(d\))
• Test wire (e.g., nichrome or constantan) clamped along a metre rule
• Flying lead with crocodile clip or sliding contact
• Switch
Step-by-Step Procedure
1. Measure Diameter: Use a micrometer screw gauge to measure the diameter \(d\) of the wire at several different positions (at least \(3\) to \(5\) places) along its length, and at different orientations to account for any non-circular cross-section. Calculate the mean diameter and find \(A = \frac{\pi d^2}{4}\).
2. Set up the Circuit: Connect the test wire in series with a power supply, ammeter, and switch. Connect a voltmeter in parallel across the measured length \(L\) of the wire.
3. Vary the Length: Attach the flying lead to set the contact length \(L\) (e.g., starting at \(0.20\text{ m}\) and increasing in steps of \(0.10\text{ m}\) up to \(0.80\text{ m}\) or \(1.00\text{ m}\)).
4. Take Measurements: Close the switch briefly, record the current \(I\) and potential difference \(V\), then open the switch.
5. Calculate Resistance: For each length, calculate \(R = \frac{V}{I}\).
Graphical Analysis
Starting from \(R = \frac{\rho L}{A}\), comparing to the straight line equation \(y = mx + c\):
• Plot \(R\) on the y-axis against \(L\) on the x-axis.
• The graph should produce a straight line passing through the origin.
• The gradient \(m = \frac{\rho}{A}\).
• Therefore, resistivity is determined by: \(\rho = \text{gradient} \times A\).
Experimental Precautions to Ensure Accuracy
• Keep currents small: Use a variable resistor or low supply voltage to keep current low, preventing the wire from heating up, which would alter its resistance.
• Switch off between readings: Minimises unwanted heating effects.
• Check for zero error: Check the micrometer for zero error before recording diameter values.
• Ensure the wire is straight: Avoid kinks when measuring \(L\) with the metre rule.
Key Takeaway
Plotting \(R\) against \(L\) gives a gradient of \(\frac{\rho}{A}\). Multiplying the gradient by the wire's cross-sectional area gives the material's resistivity \(\rho\).
5. Superconductivity
What is a Superconductor?
A superconductor is a material that loses all of its electrical resistance when cooled below a specific temperature known as the critical temperature (\(T_c\)) or transition temperature.
• At temperatures above \(T_c\), the material behaves like a normal conductor with standard resistance.
• At or below \(T_c\), the resistance drops abruptly to exactly zero (\(R = 0\)).
Key Advantages of Superconductors
When resistance is zero (\(R = 0\)):
• No electrical energy is lost as heat (\(P = I^2 R = 0\)).
• Very large currents can be sustained without overheating the material.
• Extremely strong magnetic fields can be generated.
Applications of Superconductivity
1. MRI Scanners (Magnetic Resonance Imaging): Superconducting electromagnets produce the powerful, stable magnetic fields required for medical body imaging.
2. Particle Accelerators (e.g., CERN's Large Hadron Collider): Superconducting coils steer high-energy subatomic particles around particle tracks.
3. Maglev Trains: Magnetic levitation uses superconducting magnets to float trains above tracks, eliminating mechanical friction.
4. Lossless Power Transmission: Theoretical long-distance cables that transmit electricity with zero resistive power loss.
The Main Limitation
Currently known practical superconductors require cooling down to cryogenic temperatures using liquid helium (around \(4.2\text{ K}\)) or liquid nitrogen (around \(77\text{ K}\) for high-temperature superconductors). The cost and engineering complexity of keeping materials this cold limits their widespread everyday use.
Key Takeaway
Superconductors have zero electrical resistance below their critical temperature \(T_c\), making them ideal for high-field electromagnets and lossless energy systems, provided they are kept cold enough.
Chapter Quick Review Summary
• Resistance: \(R = \frac{V}{I}\) (measured in \(\Omega\)).
• Ohm's Law: \(I \propto V\) for metal conductors at constant temperature.
• Filament Lamp: Non-ohmic; \(R\) increases as temperature rises.
• NTC Thermistor: Semiconductor; \(R\) decreases as temperature rises.
• Resistivity: \(\rho = \frac{R A}{L}\) (measured in \(\Omega\text{ m}\)); independent of sample dimensions.
• Superconductivity: Complete loss of electrical resistance below critical temperature \(T_c\).