Current and voltage rules · Total resistance · Why homes use parallel circuits
⚡ State the current rule for series and parallel circuits
🔋 State the voltage rule for series and parallel circuits
🔢 Calculate total resistance in series and parallel circuits
📐 Apply Ohm's law alongside circuit rules to solve problems
🏠 Explain why parallel circuits are used in homes
📊 Interpret and draw circuit diagrams for both circuit types
What is a Series Circuit?
In a series circuit, all components are connected in a single loop — there is only one path for current to flow. If one component breaks (or is removed), the entire circuit is broken and all components stop working.
Series circuit: A circuit where components are connected one after another in a single loop, sharing the same current path.
Current Rule — Series
Because there is only one path, the same current flows through every component. Charge cannot build up or disappear anywhere in the loop.
Itotal = I₁ = I₂ = I₃ …
This means an ammeter placed anywhere in a series circuit gives the same reading.
Voltage Rule — Series
The total potential difference (voltage) supplied by the battery is shared between all components. Each component gets a "slice" of the total voltage. The slices always add up to the supply voltage.
Vtotal = V₁ + V₂ + V₃ …
A component with higher resistance gets a larger share of the voltage.
Total Resistance — Series
Adding more resistors in series increases the total resistance because the current must pass through each one in turn. The total resistance is simply the sum of all individual resistances.
Rtotal = R₁ + R₂ + R₃ …
For example, two 10 Ω resistors in series give a total resistance of 20 Ω — double that of one alone.
What is a Parallel Circuit?
In a parallel circuit, components are connected across separate branches between the same two points. There are multiple paths for current to flow. If one branch breaks, current can still flow through the other branches — the remaining components continue to work.
Parallel circuit: A circuit where components are connected across separate branches, each with its own current path.
Voltage Rule — Parallel
All branches connect between the same two points (same high-potential and low-potential nodes), so every branch has the same potential difference across it. This is equal to the supply voltage.
Vtotal = V₁ = V₂ = V₃ …
Current Rule — Parallel
At each junction where branches split, the current divides. Each branch carries a portion of the total current. The portions must add up to the total current drawn from the supply.
Itotal = I₁ + I₂ + I₃ …
Branches with lower resistance carry a larger current (more charge takes the easier path).
Total Resistance — Parallel
Adding more branches in parallel gives the current more paths, so the overall resistance decreases. The total resistance is always less than the smallest individual resistance.
1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ …
For two resistors in parallel: Rtotal = (R₁ × R₂) ÷ (R₁ + R₂)
For example, two 10 Ω resistors in parallel give Rtotal = (10 × 10)/(10 + 10) = 100/20 = 5 Ω — half the resistance of one alone.
Ohm's Law — The Essential Tool
To solve circuit problems, you almost always need Ohm's Law alongside the circuit rules above. Ohm's Law links potential difference (V), current (I) and resistance (R).
V = I × R
Symbol
Quantity
SI Unit
Unit Symbol
V
Potential difference (voltage)
volt
V
I
Current
ampere
A
R
Resistance
ohm
Ω
You can rearrange Ohm's Law to find any quantity:
I = V ÷ R R = V ÷ I
Always apply Ohm's Law to each component separately, using the voltage across that component and the current through it.
Why Homes Use Parallel Circuits
Every circuit in a domestic home is wired in parallel. This is not an accident — there are important practical reasons.
1. Each appliance receives the full mains voltage (230 V)
Appliances are designed to operate at 230 V. In a parallel circuit, every branch (and therefore every appliance) has 230 V across it. In a series circuit the voltage would be shared, so appliances would not receive enough voltage to work correctly.
2. Appliances work independently
Turning off one appliance (opening its switch) does not affect the others, because each appliance is on its own branch. In a series circuit, switching off one device would break the loop and turn off everything.
3. Different appliances can be added or removed
Plugging in a new device simply adds another branch in parallel. The other appliances are unaffected. Adding a device in series would change the total resistance and therefore the current and voltage for all other devices.
