Current electricity class 12 notes: Electrical Resistivity & Conductivity
Unit 2 • Current ElectricityUnderstanding the intrinsic properties of materials that govern their resistance to current flow
Electrical Resistivity (ρ)
Definition:
Resistivity is an intrinsic property of a material that quantifies how strongly it opposes the flow of electric current, independent of its shape or size.
Fundamental Equation
Where:
R = Resistance (Ω)
ρ = Resistivity (Ω·m)
L = Length (m)
A = Cross-sectional area (m²)
Material Dependence
- Pure metals have lowest resistivity (Ag > Cu > Au > Al)
- Alloys have higher resistivity than pure metals
- Semiconductors have intermediate resistivity
- Insulators have very high resistivity (>1010 Ω·m)
| Material | Resistivity (Ω·m) | Type |
|---|---|---|
| Silver | 1.59×10-8 | Conductor |
| Copper | 1.68×10-8 | Conductor |
| Silicon | 2.3×103 | Semiconductor |
| Glass | 1010-1014 | Insulator |
Electrical Conductivity (σ)
Definition:
Conductivity is the reciprocal of resistivity, measuring a material’s ability to conduct electric current.
Relationship
Units: Siemens/meter (S/m) or Ω-1m-1
Microscopic Interpretation
Where:
n = charge carrier density (m-3)
e = electron charge (1.6×10-19 C)
μ = mobility (m²/V·s)
Key Points
- Good conductors have high σ (~107 S/m)
- Semiconductors have σ that increases with temperature
- Superconductors have σ → ∞ below critical temperature
- Conductivity depends on both carrier density and mobility
Temperature Dependence
Conductors
- ρ increases with temperature
- α = positive temperature coefficient
- Due to increased lattice vibrations scattering electrons
- Typical α ~ 0.004/°C for pure metals
Semiconductors
- ρ decreases with temperature
- Negative temperature coefficient
- Due to increased charge carriers
- Eg = band gap energy
Special Cases
Superconductors:
- ρ → 0 below critical temperature Tc
- No energy loss during current flow
- Used in MRI machines, maglev trains
Alloys:
- Small temperature dependence
- Used in precision resistors (e.g., manganin)
- α ≈ 0.00001/°C for manganin
Worked Example
Problem:
A copper wire (ρ = 1.68×10-8 Ω·m, α = 0.00404/°C) has:
- Length = 50m
- Diameter = 1mm
- Initial temperature = 20°C
Calculate:
- Resistance at 20°C
- Resistance at 80°C
- % increase in resistance
Solution:
(1) Resistance at 20°C:
(2) Resistance at 80°C:
(3) % increase:
Practice Problems
Problem 1
A tungsten wire (ρ = 5.6×10-8 Ω·m) has 2Ω resistance. If length is 10m, find its diameter.
Problem 2
Calculate the conductivity of aluminum (ρ = 2.65×10-8 Ω·m) in MS/m.
Problem 3
A copper wire has 1.2Ω resistance at 20°C. Find its resistance at 100°C (α = 0.00404/°C).
Problem 4
A silicon sample has electron mobility 0.15 m²/V·s and carrier density 1.5×1016 m-3. Calculate its resistivity.
Problem 5
A manganin wire (α ≈ 0) has 100Ω resistance at 20°C. What will be its resistance at 200°C?
Key Takeaways
Essential Concepts
- Resistivity (ρ) is intrinsic to material
- Conductivity (σ) = 1/ρ
- Resistance depends on both ρ and geometry
- Metals: ρ increases with temperature
- Semiconductors: ρ decreases with temperature
Common Mistakes
- Confusing resistance and resistivity
- Using wrong temperature coefficient sign
- Forgetting to convert units (mm to m, etc.)
- Assuming all materials have same temperature dependence
Next Topic: Temperature Dependence of Resistance
In the next section, we’ll explore in-depth how resistance changes with temperature for different materials.
Continue to Next Topic →
