Series and Parallel Circuits
Analyze voltage, current, equivalent resistance, and power in series and parallel circuits using conservation of charge and energy.
A lesson of Physics, in Electricity and Magnetism, about 10 minutes.
By the end of this lesson, you will be able to analyze series and parallel resistor circuits, find equivalent resistance, and apply Kirchhoff's Laws.
Series Circuits
Components connected end-to-end; single current path.
Same current through all resistors: $I_1 = I_2 = I_3 = I$
Voltage divides: $V = V_1 + V_2 + V_3$
Equivalent resistance: $R_{eq} = R_1 + R_2 + R_3$ (always > any individual $R$)
Parallel Circuits
Components connected between the same two nodes; multiple current paths.
Same voltage across all resistors: $V_1 = V_2 = V_3 = V$
Current divides: $I = I_1 + I_2 + I_3$
Equivalent resistance: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ (always < smallest individual $R$)
Example 1 — Series Circuit
$R_1 = 4$ Ω and $R_2 = 6$ Ω in series with 10 V battery.
$R_{eq} = 10$ Ω; $I = 10/10 = 1.0$ A
$V_1 = 4$ V; $V_2 = 6$ V (proportional to resistance); sum = 10 V ✓
Example 2 — Parallel Circuit
$R_1 = 4$ Ω and $R_2 = 6$ Ω in parallel with 12 V battery.
$\frac{1}{R_{eq}} = \frac{1}{4} + \frac{1}{6} = \frac{5}{12} \implies R_{eq} = 2.4$ Ω
$I_{\text{total}} = 12/2.4 = 5$ A
$I_1 = 12/4 = 3$ A; $I_2 = 12/6 = 2$ A; sum = 5 A ✓
Label the Diagram: Reading a Circuit
Every circuit you will meet is drawn with these symbols. Learning to read them is most of learning circuits.
The detail worth the trouble. The ammeter is in the loop, in series, because it measures the current passing through. The voltmeter is connected across the lamp, in parallel, because it measures the difference in potential between two points. Wire them the other way round and an ammeter across a cell is a short circuit.
Sort It: Series or Parallel
Kirchhoff's Laws
Junction Rule (KCL): the sum of currents entering a junction = sum leaving. (Conservation of charge.)
Loop Rule (KVL): the sum of all voltage changes around any closed loop = 0. (Conservation of energy.)
Why Parallel Is Used in Homes
Household circuits are wired in parallel:
- All appliances receive the same voltage (120 V)
- Adding more devices doesn't reduce voltage to others
- If one device fails, others still work
Series lights: if one bulb burns out, the entire string goes dark.