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Resistance and Resistivity

Resistance depends on geometry (L, A) and material property ρ (resistivity). σ is conductivity. SI unit of ρ: Ω·m. SI unit of σ: S m⁻¹ (siemens per metre).
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Derivation

Resistance from geometry

From V=IRV = IR and the derivation of Ohm's law:

R=VI=ELσEA=LσA=ρLAR = \frac{V}{I} = \frac{EL}{\sigma EA} = \frac{L}{\sigma A} = \rho\frac{L}{A} R=ρLA\boxed{R = \rho\frac{L}{A}}

where ρ=1/σ\rho = 1/\sigma is the resistivity of the material.

Dependence on geometry

  • R∝LR \propto L: longer conductor → more collisions → more resistance
  • R∝1/AR \propto 1/A: wider conductor → more parallel paths → less resistance

Microscopic expression for resistivity

From drift velocity: vd=eEτ/mv_d = eE\tau/m, and J=nevd=ne2Eτ/m=σEJ = nev_d = ne^2E\tau/m = \sigma E:

σ=ne2τm  ⟹  ρ=mne2τ\sigma = \frac{ne^2\tau}{m} \implies \boxed{\rho = \frac{m}{ne^2\tau}}

Units

[ρ]=Ω⋅m[\rho] = \Omega\cdot\text{m}. Typical values:

  • Copper: ρ≈1.7×10−8\rho \approx 1.7\times10^{-8} Ω·m (excellent conductor)
  • Silicon: ρ≈640\rho \approx 640 Ω·m (semiconductor)
  • Glass: ρ≈1012\rho \approx 10^{12} Ω·m (insulator)
Note
Resistivity is a material property; resistance is a device property. Two wires of the same material but different dimensions have different $R$ but the same $\rho$.