state Ohm’s law

Resistance and Resistivity

Learning Objective

State Ohm’s law and understand the relationship between resistance, resistivity, length and cross‑sectional area of a conductor.

Key Definitions

• Resistance (R): The opposition offered by a material to the flow of electric current. Measured in ohms (Ω).
• Resistivity (ρ): An intrinsic property of a material that quantifies how strongly it resists electric current. Measured in ohm‑metres (Ω·m).
• Conductivity (σ): The reciprocal of resistivity, σ = 1/ρ, measured in siemens per metre (S·m⁻¹).

Ohm’s Law

Ohm’s law relates the potential difference (V) across a conductor to the current (I) flowing through it and its resistance (R):

$$ V = I R $$

In terms of current:

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

Ohm’s law is valid for ohmic conductors where the resistance remains constant over a range of applied voltages.

Resistance of a Uniform Conductor

The resistance of a uniform cylindrical conductor can be expressed using its resistivity:

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

where:

• $\rho$ = resistivity of the material (Ω·m)
• $L$ = length of the conductor (m)
• $A$ = cross‑sectional area (m²)

Factors Affecting Resistance

1. Material: Different materials have different resistivities.
2. Length ($L$): Resistance is directly proportional to length.
3. Cross‑sectional area ($A$): Resistance is inversely proportional to area.
4. Temperature: For most conductors, resistivity increases with temperature; for semiconductors it decreases.

Typical Resistivity \cdot alues

Material	Resistivity (Ω·m)
Copper	1.68 × 10⁻⁸
Aluminium	2.82 × 10⁻⁸
Silver	1.59 × 10⁻⁸
Nickel–Chromium alloy (Nichrome)	1.10 × 10⁻⁶
Glass (dry)	10¹⁰ – 10¹⁴

Example Calculation

Calculate the resistance of a copper wire 2.0 m long with a cross‑sectional area of 0.5 mm².

Given:

$$ \rho_{\text{Cu}} = 1.68 \times 10^{-8}\ \text{Ω·m},\; L = 2.0\ \text{m},\; A = 0.5\ \text{mm}^2 = 0.5 \times 10^{-6}\ \text{m}^2 $$

Using $R = \rho L / A$:

$$ R = \frac{1.68 \times 10^{-8} \times 2.0}{0.5 \times 10^{-6}} = \frac{3.36 \times 10^{-8}}{5.0 \times 10^{-7}} = 0.0672\ \text{Ω} $$

Thus the resistance is approximately $6.7 \times 10^{-2}\ \text{Ω}$.

Suggested Diagram

Summary

• Ohm’s law: $V = IR$ for ohmic materials.
• Resistance depends on material (via resistivity), length, and cross‑sectional area: $R = \rho L/A$.
• Resistivity is a fundamental property; lower resistivity means better conductivity.