Introduction to Resistivity

In the previous chapters, we looked at resistance (\(R\)), which tells us how much a component opposes the flow of current. However, resistance depends on the shape of the object. For example, a long, thin wire has more resistance than a short, thick one, even if they are made of the same metal.

In this chapter, we focus on resistivity. Think of resistivity as a "fair way" to compare different materials. It tells us how much a specific material opposes current, regardless of its size or shape. It is a fundamental property of the material itself.

The Resistivity Formula

The resistance of a wire is determined by three physical factors: its length (\(L\)), its cross-sectional area (\(A\)), and the material it is made of (its resistivity, \( \rho \)).

The relationship is given by the formula:

\( \rho = \frac{RA}{L} \)

Where:

  • \( \rho \) (the Greek letter rho) is the resistivity measured in ohm-metres (\( \Omega \text{m} \)).
  • \( R \) is the resistance measured in ohms (\( \Omega \)).
  • \( A \) is the cross-sectional area measured in metres squared (\( \text{m}^2 \)).
  • \( L \) is the length of the wire measured in metres (\( \text{m} \)).

Understanding the relationships:

1. Resistance is proportional to length (\( R \propto L \)): If you double the length of a wire, you double the resistance because the electrons have to collide with twice as many ions.

2. Resistance is inversely proportional to area (\( R \propto \frac{1}{A} \)): If you make the wire thicker (larger area), the resistance decreases because there is more space for the electrons to flow through.

Quick Tip: Don't confuse resistance with resistivity! Resistance is for a specific object (like a 2-metre copper wire), while resistivity is for the material in general (copper).

Key Takeaway: A material with low resistivity (like copper) is a good conductor. A material with high resistivity (like rubber) is a good insulator.

Temperature and Resistance

The resistance of a material isn't just about its shape; it also changes with temperature. The way it changes depends on whether the material is a metal or a semiconductor.

1. Metal Conductors

In metals, as the temperature increases, the resistance increases. This is because the metal ions inside the wire vibrate more vigorously as they get hotter. These vibrations make it much harder for the free electrons to pass through without colliding, which slows down the current.

2. NTC Thermistors

A thermistor is a type of semiconductor. In this course, we focus on Negative Temperature Coefficient (ntc) thermistors. For these components, as the temperature increases, the resistance decreases.

Why? Even though the ions vibrate more (which would normally increase resistance), the heat provides enough energy to "release" many more charge carriers (electrons) within the semiconductor material. The huge increase in available charge carriers outweighs the effect of the vibrating ions, so current flows more easily.

Did you know? This makes NTC thermistors perfect for digital thermometers or as sensors in ovens and car engines!

Superconductivity

Imagine a material that has absolutely zero resistance. This state is called superconductivity.

Some materials, when cooled down to a very specific temperature called the critical temperature (\( T_c \)), suddenly lose all their electrical resistance. Below this temperature, current can flow forever without losing any energy as heat!

Applications of Superconductors:

  • Power Cables: To transmit electricity without any energy loss.
  • Strong Electromagnets: Used in MRI scanners and high-speed maglev trains.
  • Particle Accelerators: To create the massive magnetic fields needed to accelerate particles.

Important: The main drawback is that most materials currently known to be superconductors only work at extremely low temperatures (close to absolute zero), which is very expensive to maintain.

Required Practical 5: Determining Resistivity

You are required to know how to determine the resistivity of a wire experimentally. Here is the step-by-step process:

1. Equipment

  • A length of the test wire (e.g., constantan).
  • A micrometer (to measure diameter).
  • An ammeter and a voltmeter (or a multimeter).
  • A power supply and a ruler.

2. The Method

1. Measure the diameter (\( d \)) of the wire in several places using a micrometer and calculate an average. Use this to find the cross-sectional area: \( A = \frac{\pi d^2}{4} \).

2. Set up a circuit to measure the resistance (\( R \)) of a measured length (\( L \)) of the wire. You can do this by recording the potential difference (\( V \)) and current (\( I \)) and using \( R = \frac{V}{I} \).

3. Vary the length of the wire (e.g., from \( 10 \text{ cm} \) to \( 100 \text{ cm} \)) and record the resistance for each length.

4. Plot a graph of Resistance (\( R \)) on the y-axis against Length (\( L \)) on the x-axis.

3. Analyzing the Results

Since \( R = \frac{\rho L}{A} \), the equation of your graph is in the form \( y = mx \).
The gradient of your graph is equal to \( \frac{\rho}{A} \).
Therefore: Resistivity \( \rho = \text{gradient} \times A \).

Common Mistake: Forgetting to convert units! Always make sure your diameter is in metres (not mm) and your area is in \( \text{m}^2 \) before calculating resistivity.

Summary Review

  • Resistivity Formula: \( \rho = \frac{RA}{L} \).
  • Metals: Resistance increases when temperature increases.
  • NTC Thermistors: Resistance decreases when temperature increases.
  • Superconductivity: Zero resistance below a critical temperature (\( T_c \)).
  • Practical: Use a micrometer for diameter and a graph of \( R \) against \( L \) to find resistivity.