🔬 11.5 Factors Affecting Resistance
The resistance of a conductor depends on three main factors: its length, cross-sectional area, and the material it's made from.
R = ρ × (l/A)
Where:
- R = Resistance (Ohms, Ω)
- ρ = Resistivity (Ohm-meters, Ω·m)
- l = Length (meters, m)
- A = Cross-sectional area (square meters, m²)
Key Facts:
- Resistance is directly proportional to length (R ∝ l)
- Resistance is inversely proportional to area (R ∝ 1/A)
- Different materials have different resistivities
- Metals: 10⁻⁸ to 10⁻⁶ Ω·m
- Insulators: 10¹² to 10¹⁷ Ω·m
➡️ 11.6.1 Resistors in Series
When resistors are connected end-to-end, they are in series. The current through each resistor is the same.
Rs = R₁ + R₂ + R₃ + ...
Where:
- Rs = Total series resistance
- R₁, R₂, R₃ = Individual resistances
Key Properties:
- Same current through all resistors
- Voltage divides across resistors
- V = V₁ + V₂ + V₃
- Total resistance increases
- Used in voltage dividers
⚡ 11.6.2 Resistors in Parallel
When resistors are connected across the same two points, they are in parallel. The voltage across each resistor is the same.
1/Rp = 1/R₁ + 1/R₂ + 1/R₃ + ...
Where:
- Rp = Total parallel resistance
- R₁, R₂, R₃ = Individual resistances
Key Properties:
- Same voltage across all resistors
- Current divides among resistors
- I = I₁ + I₂ + I₃
- Total resistance decreases
- Used in home wiring
Example 11.6 - Resistance with Different Dimensions
Problem: A wire of given material having length l and area of cross-section A has a resistance of 4 Ω. What would be the resistance of another wire of the same material having length l/2 and area of cross-section 2A?
Solution:
For first wire: R₁ = ρ(l/A) = 4 Ω
For second wire: R₂ = ρ(l/2)/(2A)
R₂ = ρ × (1/4) × (l/A)
R₂ = (1/4) × R₁ = (1/4) × 4 Ω
Answer: R₂ = 1 Ω
Example 11.8 - Parallel Resistors
Problem: Three resistors of 5 Ω, 10 Ω, and 30 Ω are connected in parallel to a 12 V battery. Calculate the current through each resistor and the total current.
Solution:
I₁ = V/R₁ = 12V/5Ω = 2.4 A
I₂ = V/R₂ = 12V/10Ω = 1.2 A
I₃ = V/R₃ = 12V/30Ω = 0.4 A
Total current: I = 2.4 + 1.2 + 0.4 = 4.0 A
Total resistance: 1/Rₚ = 1/5 + 1/10 + 1/30 = 10/30
Answer: Rₚ = 3 Ω
Activity 11.3 - Exploring Resistance Factors
- Does the current depend on the length of the conductor?
- Does the current depend on the area of cross-section of the wire?
- How does material affect resistance?
Observations:
• Current decreases to half when length is doubled
• Current increases when thicker wire is used
• Different materials show different currents
| Material |
Resistivity (Ω·m) |
Type |
Common Use |
| Silver |
1.60 × 10⁻⁸ |
Conductor |
Premium electronics |
| Copper |
1.62 × 10⁻⁸ |
Conductor |
Electrical wiring |
| Nichrome |
100 × 10⁻⁶ |
Alloy |
Heating elements |
| Glass |
10¹⁰ - 10¹⁴ |
Insulator |
Electrical insulation |
| Rubber |
10¹³ - 10¹⁶ |
Insulator |
Wire coating |