Welcome to Resistivity and Superconductivity!
In your previous studies, you learned about Resistance (\( R \)). You know that a long wire has more resistance than a short one, and a thin wire has more than a thick one. But how do we compare the "resistiveness" of the materials themselves? Why is copper better than iron? This is where Resistivity comes in. In this chapter, we will also explore the strange world of Superconductivity, where resistance completely disappears!
1. What is Resistivity?
Resistance depends on the shape and size of an object. Resistivity (\( \rho \)), however, is a property of the material itself, regardless of its shape. Think of it like this: "Resistance" is how hard it is to get through a specific corridor, but "Resistivity" is how crowded the building material is in general.
The Formula
The resistance (\( R \)) of a wire is proportional to its length (\( L \)) and inversely proportional to its cross-sectional area (\( A \)). We use the constant \( \rho \) (the Greek letter rho) to link them:
\( \rho = \frac{RA}{L} \)
Where:
\( \rho \) = Resistivity in ohm-metres (\( \Omega \text{m} \))
\( R \) = Resistance in ohms (\( \Omega \))
\( A \) = Cross-sectional area in square metres (\( \text{m}^2 \))
\( L \) = Length in metres (\( \text{m} \))
Quick Tip: Don't confuse resistivity (\( \Omega \text{m} \)) with resistance per metre (\( \Omega/\text{m} \)). They are different units! Always check your units carefully.
Key Takeaway:
If you have two wires of the same material but different sizes, they will have different resistances but the same resistivity.
2. Required Practical 5: Determining Resistivity
You need to know how to find the resistivity of a wire experimentally. Don't worry if this seems like a lot of steps; it's quite logical once you see it.
The Process:
1. Measure the Diameter: Use a micrometer to measure the diameter of the test wire. Measure it at several points along the wire and at different orientations, then calculate an average.
2. Calculate Area: Use the formula for the area of a circle: \( A = \frac{\pi d^2}{4} \) (where \( d \) is the average diameter).
3. Set up the Circuit: Connect the wire in a circuit with an ammeter (in series) and a voltmeter (in parallel across the test wire).
4. Vary the Length: Use a crocodile clip to change the length (\( L \)) of the wire being tested.
5. Record Readings: For each length, record the current (\( I \)) and potential difference (\( V \)). Calculate Resistance using \( R = \frac{V}{I} \).
6. Analyze: Plot a graph of Resistance (\( R \)) on the y-axis against Length (\( L \)) on the x-axis.
The Graph:
Since \( R = \frac{\rho L}{A} \), the gradient of your \( R \text{-} L \) graph will be \( \frac{\rho}{A} \).
Therefore: \( \rho = \text{gradient} \times A \).
Common Mistake: Forgetting to convert diameter from millimetres (\( \text{mm} \)) to metres (\( \text{m} \)) before calculating the area. Always work in SI units!
3. Temperature Effects
How a material reacts to heat tells us a lot about its internal structure. In the AQA syllabus, we focus on metals and NTC thermistors.
Metals (Conductors)
In a metal, as the temperature increases, the resistance increases.
Why? Inside the metal, there is a lattice of positive ions. When it gets hotter, these ions vibrate more vigorously. This makes it much harder for the "sea" of delocalised electrons to flow through the metal because they collide with the vibrating ions more often.
NTC Thermistors (Semiconductors)
NTC stands for Negative Temperature Coefficient. For these components, as temperature increases, the resistance decreases.
Why? In a semiconductor, there aren't many free electrons at room temperature. When you heat it up, the extra energy "shakes" more electrons free from their atoms. This sudden increase in the number of charge carriers (\( n \)) far outweighs the effect of the vibrating ions, so the resistance drops.
Analogy: Imagine a busy train station. In a metal, heating it is like making everyone in the station dance wildly—it's harder to get through. In a thermistor, heating it is like opening ten extra ticket barriers—even if people are dancing, the flow improves because more paths are open.
4. Superconductivity
Superconductivity is a "super" state of matter that occurs in certain materials at very low temperatures.
What is it?
A superconductor is a material that has zero resistivity when cooled below a specific temperature. This temperature is called the critical temperature (\( T_c \)).
The Benefits:
Because there is zero resistance (\( R = 0 \)), there is zero energy loss as heat when a current flows. This means:
1. You can have incredibly high currents without the wire melting.
2. You can create very strong magnetic fields (used in MRI scanners and particle accelerators).
3. Power cables could transmit electricity across the country with 100% efficiency (if we could keep them cold enough!).
Did you know?
Most materials need to be cooled to near absolute zero (\( 0 \text{ K} \) or \( -273^{\circ}\text{C} \)) using liquid helium to become superconductors. Scientists are currently searching for "high-temperature" superconductors that work at room temperature, which would revolutionise technology!
Note: You do not need to know about "critical fields" for your exam—only the effect of temperature.
5. Quick Review & Summary
Key Formulas to Remember:
\( \rho = \frac{RA}{L} \)
\( R = \frac{V}{I} \)
\( A = \frac{\pi d^2}{4} \)
Quick Summary Table:
Material Type: Metal (Conductor)
Temp Increase Effect: Resistance Increases
Reason: More lattice vibrations increase electron collisions.
Material Type: NTC Thermistor (Semiconductor)
Temp Increase Effect: Resistance Decreases
Reason: More charge carriers are released.
Material Type: Superconductor
Temp Increase Effect: Resistance drops to ZERO at \( T_c \)
Reason: Special state of matter with no resistivity.
Don't worry if the resistivity practical seems fiddly at first. The key is practicing the calculation for the cross-sectional area and remembering that the gradient of your graph isn't the resistivity itself, but is linked to it via the area!