define resistance

Resistance and Resistivity

Learning Objectives

• Define electrical resistance and resistivity (AO1).
• Apply Ohm’s law, the power‑dissipation formulas and the resistivity relation to quantitative problems (AO2).
• Identify the variables that affect resistance and understand how they are incorporated in the syllabus.
• Analyse simple and complex DC circuits using series/parallel rules and Kirchhoff’s laws (AO2).
• Carry out practical measurements of resistance, evaluate uncertainties and recognise non‑ohmic behaviour (AO3).

1. Definition of Resistance

Electrical resistance (R) quantifies how strongly a material opposes the flow of electric charge. It is a scalar quantity (no direction) and relates the potential difference (V) applied across a conductor to the current (I) that flows through it.

Units: ohm (Ω). 1 Ω = 1 V ⁄ A.

2. Power Dissipation in a Resistor

When a current passes through a resistance, electrical energy is converted into heat. The power (rate of energy conversion) can be expressed in three equivalent forms:

where P is power (watt, W). These formulas are required for AO2 quantitative problems.

Example: A 10 Ω resistor carries 2 A. The power dissipated is

3. Resistivity and the Geometrical Dependence of Resistance

For a uniform cylindrical conductor the resistance depends on an intrinsic material property – the resistivity (ρ) – and on its geometry:

Re‑arranged, the formula can be used to find resistivity:

• ρ – resistivity (Ω·m), a material constant.
• L – length of the conductor (m).
• A – cross‑sectional area (m²).

4. Symbol Summary

5. Example Calculation – Using the Resistivity Formula

Calculate the resistance of a copper wire 2.0 m long with a cross‑sectional area of \(1.0\times10^{-6}\,\text{m}^2\). Resistivity of copper: \(\rho = 1.68\times10^{-8}\,\Omega\!\cdot\!\text{m}\).

Result: R = 0.0336 Ω.

6. Temperature Dependence of Resistance

For most metallic conductors the resistance varies linearly with temperature (within the range covered by the syllabus):

• R₀ – resistance at reference temperature \(T_{0}\) (usually 20 °C).
• α – temperature coefficient of resistance (°C⁻¹). Typical value for copper: \(3.9\times10^{-3}\,\text{°C}^{-1}\).
• T – actual temperature (°C).

Example: A copper wire has \(R_{0}=0.0336\;\Omega\) at 20 °C. Find its resistance at 50 °C.

7. Series and Parallel Combination Rules

When resistors are connected in a circuit, the equivalent resistance is obtained by the appropriate rule.

• Series: \(R{\text{eq}} = R{1}+R{2}+R{3}+\dots\)
• Parallel: \(\displaystyle\frac{1}{R{\text{eq}}}= \frac{1}{R{1}}+\frac{1}{R{2}}+\frac{1}{R{3}}+\dots\)

Worked example (parallel): Two resistors, 4 Ω and 6 Ω, are connected in parallel.

8. Kirchhoff’s Laws (AO2)

1. Kirchhoff’s Current Law (KCL): The algebraic sum of currents entering a junction equals the sum leaving; \(\sum I = 0\).
2. Kirchhoff’s Voltage Law (KVL): The algebraic sum of potential differences around any closed loop is zero; \(\sum V = 0\).

These laws are essential for analysing circuits that cannot be reduced to simple series/parallel combinations.

9. Potential Divider

A pair of series resistors can be used to obtain a required fraction of the source voltage.

where \(V{x}\) is the voltage across resistor \(R{x}\) and \(V_{\text{s}}\) is the total applied voltage.

10. Practical Measurement Techniques (AO3)

• Wheatstone (metre‑bridge) method: Adjust a known variable resistor until the galvanometer reads zero. Then \(\displaystyle R{x}=R{1}\frac{R{3}}{R{2}}\).
• Four‑wire (Kelvin) method: Separate current‑carrying and voltage‑sensing leads to eliminate lead resistance – vital for low‑Ω measurements.
• Using a digital multimeter: Select the appropriate range, zero the instrument (if required), record the reading and its uncertainty.

11. Uncertainty and Error Propagation (AO3)

When resistance is calculated from \(R=\rho L/A\), the relative uncertainty is

Example: If \(\rho\) is known to 2 %, \(L\) to 1 % and \(A\) to 3 %, then

12. Non‑Ohmic Conductors

Ohm’s law (\(V=IR\)) holds only for ohmic devices where the \(V\)–\(I\) relationship is linear. Devices such as filament lamps, diodes and thermistors show a curved characteristic; they are non‑ohmic. The definition \(R=V/I\) still applies instantaneously, but the resistance varies with voltage or temperature.

13. Suggested Diagram

14. Quick‑Check Questions

1. State Ohm’s law and explain the meaning of each symbol.
2. How does doubling the length of a wire affect its resistance, assuming material, temperature and cross‑section remain unchanged?
3. If two wires of the same material have the same resistance, but one is twice as long as the other, what is the ratio of their cross‑sectional areas?
4. Write the temperature‑coefficient formula and calculate the resistance of a 10 Ω copper resistor at 70 °C (\(\alpha = 3.9\times10^{-3}\,\text{°C}^{-1}\), \(R_{0}=10\;\Omega\) at 20 °C).
5. Two resistors (8 Ω and 12 Ω) are connected in series with a 24 V battery. What voltage appears across the 12 Ω resistor?
6. Apply Kirchhoff’s voltage law to a loop containing a 5 V battery, a 2 Ω resistor and an unknown resistor \(R\). If the measured current is 1 A, find \(R\).
7. Describe how you would use a Wheatstone bridge to determine an unknown resistance and list two sources of experimental error.

15. Missing Topics Checklist (What to Study Next)

• Power dissipation in resistors and its applications (heating, fuses, etc.).
• Series and parallel circuit analysis, including equivalent resistance for mixed networks.
• Kirchhoff’s current and voltage laws for complex DC circuits.
• Potential‑divider principle and practical uses (voltage reference, sensor biasing).
• Practical resistance‑measurement techniques (Wheatstone bridge, four‑wire method, multimeter use).
• Experimental skills: uncertainty analysis, error propagation, and data presentation.
• Non‑ohmic behaviour and how to recognise it from V‑I graphs.