Short on time? This page is the recap. For the derivations, figures and worked examples, use the full study guide.
Recap
- Series: charge through one element must go through the others, with no other path. Same current in each.
- Parallel: elements share the same two end points, so charge can take any one path. Same potential difference across each.
- Reduce a network from the inside out, find the total current, then work back out to each resistor.
- Adding a parallel path lowers R_eq and raises the total current.
- Ideal wires and batteries have no resistance. You may ignore wire resistance only if the circuit has other resistance.
- A real battery is an ideal emf in series with an internal resistance r. The emf is the terminal voltage when no current flows.
- Unless told otherwise, batteries, wires and meters are ideal.
Key relationships
| Situation | Relationship |
|---|---|
| Resistors in series | R_eq = R₁ + R₂ + … |
| Resistors in parallel | 1/R_eq = 1/R₁ + 1/R₂ + … |
| Two in parallel | R_eq = R₁R₂/(R₁ + R₂) |
| n identical in parallel | R_eq = R/n |
| Single loop with a real battery | I = ℰ/(R + r) |
| Terminal voltage | ΔV = ℰ − Ir |
| Meter | Connected | Ideal resistance | A real one… |
|---|---|---|---|
| Ammeter | in series | zero | adds resistance, so current falls slightly |
| Voltmeter | in parallel | infinite | adds a parallel path, which usually lowers the reading |
Mistakes to avoid
- Calling elements parallel because they look side by side. Check the junctions at both ends.
- Forgetting to invert after adding reciprocals.
- Using “product over sum” for three resistors. It works for pairs only.
- Assuming terminal voltage equals emf when a current is drawn.
- Treating the battery as a constant-current source. It fixes ℰ, not I.
- Ammeter across an element. That creates a short circuit.
- Ignoring wire resistance in a short circuit. Then r and the wire set the current.
Quick self-check
- What is 12 Ω in parallel with 6.0 Ω? (4.0 Ω)
- A 3.0 Ω resistor is added in parallel to a circuit fed by an ideal battery. Does the battery current rise or fall? (Rise: R_eq falls)
- A battery of emf 9.0 V and internal resistance 1.5 Ω delivers 2.0 A. What is its terminal voltage? (6.0 V)
- Why does an ideal voltmeter have infinite resistance? (So no current passes through it and the circuit is unchanged)
Next: practice questions.