Resistivity and charge carriers (I = nqvA)

Physics (XPH11) · Pearson Edexcel International AS Level · 5 min read · Updated: 7 Sept 2026

Introduction to Resistivity and Charge Carriers

In your previous studies, you learned about resistance (\(R\)), which tells us how much a specific component opposes the flow of current. But why does a thick copper wire have less resistance than a thin one? And why does a long wire have more resistance than a short one? In this chapter, we dive deeper into the material properties that determine resistance and look at the "hidden" movement of particles inside a wire using the transport equation (\(I = nqvA\)).

Understanding these concepts is the key to knowing why we use copper for house wiring, why silicon is used in electronics, and how sensors like thermistors work!

1. Resistivity: The Property of the Material

While resistance (\(R\)) depends on the shape and size of an object, resistivity (\(\rho\)) is a property of the material itself. Think of it this way: resistance is like how hard it is to run through a specific corridor; resistivity is like how "cluttered" the building material is, regardless of the corridor's size.

The Resistivity Equation

The resistance of a uniform conductor is determined by three factors: its length (\(l\)), its cross-sectional area (\(A\)), and the resistivity of the material (\(\rho\)).

\(R = \frac{\rho l}{A}\)

\(R\): Resistance, measured in Ohms (\(\Omega\))
\(\rho\): Resistivity, measured in Ohm-metres (\(\Omega \text{m}\))
\(l\): Length of the conductor, measured in metres (\(\text{m}\))
\(A\): Cross-sectional area, measured in square metres (\(\text{m}^2\))

Quick Tip: If you rearrange the formula to find the units for resistivity, you get \(\rho = \frac{RA}{l}\), which gives units of \((\Omega \times \text{m}^2) / \text{m} = \Omega \text{m}\). Students often accidentally write \(\Omega / \text{m}\), so be careful!

Understanding the Relationships

1. Length (\(l\)): If you double the length of a wire, you double the resistance (\(R \propto l\)). It's like making a runner go twice as far through the same obstacles.
2. Area (\(A\)): If you double the cross-sectional area, you halve the resistance (\(R \propto \frac{1}{A}\)). A wider wire provides more "lanes" for electrons to flow through.
3. Material (\(\rho\)): Conductors (like copper) have very low resistivity, while insulators (like rubber) have very high resistivity.

2. Core Practical 7: Determining Resistivity

In your exam, you may be asked to describe how to determine the resistivity of a material (usually a long piece of metal wire). Here is a step-by-step guide:

Measure the diameter: Use a micrometer screw gauge to measure the diameter of the wire at several points and orientations. Calculate the average diameter (\(d\)).
Calculate Area: Use \(A = \pi r^2\) or \(A = \frac{\pi d^2}{4}\).
Set up the circuit: Connect the wire to a power supply with an ammeter in series and a voltmeter in parallel (or use a digital multimeter/ohmmeter).
Vary the length: Use a meter ruler to measure the length (\(l\)) of the wire being tested. Record the resistance (\(R\)) for different lengths (e.g., \(0.20 \text{m}, 0.40 \text{m}, 0.60 \text{m} \dots\)).
Plot a graph: Plot Resistance (\(R\)) on the y-axis and length (\(l\)) on the x-axis.
Calculate \(\rho\): Since \(R = \frac{\rho}{A} \times l\), the gradient of your graph is equal to \(\frac{\rho}{A}\). Therefore, \(\rho = \text{gradient} \times A\).

Don't worry if your graph doesn't go perfectly through the origin; a small y-intercept might be caused by contact resistance at the clips!

3. Charge Carriers and the Equation \(I = nqvA\)

Electric current is the flow of charge carriers (usually electrons in metals). To understand how current relates to the physical movement of these particles, we use the transport equation:

\(I = nqvA\)

What do the symbols mean?

\(I\): Current in Amperes (\(\text{A}\)).
\(n\): Number density of charge carriers. This is the number of free charge carriers per unit volume (measured in \(\text{m}^{-3}\)).
\(q\): The charge of each carrier in Coulombs (\(\text{C}\)). For an electron, \(q = e = 1.60 \times 10^{-19} \text{C}\).
\(v\): Drift velocity of the carriers (\(\text{m s}^{-1}\)). This is the average speed the electrons travel along the wire.
\(A\): Cross-sectional area of the conductor (\(\text{m}^2\)).

