D. Physics: Principles and Applications of Electricity · Electricity: Charges, Circuits and EnergyParallel Circuits and Comparing Series and Parallel
D2.6 D2.5 A1.2About 75 minutes
Learning goals- I can construct and draw a parallel circuit and measure the current and potential differences in it.
- I can explain why every branch of a parallel circuit has the same potential difference as the source.
- I can show that the branch currents add up to the total current.
- I can calculate the total resistance of resistors in parallel and explain why it is less than the smallest branch resistance.
- I can compare series and parallel circuits and explain why homes are wired in parallel.
Words to know
parallel circuit
A circuit with two or more paths (branches) for the current. Each load is on its own branch, connected across the source.
branch
One of the separate paths in a parallel circuit.
junction
A point where wires split into branches or join back together, shown by a dot.
Kirchhoff's current law
The total current into a junction equals the total current out of it.
total resistance in parallel
Found from 1/R_T = 1/R₁ + 1/R₂ + ... It is always less than the smallest branch resistance.
Why don't your lights go off when the toaster pops?
In your home, you can switch off the kitchen light and the fridge keeps running. One bulb in a ceiling fixture can burn out while the others stay lit. That is because home circuits are wired in parallel: every device has its own path to the source. Today you will find out how current, potential difference and resistance behave when there is more than one path.
Rule 1: the same potential difference across every branch
Each branch connects directly to both terminals of the source. So each coulomb, whichever branch it takes, gives up all the energy it got from the source in that one branch:
V_T = V₁ = V₂ = V₃ = ...
In a home, that is why every outlet and every light gets the same 120 V.
Rule 2: the branch currents add up
When the current reaches a junction, it splits. Charge is not created or lost, so:
I_T = I₁ + I₂ + I₃ + ...
This is Kirchhoff's current law. Each branch's current depends on its own resistance: I = V ÷ R. The branch with less resistance gets more current.
Rule 3: adding branches lowers the total resistance
Every new branch is a new path for the current, so the circuit lets more current flow from the source. More current for the same voltage means less total resistance. The rule is:
1/R_T = 1/R₁ + 1/R₂ + 1/R₃ + ...
The total resistance of a parallel circuit is always less than the smallest branch resistance.
Check with our circuit: 1/R_T = 1/6.0 + 1/12 = 2/12 + 1/12 = 3/12, so R_T = 12/3 = 4.0 Ω. Then I_T = 12 V ÷ 4.0 Ω = 3.0 A. That matches the main ammeter!
Shortcut for two equal resistors: R_T is half of one of them (two 10 Ω in parallel = 5 Ω).
Worked exampleExample 1: Solve a parallel circuit
A 12 V battery is connected to three resistors in parallel: 12 Ω, 6.0 Ω and 4.0 Ω. Find each branch current, the total current and the total resistance.
- Each branch has 12 V across it.
- I₁ = 12 ÷ 12 = 1.0 A; I₂ = 12 ÷ 6.0 = 2.0 A; I₃ = 12 ÷ 4.0 = 3.0 A
- I_T = 1.0 + 2.0 + 3.0 = 6.0 A
- R_T = V_T ÷ I_T = 12 ÷ 6.0 = 2.0 Ω
- Check with the formula: 1/12 + 1/6 + 1/4 = 1/12 + 2/12 + 3/12 = 6/12, so R_T = 12/6 = 2.0 Ω ✓
I₁ = 1.0 A, I₂ = 2.0 A, I₃ = 3.0 A, I_T = 6.0 A, R_T = 2.0 Ω (less than the smallest resistor, 4.0 Ω).
Worked exampleExample 2: Find a missing branch
Two lamps are in parallel with a 6.0 V battery. The total current is 1.5 A and lamp 1 carries 0.50 A. Find the current through lamp 2 and the resistance of each lamp.
- I₂ = I_T − I₁ = 1.5 − 0.50 = 1.0 A
- Each lamp has 6.0 V across it.
