Resistivity, Potential Dividers, e.m.f. and Internal Resistance

Q: What is the formula for calculating the resistance of a wire from its physical dimensions?

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

Physics (YPH11) · Pearson Edexcel International A Level · 5 min read · Updated: 7 Sept 2026

Introduction to Advanced Resistance and Voltage

In your earlier studies, you learned that resistance is simply how much a component opposes current. Now, we are going to dive deeper! We will explore resistivity (the "DNA" of a material's resistance), how we can use potential dividers to control voltages in gadgets, and the "hidden" resistance inside batteries known as internal resistance.

Don't worry if these terms sound a bit technical—we will break them down step-by-step with simple analogies and clear formulas.

1. Resistivity: The Material's Identity

While resistance (\(R\)) depends on the shape of an object, resistivity (\(\rho\)) is a property of the material itself. Think of it like this: "Resistance" is how hard it is to walk through a specific hallway, but "Resistivity" is how crowded the building material is in general.

The Resistivity Formula

The resistance of a wire depends on three things: its length (\(l\)), its cross-sectional area (\(A\)), and the material it is made of (\(\rho\)).

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

Where:
- \(R\) is Resistance (measured in Ohms, \(\Omega\))
- \(\rho\) (the Greek letter 'rho') is Resistivity (measured in Ohm-metres, \(\Omega \text{m}\))
- \(l\) is the length of the wire (metres, \(\text{m}\))
- \(A\) is the cross-sectional area (metres squared, \(\text{m}^2\))

Why does it work this way?

- Length (\(l\)): Doubling the length doubles the resistance because electrons have to collide with twice as many ions.
- Area (\(A\)): Doubling the area halves the resistance. It’s like widening a corridor; it's much easier for people (electrons) to flow through.

The Microscopic View: \(I = nqvA\)

To understand why different materials have different resistivities, we look at the transport equation:

\(I = nqvA\)

Where:
- \(I\) is Current (\(\text{A}\))
- \(n\) is the number density of charge carriers (how many free electrons there are per cubic metre)
- \(q\) is the charge of a single carrier (for an electron, this is \(1.60 \times 10^{-19} \text{ C}\))
- \(v\) is the drift velocity (the slow average speed of electrons)
- \(A\) is the cross-sectional area

Key Insight: Metals have a very high \(n\) (lots of free electrons), so they have low resistivity. Insulators have a very low \(n\), so they have high resistivity.

Quick Review: Resistivity (\(\rho\)) is a constant for a material at a specific temperature. Resistance (\(R\)) changes if you stretch or thicken the wire.

2. Temperature and Resistance

The way resistance changes with temperature depends on what the material is made of:

- Metals: When they get hotter, the metal ions vibrate more. These lattice vibrations get in the way of flowing electrons, so resistance increases.
- NTC Thermistors: (Negative Temperature Coefficient). As they get hotter, more conduction electrons are released (the \(n\) in our equation increases). This effect is so strong that it overcomes the vibrations, and resistance decreases.

3. Potential Dividers

A potential divider is a simple circuit that uses two or more resistors in series to "split" the voltage from a source. This allows you to get a specific output voltage (\(V_{out}\)) that is smaller than the input voltage (\(V_{in}\)).

The Potential Divider Equation

If you have two resistors, \(R_1\) and \(R_2\), in series, and you take the output across \(R_2\):

\(V_{out} = \frac{R_2}{R_1 + R_2} \times V_{in}\)

Pro-tip: The resistor with the larger share of the resistance gets the larger share of the voltage!

Sensors in Potential Dividers

We often replace one of the fixed resistors with a sensor to make a circuit that responds to the environment:

- LDR (Light Dependent Resistor): Resistance decreases as light intensity increases. (Mnemonic: Light Up, Resistance Down - LURD). Used in streetlights.
- Thermistor: Resistance decreases as temperature increases. Used in digital thermometers.

Example: In a night-light circuit, an LDR is used. When it gets dark, the LDR's resistance increases, so it takes a larger share of the voltage, turning the light on.

4. E.M.F. and Internal Resistance

Have you ever noticed that a battery feels warm after use? That's because batteries aren't perfect—they have their own internal resistance.