Did you know? Even though a light turns on "instantly," the actual drift velocity (\(v\)) of electrons is incredibly slow—often less than a millimetre per second! The "speed of electricity" is actually the speed at which the electric field travels through the wire, which is close to the speed of light.

4. Explaining Resistivity in Different Materials

The value of \(n\) (number density) is what determines whether a material is a conductor, semiconductor, or insulator.

Conductors (Metals)

Metals have a very high value of \(n\) (roughly \(10^{28} \text{m}^{-3}\)). Because there are so many free electrons, even a very slow drift velocity can create a large current. This results in low resistivity.

Semiconductors

Semiconductors (like silicon) have a much lower \(n\) than metals. However, their resistivity can change significantly. If you give them energy (like heat), more charge carriers are released, \(n\) increases, and the resistivity decreases.

Insulators

Insulators have almost no free charge carriers (\(n\) is nearly zero), so no matter how much voltage you apply, the current \(I\) remains negligible. This results in very high resistivity.

Key Takeaway: The difference in resistivity between materials is primarily due to the vast range of \(n\) values available in nature.

5. Temperature and Resistance

Why does resistance change when things get hot? It's a battle between two effects:

In Metals:

When a metal gets hotter, the positive ions in the lattice vibrate more. These vibrating ions collide more frequently with the flowing electrons. This gets in the way of the drift velocity, making it harder for current to flow. Since \(n\) stays roughly constant in metals, the increased collisions mean resistance increases as temperature increases.

In Negative Temperature Coefficient (NTC) Thermistors:

As temperature increases, the thermal energy is enough to "liberate" (free) many more charge carriers. In these materials, the increase in \(n\) is much more significant than the increase in lattice vibrations. Because \(n\) increases, the resistance decreases as temperature increases.

Note: For more on how this is used in circuits, see the chapter on "Potential dividers, thermistors and LDRs".

Summary Checklist

Can you define resistivity and state its unit (\(\Omega \text{m}\))?
Can you use \(R = \frac{\rho l}{A}\) to calculate unknown values?
Can you describe the method for Core Practical 7, including how to reduce uncertainty?
Do you understand that \(I = nqvA\) explains how current depends on the density and speed of charge carriers?
Can you explain why resistance increases with temperature in metals (vibrations) but decreases in NTC thermistors (more carriers)?

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is resistivity and what is its SI unit?

Resistivity (\(\rho\)) is an intrinsic property of a material that quantifies how much it opposes the flow of electric current, measured in Ohm-metres (\(\Omega \text{m}\)).

What is the formula relating resistance, resistivity, length, and cross-sectional area?

The formula is \(R = \frac{\rho l}{A}\), where \(R\) is resistance, \(\rho\) is resistivity, \(l\) is length, and \(A\) is cross-sectional area.

In Core Practical 7, how is the resistivity of a wire determined from a graph?

By plotting a graph of Resistance (\(R\)) against length (\(l\)), the resistivity is calculated as \(\rho = \text{gradient} \times A\).

What does each symbol represent in the transport equation \(I = nqvA\)?

\(I\) is current, \(n\) is number density of charge carriers, \(q\) is the charge of each carrier, \(v\) is drift velocity, and \(A\) is cross-sectional area.

What is drift velocity (\(v\)) and how does it compare to the speed of light?

Drift velocity is the slow average speed at which charge carriers travel through a conductor, often less than 1 mm/s, which is much slower than the speed at which electric fields propagate.

How does the number density (\(n\)) differ between conductors, semiconductors, and insulators?

Conductors have a very high \(n\), semiconductors have a lower \(n\) that can increase with energy, and insulators have a nearly zero \(n\).

Why does the resistance of a metal increase as its temperature increases?

Increasing temperature causes metal ions to vibrate more, leading to more frequent collisions with electrons, which reduces their drift velocity and increases resistance.

Why does the resistance of an NTC thermistor decrease as temperature increases?

Thermal energy liberates more charge carriers, significantly increasing the number density (\(n\)), which outweighs the effect of increased lattice vibrations.

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