TL;DR:
In series, the same current flows through every part, the voltages add up to the supply, and resistances add:
R = R1 + R2.In parallel, every branch gets the full supply voltage, the branch currents add up, and
1/R = 1/R1 + 1/R2.Two 100 Ω resistors make 200 Ω in series and 50 Ω in parallel.
Houses are wired in parallel so each appliance gets full voltage and works on its own.
You have two identical resistors and a battery. There are only two sensible ways to connect them: one after the other in a single loop, or side by side so each has its own path. The two layouts behave very differently, and almost every circuit problem in a physics course is a mix of them.
Two ways to connect parts
In a series circuit, parts are joined end to end, so there is only one path for charge to follow. If you remove one part, the loop is broken and everything stops.
In a parallel circuit, the parts are joined across the same two points, so the current has a choice of paths. Remove one branch and the others carry on.
To tell them apart, trace the path of the charge with your finger. If your finger must go through both parts in turn, they are in series. If it can split and rejoin, they are in parallel.
Why the rules come out the way they do
Two ideas explain everything. First, charge is not used up, so the current that goes into a junction equals the current that comes out. Second, voltage is energy per charge, so going around any closed loop, the energy gained at the battery equals the energy spent in the parts.
In series there is a single path with no junctions, so the current is the same everywhere. The supply voltage is shared out between the parts, so the part voltages add up to the supply. With Ohm's law V = IR, the same current times each resistance gives V = I·R1 + I·R2 = I(R1 + R2). The total resistance is R1 + R2.
In parallel, each branch connects straight across the supply, so each branch has the full supply voltage. The currents split at the junction and add back up: I = V/R1 + V/R2. Dividing by V gives 1/R = 1/R1 + 1/R2. Adding a second path gives the current more room to flow, so the total resistance gets smaller, not larger.
Quantity | Series | Parallel |
|---|---|---|
Current | Same through every part | Splits; branch currents add |
Voltage | Shared; part voltages add to the supply | Same across every branch |
Total resistance |
|
|
If one part is removed | All stop | Other branches keep working |
Worked example: two 100 Ω resistors on a 3 V supply
Say two AA cells give a 3 V supply, and you have two 100 Ω resistors. First connect them in series.
Total resistance:
R = 100 + 100 = 200 Ω.Current:
I = V/R = 3 / 200 = 0.015 A, which is 15 mA, the same through both.Voltage across each:
V = IR = 0.015 × 100 = 1.5 V.Check:
1.5 + 1.5 = 3 V, the supply.
Now connect the same two resistors in parallel across the same supply.
Total resistance:
1/R = 1/100 + 1/100 = 2/100, soR = 50 Ω.Voltage across each branch: the full
3 V.Current in each branch:
I = 3 / 100 = 0.03 A, which is 30 mA.Total current:
0.03 + 0.03 = 0.06 A. Check with the combined resistance:3 / 50 = 0.06 A. They agree.
Notice the pattern. The parallel pair draws four times the current of the series pair, since 0.06 ÷ 0.015 = 4. Same parts, same battery, very different behaviour.
Stay with batteries: Try circuits like this only with small batteries and a resistor or bulb in the loop. Never connect a wire straight across a battery, which can make it hot enough to burn you. Never experiment with wall outlets or mains power, which can kill.
Why houses are wired in parallel
If the lamps and sockets in a house were in series, they would share the supply voltage, so each would get only a slice of it. Turning one lamp off would break the loop and switch off every other one.
In parallel, every appliance is connected across the same supply, so each gets the full voltage it was designed for. Each one can be switched on and off on its own. The cost is that currents add, so each extra appliance draws more current from the circuit, which is why circuits have breakers or fuses to cut off an overload.
How to check your answer
Size check. Series resistance must be larger than the biggest single resistor. Parallel resistance must be smaller than the smallest one. Here, 200 is above 100 and 50 is below 100.
Voltage check. In series the part voltages must add to the supply. In parallel they must all equal it.
Current check. In parallel the branch currents must add to the total. In series they must all be equal.
Units. Volts divided by ohms gives amps.
Mistakes that lose marks
Adding resistances directly in a parallel circuit, or using the reciprocal formula for a series one.
Forgetting to flip
1/Rback toRat the end. If you get 0.02, that is 1/R, so R is 50.Saying the current is used up as it passes through a resistor. It is the energy that is spent, not the charge.
Using the supply voltage on a series part instead of that part's share.
Treating a circuit as all series or all parallel when it is a mix. Reduce the parallel pieces first, then add the series ones.
Where you will use it next
Mixed circuits are solved by replacing each parallel group with its single equivalent resistance, then combining. The same ideas return in power calculations (P = IV), in how batteries are combined, and in the energy-per-charge picture in what voltage actually is.
Frequently asked questions
Why is the parallel resistance smaller than either resistor?
Each branch gives the current an extra path, so for the same voltage more current flows. More current for the same voltage means lower total resistance.
What is a shortcut for two equal resistors in parallel?
Divide one resistance by two. For two 100 Ω resistors that gives 50 Ω. For two unequal resistors you can use R1·R2 / (R1 + R2).
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