Recall and use the equation for electrical energy E = I V t

4.2.4 Resistance

Learning Objective

Recall and use the equation for electrical energy

\$E = I \cdot t\$

where E is the energy transferred (J), I is the current (A), V is the potential difference (V) and t is the time (s).

Key Concepts

• Resistance (\(R\)) opposes the flow of electric current.
• Ohm’s Law: \(V = I R\).
• Resistivity (\(\rho\)) is a material property: \(R = \rho \dfrac{L}{A}\) where \(L\) is length and \(A\) is cross‑sectional area.
• Electrical power: \(P = IV = I^{2}R = \dfrac{V^{2}}{R}\).
• Energy transferred in a circuit: \(E = P t = I \cdot t\).

Symbols and Units

Deriving the Energy Equation Using Resistance

1. Start with Ohm’s Law: \(V = I R\).
2. Substitute \(V\) into the power formula: \(P = I V = I (I R) = I^{2} R\).
3. Energy is power multiplied by time: \(E = P t = I^{2} R t\).
4. Alternatively, using \(V = I R\) directly in the energy formula gives the same result:

   \$E = I \cdot t = I (I R) t = I^{2} R t.\$

Worked Example

Problem: A heater of resistance \(20\;\Omega\) is connected to a \(240\;{\rm V}\) supply. Calculate the energy used after 5 minutes.

Solution:

1. Find the current using Ohm’s Law:

   \$I = \frac{V}{R} = \frac{240\;\text{V}}{20\;\Omega} = 12\;\text{A}.\$

2. Convert time to seconds: \(5\;\text{min} = 5 \times 60 = 300\;\text{s}\).
3. Use the energy equation:

   \$E = I \cdot t = 12\;\text{A} \times 240\;\text{V} \times 300\;\text{s} = 864\,000\;\text{J}.\$

4. Optionally express in kilojoules: \(E = 864\;\text{kJ}\).

Common Mistakes to Avoid

• Confusing resistance (\(R\)) with reactance or impedance – for IGCSE only pure resistance is considered.
• Forgetting to convert minutes or hours to seconds when using the energy formula.
• Using the wrong unit for energy; the SI unit is the joule (J), not watt‑hours unless explicitly required.
• Mixing up the symbols: \(V\) is voltage, not volume; \(I\) is current, not intensity.

Practice Questions

1. A lamp has a resistance of \(30\;\Omega\) and is connected to a \(120\;{\rm V}\) source.

   • Calculate the current flowing through the lamp.
   • Determine the power consumed.
   • How much energy does the lamp use in 2 hours?

2. A circuit contains a resistor of \(10\;\Omega\) and a current of \(3\;\text{A}\) flows through it for 45 seconds.

   • Find the voltage across the resistor.
   • Calculate the energy transferred.

3. Explain why a material with a high resistivity is a good insulator, using the formula \(R = \rho \dfrac{L}{A}\).

Summary

Understanding resistance allows us to apply Ohm’s Law and the energy equation \(E = I \cdot t\) to a wide range of practical problems. By mastering the relationships between current, voltage, resistance, power and energy, students can confidently analyse electrical circuits and predict the performance of devices.