In electrical materials, what is the mathematical relationship between resistivity (ρ) and conductivity (σ) of a conductor?

Introduction / Context:
Resistivity and conductivity are fundamental properties of materials that describe how easily electric charge can flow through them. Resistivity tells us how strongly a material opposes current, while conductivity tells us how readily it allows current to flow. This question tests your understanding of the mathematical relationship between these two quantities.

Given Data / Assumptions:

• Resistivity (ρ) is a material property with units of ohm metre (Ω·m).
• Conductivity (σ) is also a material property, with units of siemens per metre (S/m).
• We assume a homogeneous, isotropic conductor and direct current conditions.
• The relationship between resistance, resistivity, conductivity and geometry is known.

Concept / Approach:
For a uniform conductor of length L and cross-sectional area A, the resistance R is given by R = ρ * (L / A). The conductivity σ is defined as the reciprocal of resistivity: σ = 1 / ρ. This means that materials with high resistivity have low conductivity, and materials with low resistivity have high conductivity. The relationship ρ = 1 / σ expresses this inverse connection clearly. Any formula suggesting equality, squares or a simple multiple is incorrect.

Step-by-Step Solution:
Step 1: Recall that resistance R is related to resistivity ρ by R = ρ * (L / A). Step 2: Conductivity σ is defined such that current density J is proportional to electric field E via J = σ * E. Step 3: Using Ohm's law and definitions, one can show that σ = 1 / ρ for a given material. Step 4: Rearranging this gives ρ = 1 / σ, showing that resistivity and conductivity are reciprocals. Step 5: Therefore, the correct relationship is ρ = 1 / σ, meaning that high resistivity implies low conductivity and vice versa.

Verification / Alternative check:
Consider a copper wire, which is a good conductor. It has low resistivity (for example, around 1.7 × 10^-8 Ω·m) and high conductivity (around 5.9 × 10^7 S/m). If you multiply ρ and σ for a given material, the product ρ * σ is approximately 1, confirming that they are reciprocals. For an insulator like rubber, resistivity is very high and conductivity is very low, again consistent with ρ = 1 / σ. Reference tables for materials always list resistivity and conductivity as inverse quantities.

Why Other Options Are Wrong:
Resistivity and conductivity are equal: ρ = σ: This is incorrect because good conductors have low ρ but high σ; they are not equal in magnitude or units.

Resistivity is the square of conductivity: ρ = σ^2: There is no theoretical or practical basis for a square relationship in basic material properties.
Resistivity is twice the conductivity: ρ = 2σ: Again, this is arbitrary and incorrect; the relationship is inverse, not a constant multiple.

Common Pitfalls:
A common error is to think that a better conductor must have both high conductivity and high resistivity, forgetting that these are opposites. Another confusion comes from mixing up resistance (which depends on length and area) with resistivity (a pure material property). Keep in mind that resistivity and conductivity are reciprocal material constants: if one goes up, the other goes down.

Final Answer:
The correct relationship is that Resistivity is the reciprocal of conductivity: ρ = 1 / σ.