Resistors in Series

When resistors are connected in series, they are joined end-to-end along a single path so that the same electric current flows through every one of them in turn. Think of it like a queue at a single turnstile: every person must pass through each gate in sequence, and no one can take a shortcut. Because there is only one route for charge to travel, the current is identical at every point in the circuit. This single-path arrangement is the defining feature of a series connection. The trade-off for having one continuous path is that each resistor adds its own opposition to current flow. The total resistance a battery 'feels' is simply the sum of all the individual resistances along the path. This combined value is called the equivalent resistance (or total resistance), R_eq. It is the value of a single resistor that would produce exactly the same current draw from the battery as the entire series chain does. Equivalent resistance always increases as more resistors are added in series. Series circuits also share the supply voltage across the resistors in proportion to their individual resistances. A larger resistor takes a larger share of the voltage — a concept called a voltage divider. Understanding series resistor combinations is fundamental to analysing real circuits, from household wiring to electronic devices.

Key takeaways
  • Single current path
  • Voltage drops add up
  • Deriving equivalent resistance
  • Practical implications

Single current path

In a series circuit, all components are connected along one continuous loop with no branching. Consequently, the same current I passes through every resistor. This follows directly from the conservation of charge: charge cannot accumulate at any junction because there are no junctions — every charge carrier that enters one resistor must exit it and immediately enter the next.

Voltage drops add up

Applying Ohm's Law (V = IR) to each resistor individually gives voltage drops V₁ = IR₁, V₂ = IR₂, and so on. By Kirchhoff's Voltage Law, the sum of all these drops must equal the total supply voltage: V_total = V₁ + V₂ + V₃ + … This is the energy perspective: the battery supplies energy per unit charge (voltage) that is consumed bit by bit across each resistor.

Deriving equivalent resistance

Since V_total = IR_eq and also V_total = IR₁ + IR₂ + IR₃ + …, we can factor out the common current I: IR_eq = I(R₁ + R₂ + R₃ + …). Dividing both sides by I gives R_eq = R₁ + R₂ + R₃ + … The equivalent resistance is simply the arithmetic sum of all individual resistances. No matter how many resistors you add in series, R_eq is always greater than the largest single resistor in the chain.

Practical implications

Because equivalent resistance increases with each added resistor, the total current drawn from the battery decreases as the series chain grows longer. A broken wire or a blown component in a series circuit creates infinite resistance at that point, stopping current everywhere — this is why old-style fairy lights would all go out when one bulb failed. Series arrangements are useful when a precise voltage division is needed, such as in sensor circuits or attenuator networks.

The Key Equation

Req=R1+R2+R3+⋯+RnR_{\text{eq}} = R_1 + R_2 + R_3 + \cdots + R_n
  • R_{\text{eq}} — Equivalent resistance (Ohm (Ω)): The single resistance value that replaces the entire series combination, producing the same current for the same applied voltage.
  • R_1, R_2, \ldots, R_n — Individual resistances (Ohm (Ω)): The resistance of each separate resistor in the series chain, numbered 1 through n.
  • n — Number of resistors (Dimensionless): The total count of resistors connected in series.

Worked Example

Problem. Three resistors of 12 Ω, 8 Ω, and 20 Ω are connected in series to a 24 V battery. Find (a) the equivalent resistance, (b) the current flowing through the circuit, and (c) the voltage drop across the 8 Ω resistor.

Req=40 Ω,I=0.6 A,V2=4.8 VR_{\text{eq}} = 40\,\Omega,\quad I = 0.6\,\text{A},\quad V_2 = 4.8\,\text{V}

Answer. Equivalent resistance = 40 Ω; current = 0.6 A; voltage across 8 Ω resistor = 4.8 V.

Figure
Diagram
Resistors in Series — Circuit Diagram+−24 VR₁=12ΩR₂=8ΩR₃=20ΩI=0.6AV₁=7.2VV₂=4.8VV₃=12VR_eq = R₁ + R₂ + R₃ = 12 + 8 + 20 = 40 Ω
Origins · 1 minGeorg Simon Ohm and how this was discovered
GS
Georg Simon Ohm
1789–1854 · German

Georg Simon Ohm formulated Ohm's Law in 1827, establishing the proportional relationship between voltage, current, and resistance (V = IR). This law is the foundation for analysing resistor networks, including the derivation of equivalent resistance in series combinations. His meticulous experimental work with wires of different lengths and thicknesses gave physics its precise, quantitative language for electrical resistance. Ohm's groundbreaking 1827 paper was initially ridiculed by German academics, and he was forced to resign his teaching position. It took more than a decade before the scientific community — particularly in Britain — recognised the importance of his work. He was eventually awarded the Copley Medal in 1841 and the unit of resistance was named the 'ohm' in his honour in 1861.

  • Resistors in Parallel — The natural counterpart to series combinations — parallel resistors provide multiple current paths and reduce equivalent resistance. Understanding both arrangements is essential for analysing mixed networks.
  • Ohm's Law — Ohm's Law (V = IR) is applied at every step when finding voltage drops across individual series resistors and calculating the current through the circuit.
  • Kirchhoff's Voltage Law — Kirchhoff's Voltage Law states that voltage drops around a closed loop sum to zero, which directly justifies why individual voltage drops add up to the supply voltage in a series circuit.
  • Voltage Dividers — A voltage divider is a direct application of series resistors, using the proportional voltage drops across each resistor to produce a desired fraction of the supply voltage.