PhysicsApplying Ohm's Law to Series and Parallel Circuits
Atlas stands at a cluttered electronics workbench covered with batteries, jumper wires, resistors, and a glowing multimeter, tracing a circuit diagram with one finger while explaining current flow to a curious student nearby.
- Explain what Ohm's Law states and identify the three quantities it relates.
- Calculate total resistance for resistors connected in series and in parallel.
- Predict the current through each branch of a parallel circuit using Ohm's Law.
- Compare how voltage distributes across series resistors versus parallel resistors.
- Solve multi-step problems involving combined series and parallel resistor networks.
Key terms
- Ohm's law
- The relationship V = IR linking voltage, current, and resistance in a resistive component.
- Equivalent resistance
- The single resistance that could replace a network while drawing the same current at the same voltage.
- Series connection
- A single-path arrangement where the same current flows through every component and voltages add.
- Parallel connection
- A multi-branch arrangement where each branch shares the same voltage and branch currents add.
- Node
- A junction point in a circuit where two or more components connect and currents combine or split.
Combining Resistors Correctly
The rules for combining resistors follow directly from how charge and voltage behave. In series there is one path, so the same current passes through each resistor and the total resistance is the simple sum R_total = R₁ + R₂ + …. In parallel each branch spans the same two nodes, so every branch sees the full source voltage and the reciprocals add: 1/R_total = 1/R₁ + 1/R₂ + …. A crucial consequence is that the parallel equivalent is always smaller than the smallest branch resistor, because each added path gives current another route.
A Strategy for Mixed Networks
Real circuits combine series and parallel sections, so a systematic method prevents errors. First identify which resistors share a single current path (series) and which share two nodes (parallel). Collapse each parallel group into one equivalent resistor using the reciprocal formula, then sum the resulting series resistors to find R_total. Use Ohm's law I = V/R_total to find the total current, then work backward with V = IR to recover the voltage across and current through each original component. Reducing the network one block at a time keeps the bookkeeping manageable.
Worked examples
Three resistors of 4 Ω, 6 Ω, and 2 Ω are in series across a 12 V battery. Find the current.
- Add the series resistances: R_total = 4 + 6 + 2 = 12 Ω.
- Apply Ohm's law for the whole loop: I = V / R_total.
- Substitute: I = 12 V / 12 Ω.
- The same current flows through every series resistor, so I = 1.0 A.
Answer: 1.0 A through every resistor.
A 6 Ω and a 3 Ω resistor in parallel are in series with a 2 Ω resistor across a 12 V battery. Find the total current.
- Combine the parallel pair: 1/R_p = 1/6 + 1/3 = 1/6 + 2/6 = 3/6, so R_p = 2 Ω.
- Add the series resistor: R_total = 2 Ω + 2 Ω = 4 Ω.
- Apply Ohm's law: I = V / R_total = 12 V / 4 Ω.
- Divide to get the total current I = 3.0 A.
Answer: 3.0 A from the battery (all of it through the 2 Ω resistor).
Activity
Build a circuit by placing resistors into series or parallel slots, then calculate the missing voltage, current, or resistance values for each configuration.
Practice
Two resistors of 10 Ω and 15 Ω are connected in parallel; calculate their equivalent resistance to one decimal place.
Explain why holiday lights wired in series all go dark when one bulb burns out but parallel ones do not.
Common mistakes to avoid
- Parallel resistances add the same way series ones do.Parallel resistors combine by adding reciprocals, giving an equivalent always smaller than the smallest branch, not a simple sum.
- Each branch in a parallel circuit gets a share of the voltage.Every parallel branch receives the full source voltage; it is the current, not the voltage, that splits among branches.
Check your understanding
Three resistors — 4 Ω, 6 Ω, and 2 Ω — are connected in series to a 12 V battery. What is the current flowing through the 6 Ω resistor?
A 12 Ω resistor and a 6 Ω resistor are connected in parallel across a 12 V source. A student claims the total resistance is 18 Ω because 12 + 6 = 18. What is the actual total resistance, and why is the student wrong?
In a parallel circuit with two branches, branch A has resistance 10 Ω and branch B has resistance 5 Ω. The supply voltage is 20 V. How much current flows through branch A?
A 6 Ω resistor and a 3 Ω resistor are connected in parallel, and that parallel combination is connected in series with a 2 Ω resistor. The whole network is powered by a 12 V battery. What is the current through the 2 Ω resistor?
Recap
Ohm's law V = IR relates voltage, current, and resistance for any resistor or whole network. Series resistances add and share one current, parallel resistances combine by reciprocals and share one voltage, and mixed networks are solved by collapsing blocks step by step.
Reflect
Why does your home use parallel wiring so each appliance works independently?