- R₁ = 6.0 ÷ 0.50 = 12 Ω; R₂ = 6.0 ÷ 1.0 = 6.0 Ω
I₂ = 1.0 A, R₁ = 12 Ω, R₂ = 6.0 Ω.
| Series | Parallel |
|---|
| Paths for current | One | Two or more (branches) |
| Current | Same everywhere: I_T = I₁ = I₂ | Splits and adds: I_T = I₁ + I₂ |
| Potential difference | Shared: V_T = V₁ + V₂ | Same on every branch: V_T = V₁ = V₂ |
| Total resistance | R_T = R₁ + R₂ (bigger than any one) | 1/R_T = 1/R₁ + 1/R₂ (smaller than any one) |
| Adding a lamp | Total current drops; all lamps dim | Total current rises; other lamps stay the same brightness |
| One lamp burns out | All go out | The others stay on |
| Used for | Switches, fuses, simple strings | Home wiring, car lights, power bars |
Why homes use parallel circuits (and why breakers trip)
- Every device gets the full 120 V, and each can be switched on and off independently.
- But every device you switch on adds a branch, so the total current in the circuit's wires increases.
- Each home circuit has a breaker (often 15 A) in series with the whole circuit. If too many devices are on, the total current passes the limit and the breaker trips, protecting the wires from overheating.
Hands-on activityParallel circuit lab (virtual or real)
You need: the free PhET simulation "Circuit Construction Kit: DC", or a real kit (a holder for 2 AA cells, 2 or 3 small bulbs in holders, wires, a multimeter).
- Build a circuit with one bulb. Record the brightness, the current from the battery, and the potential difference across the bulb.
- Add a second bulb on its own branch. Record the current in the main wire and in each branch, and the potential difference across each bulb.
- Add a third branch and repeat.
- Remove one bulb. What happens to the others?
- Build a series circuit with the same bulbs and compare the brightness side by side. Write a comparison paragraph.
What to look for: in parallel, each bulb has the full battery voltage and stays equally bright as bulbs are added, while the main current increases (the branch currents add). Removing one bulb leaves the others lit. The parallel bulbs are brighter than the same bulbs in series.
Safety (grown-up note): Use only AA batteries. Never connect a wire across a battery's terminals. Note that real batteries run down faster with many parallel branches because the total current is larger.
Just for funParallel lines never meet, but parallel circuits meet twice: once where the branches split, and once where they join back up. Very sociable for something called "parallel".
Your turn
1Three lamps are in parallel with a 9.0 V battery. What is the potential difference across each lamp?
9.0 V each. Every branch has the source's full potential difference.
2In a parallel circuit, branch currents are 0.30 A and 0.50 A. What is the total current?
I_T = 0.30 + 0.50 = 0.80 A.
3Find the total resistance of 20 Ω and 20 Ω in parallel.
Two equal resistors in parallel: half of one, 10 Ω. (1/R_T = 1/20 + 1/20 = 2/20, so R_T = 10 Ω.)
4Find the total resistance of 30 Ω and 60 Ω in parallel.
1/R_T = 1/30 + 1/60 = 2/60 + 1/60 = 3/60, so R_T = 20 Ω.
5A 12 V battery runs a 4.0 Ω resistor and a 6.0 Ω resistor in parallel. Find each branch current and the total current.
I₁ = 12 ÷ 4.0 = 3.0 A; I₂ = 12 ÷ 6.0 = 2.0 A; I_T = 5.0 A.
6Why does adding a branch to a parallel circuit increase the total current?
It adds another path, so the total resistance decreases. With the same voltage, I = V ÷ R gives more current.
7Give two reasons homes are wired in parallel.
Each device gets the full 120 V, and devices can be turned on and off independently (one failing does not stop the others).
More practice: warm-up
1What does a dot on a circuit diagram mean?
A junction: wires are connected there.
