Resistance in Circuits
Resistance controls how much current flows for a given voltage. Understanding series and parallel combinations is fundamental to circuit analysis.
Series and Parallel Combinations
Series: R_eq = R₁ + R₂ + R₃
Parallel: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃
Two resistors parallel shortcut:
R_eq = (R₁ × R₂) / (R₁ + R₂)
Example: 100Ω and 150Ω in parallel:
R_eq = (100×150)/(100+150) = 60 ΩResistivity Formula
R = ρ × L / A
ρ = resistivity (Ω·m)
L = wire length (m)
A = cross-sectional area (m²)
Copper (ρ=1.68×10⁻⁸): 10m of 1mm² wire:
R = 1.68×10⁻⁸ × 10 / 10⁻⁶ = 0.168 ΩTemperature Effect
R(T) = R₀ × [1 + α(T - T₀)]
α = temperature coefficient (/°C)
Copper: α = 0.00393/°C
Nichrome: α = 0.0004/°C (used in heaters)
Copper wire at 80°C vs 20°C:
R₈₀ = R₂₀ × [1 + 0.00393×60] = 1.236 R₂₀ (+23.6%)Calculate resistance: Free Resistance Calculator
Ohm's Law Quick-Reference Table
| Voltage (V) | Resistance (Ω) | Current (A) | Power (W) |
|---|---|---|---|
| 1.5 (AA battery) | 10 | 0.15 | 0.225 |
| 5 (USB) | 5 | 1.0 | 5.0 |
| 12 (car battery) | 2 | 6.0 | 72 |
| 120 (US mains) | 60 | 2.0 | 240 |
| 230 (EU mains) | 52 | 4.4 | 1,012 |
How Ohm's Law Works
Ohm's Law (V = IR) states that the voltage across a conductor is proportional to the current through it, with resistance as the proportionality constant. Rearranged: I = V/R and R = V/I. Power combines with Ohm's Law to give the Joule heating equations: P = IV = I²R = V²/R. These four relationships (V, I, R, P) form the foundation of all DC circuit analysis.
Ohm's Law applies to linear (ohmic) conductors — materials whose resistance stays constant with changing voltage, such as resistors. Non-ohmic devices like diodes, transistors, and light bulbs (where resistance changes with temperature) require more complex models. Understanding Ohm's Law is essential for designing safe electrical wiring, calculating fuse ratings, sizing power supplies, and troubleshooting circuits.
Common Mistakes
- Forgetting units: Voltage in volts, current in amperes, resistance in ohms. Milliamperes and kilohms are common in electronics — convert first.
- Applying Ohm's Law to non-ohmic components: LEDs, diodes, and batteries do not obey V = IR with a constant R. Use their characteristic curves instead.
- Series vs. parallel confusion: Resistors in series add directly (R_total = R₁ + R₂). In parallel: 1/R_total = 1/R₁ + 1/R₂. Mixing these up gives completely wrong circuit behaviour.
Frequently Asked Questions
Resistance R = ρL/A, where ρ is resistivity, L is length, and A is cross-sectional area. A thicker wire has larger A, so resistance decreases. This is why high-current applications (house wiring, motor leads, battery cables) use thicker conductors — to reduce both resistance heating and voltage drop.
For DC circuits, Ohm's Law applies directly. In AC circuits, reactive components (capacitors and inductors) introduce impedance (Z), which is frequency-dependent. The generalised form is V = IZ, where Z is a complex number combining resistance and reactance. For purely resistive AC circuits, Z = R and Ohm's Law applies unchanged.
I = P/V. A 1,000 W microwave on a 120 V circuit draws 1000/120 ≈ 8.3 A. On 230 V, the same power draws 1000/230 ≈ 4.3 A. This is why 230 V systems can use thinner wires for the same power delivery — the lower current reduces I²R heating losses.