Recall and use the equation for electrical power P = I V

Topic 4.2.4 – Resistance

Learning Objectives

• Recall and use the fundamental equations:

  • Ohm’s law (re‑arranged): R = V ⁄ I
  • Resistance of a uniform conductor: R = \rho \dfrac{L}{A}
  • Electrical power: P = I V (and the derived forms P = I²R and P = V²⁄R)

• Explain qualitatively how resistance varies with length, cross‑sectional area, material and temperature.
• Interpret current–voltage (I‑V) graphs for ohmic and non‑ohmic devices.
• Carry out the standard AO3 practical to determine an unknown resistance using a voltmeter and an ammeter.

1. What is Resistance?

Resistance (R) is a property of a conductor that opposes the flow of electric charge. It is measured in ohms (Ω). The higher the resistance, the smaller the current for a given voltage.

2. Factors that Influence Resistance

• Material (resistivity \(\rho\)) – each material has a characteristic resistivity (Ω·m). Metals have low \(\rho\); insulators have high \(\rho\).
• Length (\(L\)) – directly proportional to resistance.

  
  Proportionality: \(R \propto L\) (for a given material and area).

• Cross‑sectional area (\(A\)) – inversely proportional to resistance.

  
  Proportionality: \(R \propto \dfrac{1}{A}\) (for a given material and length).

• Temperature – for most metals, resistance increases with temperature (positive temperature coefficient). For many semiconductors it decreases (negative coefficient).

3. Resistivity and the Resistance Formula

The intrinsic property of a material is its resistivity \(\rho\) (Ω·m). For a uniform conductor of length \(L\) and cross‑sectional area \(A\):

\$R = \rho \frac{L}{A}\$

From this equation the proportionalities in section 2 follow directly.

4. Ohm’s Law – Relating V, I and R

For an ohmic conductor (one that obeys Ohm’s law):

\$V = I R \qquad\Longleftrightarrow\qquad R = \frac{V}{I}\$

This is the equation that must be memorised for the IGCSE exam.

5. Current–Voltage (I‑V) Graphs

• Ohmic device – straight line through the origin. Gradient = \(R\). Slope \(= \dfrac{V}{I} = R\).
• Non‑ohmic device (e.g. filament lamp) – curve that is not a straight line. The resistance changes with temperature, so the gradient varies along the curve.

When sketching an I‑V graph, label the axes clearly and indicate the region that corresponds to the given resistance.

6. Electrical Power

Power is the rate at which electrical energy is converted to another form (heat, light, mechanical). The basic definition is:

\$P = I V\$

By substituting Ohm’s law you obtain two useful alternatives:

• Using \(V = I R\): \(P = I^{2} R\) (useful when current is known)
• Using \(I = \dfrac{V}{R}\): \(P = \dfrac{V^{2}}{R}\) (useful when voltage is known)

Example 1 – Power from a known current

A lamp draws \(0.50\;\text{A}\) from a \(12\;\text{V}\) battery.

1. Find the filament resistance: \(R = \dfrac{V}{I} = \dfrac{12}{0.50} = 24\;\Omega\).
2. Calculate the power: \(P = I^{2}R = (0.50)^{2}\times24 = 6.0\;\text{W}\).

Result: \(R = 24\;\Omega\), \(P = 6\;\text{W}\).

Example 2 – Length of a copper wire

Given: \(\rho_{\text{Cu}} = 1.68\times10^{-8}\;\Omega\!\cdot\!m\), \(R = 2.0\;\Omega\), \(A = 1.0\;\text{mm}^{2}\).

1. Convert area: \(A = 1.0\times10^{-6}\;m^{2}\).
2. Re‑arrange \(R = \rho L/A\) for \(L\): \(L = \dfrac{RA}{\rho}\).
3. Substitute: \(L = \dfrac{(2.0)(1.0\times10^{-6})}{1.68\times10^{-8}} \approx 1.19\times10^{2}\;m\).

Result: about 119 m of copper wire is required.

7. Summary Table of Key Formulas

8. AO3 Practical – Determining an Unknown Resistance

1. Connect the unknown resistor in series with an ammeter. Connect a voltmeter across the resistor (parallel).
2. Switch on the circuit and record the steady‑state readings \(I\) (A) and \(V\) (V).
3. Calculate the resistance using Ohm’s law: \(R = \dfrac{V}{I}\).
4. Repeat with at least two different voltage settings to check that the resistor is ohmic (the calculated \(R\) should be the same within experimental error).

Remember: the ammeter must be in series with the whole circuit, the voltmeter must be in parallel with the resistor.

9. Common Misconceptions

• Resistivity vs. Resistance – \(\rho\) is a property of the material only; \(R\) also depends on the geometry (L, A).
• Power formula scope – \(P = I V\) is valid for DC and for AC when RMS values are used. Peak values require extra factors (e.g., \(\sqrt{2}\) for a sinusoid).
• Temperature effects ignored – For high‑current devices (filament lamps, heaters) resistance can change markedly with temperature; this must be considered in calculations.
• All devices are ohmic – Many practical components (diodes, filament lamps) do not follow a straight‑line I‑V relationship.

10. Quick Revision Checklist

• Write down the three power equations and state when each is most convenient.
• State and rearrange the two forms of Ohm’s law: \(V = I R\) and \(R = V/I\).
• Recall the proportionalities: \(R \propto L\) and \(R \propto 1/A\) for a given material.
• Sketch an I‑V graph for an ohmic resistor and label the gradient as the resistance.
• Describe the step‑by‑step set‑up for measuring an unknown resistance with a voltmeter and ammeter.
• Explain qualitatively how temperature affects resistance for metals and for semiconductors.