Series Circuits: Current, Voltage, and Resistance

In a series circuit the components are connected end to end in a single loop, so the same current flows through every component, the total potential difference is shared between them (V = V₁ + V₂ + …), and the total resistance is the sum of the individual resistances, R = R₁ + R₂ + …. In brief: adding more resistors in series increases the total resistance and reduces the current. The component with the largest resistance gets the largest share of the voltage, V_n = IR_n. If one component fails (opens), the current stops everywhere in the circuit.

String lights on a Christmas tree (of the old type), a fuse in a circuit and a switch are examples of series connections. This guide states the rules for current, voltage and resistance in a series circuit, explains the voltage divider idea, compares series and parallel behavior in a table, works through a numerical example and corrects common mistakes.

What are the rules for series circuits?

• Current: the same everywhere: I = I₁ = I₂ = I₃. An ammeter reads the same value at any point in the loop.
• Voltage: the supply voltage is divided among the components: V_total = V₁ + V₂ + V₃. The p.d. across each resistor is V_i = IR_i, so larger resistors have larger voltages.
• Resistance: R_total = R₁ + R₂ + R₃. The total is always greater than the largest individual resistance.
• Power: P_total = I²R_total, the sum of the powers in each resistor. The resistor with the largest resistance dissipates the most power.
• Failure: if one component breaks the loop, the current stops in all components.

Why do the rules hold?

Charge cannot accumulate at any point in a single loop, so the current (charge per second) must be the same through every component (conservation of charge). The energy given to each coulomb by the source (V_total) must equal the sum of the energies transferred in the components (conservation of energy, Kirchhoff’s voltage law). Adding V = IR for each, with the same I, gives V_total = I(R₁ + R₂ + …), which is the rule for the total resistance. Resistors in series act like a longer wire: increasing the length of a conductor increases its resistance.

The voltage division formula is V_n = V_total × R_n ÷ R_total, used in potential dividers.

Series circuit properties

Worked example: three resistors in series

Three resistors of 10 Ω, 20 Ω and 30 Ω are connected in series to a 12 V battery.

1. Total resistance: R = 10 + 20 + 30 = 60 Ω.
2. Current: I = V ÷ R = 12 ÷ 60 = 0.20 A, through all three resistors.
3. Voltages: V₁ = 0.20 × 10 = 2.0 V; V₂ = 0.20 × 20 = 4.0 V; V₃ = 0.20 × 30 = 6.0 V. Sum = 12 V ✓.
4. Powers: P₁ = 2.0 × 0.20 = 0.40 W; P₂ = 4.0 × 0.20 = 0.80 W; P₃ = 6.0 × 0.20 = 1.2 W. Total = 2.4 W = 12 × 0.20 ✓.
5. Voltage divider check: V₃ = 12 × 30 ÷ 60 = 6.0 V ✓.
6. Add a fourth resistor of 60 Ω: the total becomes 120 Ω and the current halves to 0.10 A.

How were the rules established?

Georg Ohm’s work in 1827 on the relation between voltage, current and resistance of wires led to the rules for series circuits. Gustav Kirchhoff formulated his current and voltage laws in 1845, which generalize them to arbitrary networks. Old-style Christmas lights, which were wired in series, led to the well-known problem that one failed bulb darkened the entire string, and later designs added shunts that short-circuit the failed bulb.

Which mistakes are common?

Mistake 1: Dividing the current between components in series

The current is the same. It is the voltage that is divided.

Mistake 2: Thinking that the total resistance is smaller than any single resistor

That is true for parallel connections. In series, the total is larger than any of them.

Mistake 3: Forgetting to include all resistances

Include the internal resistance of the battery and the resistance of ammeters if these are not negligible.

Mistake 4: Assuming equal voltage across unequal resistors

V = IR gives larger voltages for larger resistances with the same current.