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 all of them; there is no other path. The current is the same in each.
- Parallel: charge takes one of several paths between the same two points. The potential difference is the same across each path.
- A group of resistors can be replaced by one equivalent resistance R_eq that draws the same current for the same potential difference.
- More parallel paths means lower R_eq.
- Wire resistance can be ignored only when other elements in the circuit have much larger resistance.
- emf ℰ: the terminal potential difference of a battery when there is no current in it.
- Ideal ammeter: zero resistance, in series. Ideal voltmeter: infinite resistance, in parallel. Real meters change the circuit slightly (qualitative only).
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| R_eq = R₁ + R₂ + … | Ω | combine resistors in series |
| 1/R_eq = 1/R₁ + 1/R₂ + … | Ω; invert at the end | combine resistors in parallel |
| R_eq = R/n | n identical resistors in parallel | shortcut for equal branches |
| ΔV_terminal = ℰ − Ir | ℰ and ΔV in V, I in A, r in Ω | find the voltage across a real battery |
| I = ℰ/(R + r) | single external resistance R | find the current with internal resistance |
Assumptions behind the numbers
- Batteries, wires and meters are ideal unless the question says otherwise.
- Resistances are constant (ohmic), so ΔV = IR applies to each resistor.
- No circuit has batteries of different emf connected in parallel.
Mistakes to avoid
- Not inverting 1/R_eq at the end.
- Calling elements parallel because of the drawing. Both ends must share the same two points.
- Splitting current equally between unequal parallel branches.
- Treating ℰ as the terminal voltage when there is a current.
- Putting an ammeter in parallel (it would short the element) or a voltmeter in series (it would open the loop).
- Saying a real voltmeter reads high. Its parallel branch lowers the resistance, so it reads slightly low.
Quick self-check
- What is the equivalent resistance of 12 Ω and 6.0 Ω in parallel? (4.0 Ω)
- A cell of emf 1.5 V and internal resistance 0.20 Ω carries 0.50 A. What is its terminal voltage? (1.4 V)
- A 20 Ω resistor is in series with two 20 Ω resistors in parallel. What is R_eq? (30 Ω)
- A third bulb is added in parallel with two bulbs across an ideal battery. Does the first bulb get dimmer? (No. Its potential difference is still the emf.)
Next: practice questions.