2Write the current rule for parallel circuits.
3Is the total resistance in parallel bigger or smaller than the smallest branch?
4One bulb burns out in a parallel circuit. What happens to the others?
5In which type of circuit is the potential difference the same across every load?
More practice: on your own
1A 6.0 V battery is connected to three 18 Ω resistors in parallel. Find R_T and I_T.
1/R_T = 3/18, so R_T = 6.0 Ω. I_T = 6.0 ÷ 6.0 = 1.0 A (each branch 0.33 A).
2Two branches carry a total of 2.4 A from a 12 V battery. One branch has a 10 Ω resistor. What is the resistance of the other branch?
I₁ = 12 ÷ 10 = 1.2 A. I₂ = 2.4 − 1.2 = 1.2 A. R₂ = 12 ÷ 1.2 = 10 Ω.
3A kitchen circuit (120 V, 15 A breaker) has a 1200 W kettle drawing 10 A and a toaster drawing 7.5 A. What happens if both are on? Explain using Kirchhoff's current law.
The branch currents add: 10 + 7.5 = 17.5 A, which is more than 15 A, so the breaker trips.
4Compare the brightness of two identical lamps connected (a) in series and (b) in parallel to the same battery. Explain.
Parallel lamps are brighter. In parallel, each lamp gets the full battery voltage; in series, each gets only half, so less current flows and less energy per second is transformed.
5A student puts a switch in one branch of a parallel circuit with two lamps. What does the switch control?
Only the lamp in that branch. The other branch stays connected to the battery, so its lamp stays on.
Challenge
1A 12 V battery is connected to a 2.0 Ω resistor in series with a parallel pair of 6.0 Ω and 3.0 Ω. Find the total resistance and the current from the battery.
Parallel pair: 1/R = 1/6 + 1/3 = 1/6 + 2/6 = 3/6, so R = 2.0 Ω. Total: R_T = 2.0 + 2.0 = 4.0 Ω. I = 12 ÷ 4.0 = 3.0 A. (Each part of the circuit then gets 6.0 V: 3.0 A × 2.0 Ω.)
2Show with a calculation that adding a 1000 Ω branch in parallel with a 10 Ω resistor still lowers the total resistance, even though 1000 Ω is a big resistance.
1/R_T = 1/10 + 1/1000 = 0.100 + 0.001 = 0.101, so R_T ≈ 9.9 Ω, which is less than 10 Ω. Any extra path, however poor, lets some more current flow, so R_T always decreases.
3Car headlights are wired in parallel with each other, not in series. Give two reasons, and say where the headlight switch must go.
In parallel, each headlight gets the full 12 V (full brightness), and if one burns out the other stays on, which matters for safety. The switch must be in the main line (in series with both branches) so one switch controls both lights.
Quiz
1. In a parallel circuit, the potential difference across each branch is:
2. Branch currents of 1.2 A, 0.8 A and 2.0 A give a total current of:
3. Two 8.0 Ω resistors in parallel have a total resistance of:
4. Adding another branch to a parallel circuit makes the total resistance:
5. Why are homes wired in parallel?
6. A 6.0 Ω and a 3.0 Ω resistor are in parallel with a 6.0 V battery. The total current is:
7. Which statement about series and parallel is true?
Big idea: This lesson completes D2.6: students build and compare series and parallel circuits, using measured and calculated current, potential difference and resistance.
Common difficulties:
- Mixing up the rules. Series: current same, voltage shared. Parallel: voltage same, current shared.
- Total resistance in parallel. Many students add the resistances. Ask: "Should R_T be smaller than the smallest one?"
- Fractions: 1/R_T = 3/12 means R_T = 12/3, not 3/12. Remind them to flip at the end.
How to help: Look at your electrical panel together (without opening it) and count the breakers: each is one parallel circuit. Talk about why running a kettle and a toaster on the same kitchen circuit can trip a breaker.