4. Individual fuses and switches
Each parallel branch can have its own fuse and switch, allowing individual control and protection of each circuit without affecting the others.
Summary: Parallel wiring gives every appliance the full supply voltage, allows independent operation, and enables individual control — all essential for safe and practical domestic use.
Comparing Series and Parallel — Quick Reference
Property
Series Circuit
Parallel Circuit
Current
Same through all components
Splits at each junction
Voltage
Shared (adds up to supply)
Same across all branches
Total resistance
R = R₁ + R₂ + … (increases)
1/R = 1/R₁ + 1/R₂ + … (decreases)
Effect of removing one component
All stop working
Others continue working
Common use
Simple indicator lights, some sensors
Domestic wiring, most real circuits
Potential difference (p.d.): The work done per unit charge as it moves between two points in a circuit. Measured in volts (V).
Current (I): The rate of flow of charge past a point. Measured in amperes (A).
Example 1 — Series Circuit: Find missing voltage and current
A series circuit contains a 12 V battery, a 4 Ω resistor (R₁) and an 8 Ω resistor (R₂). Find: (a) the total resistance, (b) the current in the circuit, (c) the voltage across each resistor.
1 Find total resistance using the series rule: Rtotal = R₁ + R₂ = 4 + 8 = 12 Ω
2 Find the current using Ohm's Law (V = I × R, so I = V ÷ R): I = Vtotal ÷ Rtotal = 12 ÷ 12 = 1 A This same current flows through every component in series.
3 Find voltage across R₁ (V = I × R): V₁ = 1 × 4 = 4 V
4 Find voltage across R₂: V₂ = 1 × 8 = 8 V
5 Check: V₁ + V₂ = 4 + 8 = 12 V ✓ (equals the supply voltage)
Rtotal = 12 Ω | I = 1 A | V₁ = 4 V | V₂ = 8 V
Example 2 — Parallel Circuit: Find branch currents and total current
A 6 V battery is connected to two resistors in parallel: R₁ = 3 Ω and R₂ = 6 Ω. Find the current through each resistor and the total current from the battery.
1 In a parallel circuit, the voltage across each branch equals the supply voltage: V₁ = V₂ = 6 V
2 Find current through R₁ using Ohm's Law: I₁ = V₁ ÷ R₁ = 6 ÷ 3 = 2 A
3 Find current through R₂: I₂ = V₂ ÷ R₂ = 6 ÷ 6 = 1 A
4 Apply the parallel current rule to find total current: Itotal = I₁ + I₂ = 2 + 1 = 3 A
5 Check using total resistance: 1/Rtotal = 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2 → Rtotal = 2 Ω I = V ÷ R = 6 ÷ 2 = 3 A ✓
I₁ = 2 A | I₂ = 1 A | Itotal = 3 A
Example 3 — Total Resistance in Parallel
Three resistors of 6 Ω, 12 Ω and 4 Ω are connected in parallel. Calculate the total resistance of the combination.
3 Find a common denominator (12): 1/Rtotal = 2/12 + 1/12 + 3/12 = 6/12 = 1/2
4 Take the reciprocal to find Rtotal: Rtotal = 2 Ω
5 Check: 2 Ω is less than the smallest individual resistance (4 Ω) ✓
Rtotal = 2 Ω
Example 4 — Mixed Circuit Analysis
A 9 V battery is connected to R₁ = 3 Ω in series with a parallel combination of R₂ = 4 Ω and R₃ = 4 Ω. Find: (a) the total resistance, (b) the total current, (c) the voltage across the parallel section.
1 Find resistance of the parallel combination (R₂ and R₃ both 4 Ω): Rparallel = (4 × 4) ÷ (4 + 4) = 16 ÷ 8 = 2 Ω
2 Now add R₁ in series with the parallel block: Rtotal = R₁ + Rparallel = 3 + 2 = 5 Ω
3 Find total current using Ohm's Law: Itotal = V ÷ Rtotal = 9 ÷ 5 = 1.8 A
4 Find voltage across R₁: VR₁ = Itotal × R₁ = 1.8 × 3 = 5.4 V
5 Find voltage across the parallel section: Vparallel = Itotal × Rparallel = 1.8 × 2 = 3.6 V Check: 5.4 + 3.6 = 9 V ✓
Rtotal = 5 Ω | Itotal = 1.8 A | Vparallel = 3.6 V
Q1. Three resistors of 5 Ω, 10 Ω and 15 Ω are connected in series. What is the total resistance?