Definitions

- e.m.f. (\(\varepsilon\)): Electromotive Force. This is the total energy supplied by the source per unit charge. It is the voltage measured when no current is flowing.
- Terminal P.D. (\(V\)): The actual voltage delivered to the external circuit.
- Internal Resistance (\(r\)): The resistance inside the battery itself.
- Lost Volts (\(Ir\)): The voltage "wasted" inside the battery due to internal resistance.

The Master Equation

\(\varepsilon = I(R + r)\) or \(\varepsilon = V + Ir\)

Where:
- \(\varepsilon\) is the e.m.f. (\(\text{V}\))
- \(V\) is the terminal potential difference (\(\text{V}\))
- \(I\) is the current (\(\text{A}\))
- \(R\) is the external load resistance (\(\Omega\))
- \(r\) is the internal resistance (\(\Omega\))

Common Mistake: Thinking e.m.f. is a force. It’s not! It is measured in Volts (Joules per Coulomb).

5. Core Practicals in this Chapter

For your Unit 3 and Unit 6 exams, you must be familiar with these two procedures:

Core Practical 7: Measuring Resistivity

- Method: Measure the resistance of a wire at different lengths (\(l\)) using an ohmmeter or ammeter/voltmeter. Measure the diameter using a micrometer screw gauge to calculate the area (\(A = \pi (\frac{d}{2})^2\)).
- Graph: Plot \(R\) on the y-axis and \(l\) on the x-axis.
- Result: The gradient of the graph will be \(\frac{\rho}{A}\). Multiply the gradient by the area to find resistivity \(\rho\).

Core Practical 8: Finding e.m.f. and Internal Resistance

- Method: Connect a cell to a variable resistor (rheostat) and measure the terminal p.d. (\(V\)) and current (\(I\)) as you change the resistance.
- Graph: Plot \(V\) on the y-axis and \(I\) on the x-axis.
- Equation: Rearranging \(\varepsilon = V + Ir\) gives \(V = -rI + \varepsilon\).
- Result: This matches the math equation \(y = mx + c\).
- The y-intercept is the e.m.f. (\(\varepsilon\)).
- The gradient is the negative internal resistance (\(-r\)).

Summary - Key Takeaways

- Resistivity (\(\rho\)) is a material property; Resistance (\(R\)) depends on dimensions.
- \(I = nqvA\) explains why metals conduct better than semiconductors.
- Potential Dividers distribute voltage based on the ratio of resistances.
- e.m.f. is the total energy available; Terminal P.D. is what the circuit actually gets after "lost volts" (\(Ir\)) are subtracted.

Quick check

Can you answer these now?

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

What is the difference between resistance and resistivity?

Resistance (\(R\)) depends on the specific shape and dimensions of an object, whereas resistivity (\(\rho\)) is an intrinsic property of the material itself.

What is the formula for calculating the resistance of a wire from its physical dimensions?

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

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

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

How does temperature affect the resistance of metals versus NTC thermistors?

In metals, resistance increases with temperature due to lattice vibrations; in NTC thermistors, resistance decreases because more charge carriers (\(n\)) are released.

What is the purpose of a potential divider circuit?

A potential divider uses resistors in series to split an input voltage (\(V_{in}\)) into a specific, smaller output voltage (\(V_{out}\)).

What is the potential divider equation for \(V_{out}\) across resistor \(R_2\)?

\(V_{out} = \frac{R_2}{R_1 + R_2} \times V_{in}\)

Define Electromotive Force (e.m.f.) and Lost Volts.

e.m.f. (\(\varepsilon\)) is the total energy supplied per unit charge by the source; 'Lost Volts' (\(Ir\)) is the energy wasted per unit charge due to internal resistance.

What is the master equation relating e.m.f., terminal p.d., and internal resistance?

\(\varepsilon = V + Ir\) or \(\varepsilon = I(R + r)\)

In Core Practical 7, how is resistivity (\(\rho\)) determined from a graph?

By plotting Resistance (\(R\)) against length (\(l\)), the gradient is \(\frac{\rho}{A}\); therefore, \(\rho = \text{gradient} \times A\).

How do you find e.m.f. and internal resistance from a graph of terminal p.d. (\(V\)) against current (\(I\))?

The y-intercept represents the e.m.f. (\(\varepsilon\)), and the gradient represents the negative internal resistance (\(-r\)).

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