Q2. In a parallel circuit, a 12 V battery powers two branches. What is the potential difference across each branch?
Q3. A series circuit has a 9 V battery and two resistors. The voltage across R₁ is 3 V. What is the voltage across R₂?
Q4. Two resistors, 6 Ω and 3 Ω, are connected in parallel. What is their combined resistance?
Q5. Give one reason why domestic appliances are connected in parallel rather than in series.
Challenge Q1 — Series Circuit Calculation
A series circuit consists of a battery of e.m.f. 15 V (internal resistance negligible), R₁ = 2 Ω, R₂ = 5 Ω and R₃ = 8 Ω.
(a) Calculate the total resistance. (b) Calculate the current in the circuit. (c) Calculate the voltage across R₂. (d) An engineer removes R₃ and replaces it with a wire (0 Ω). State and explain what happens to the current.
(a) Rtotal = 2 + 5 + 8 = 15 Ω
(b) I = V ÷ R = 15 ÷ 15 = 1 A
(c) VR₂ = I × R₂ = 1 × 5 = 5 V
(d) New Rtotal = 2 + 5 + 0 = 7 Ω. New I = 15 ÷ 7 ≈ 2.14 A. The current increases because the total resistance decreases — removing the 8 Ω component means less resistance opposes the flow of charge.
Challenge Q2 — Parallel Circuit Analysis
Three resistors R₁ = 10 Ω, R₂ = 10 Ω and R₃ = 5 Ω are connected in parallel across a 20 V supply.
(a) Calculate the total resistance of the parallel combination. (b) Calculate the current through each resistor. (c) Calculate the total current supplied by the battery. (d) Show that your answer to (c) is consistent with the total resistance from part (a).
A student claims: "Series circuits are simpler, so houses should be wired in series to save money on wire."
Evaluate this claim. In your answer, refer to: voltage requirements of appliances, the effect of one appliance breaking, individual control of appliances, and any other relevant physics. (6 marks)
Model Answer (6 mark points):
1. Voltage: In a series circuit the supply voltage is shared between appliances. Each appliance would receive less than 230 V and would not function correctly (e.g. a 230 V lamp wired in series with other appliances would be dim or not work at all).
2. Parallel voltage advantage: In a parallel circuit every branch has the same potential difference as the supply (230 V), so every appliance receives exactly the voltage it is designed for.
3. Independence — failure: In a series circuit, if one appliance breaks (open circuit) the loop is broken and all appliances stop working. In parallel, if one branch fails, current continues to flow in the other branches — other appliances are unaffected.
4. Independence — switching: In a series circuit, switching off one appliance turns off everything. In parallel, each appliance has its own switch and can be turned on or off independently.
5. Adding appliances: In series, adding more appliances increases total resistance, reducing current and changing voltages across all other appliances. In parallel, adding a device simply adds a new branch — existing appliances are unaffected.
6. Conclusion: The student's claim is incorrect. Although series circuits use less wire, the practical disadvantages (shared voltage, no independent control, complete failure if one component breaks) make them entirely unsuitable for domestic wiring. The benefits of parallel circuits far outweigh the extra cost of wiring.
Challenge Q4 — Mixed Series-Parallel
A 24 V battery (negligible internal resistance) is connected to R₁ = 6 Ω in series with a parallel combination of R₂ = 12 Ω and R₃ = 12 Ω.
(a) Find the resistance of the parallel combination. (b) Find the total circuit resistance. (c) Find the total current from the battery. (d) Find the voltage across the parallel combination. (e) Find the current through R₂ and through R₃.