Resistors in Parallel
When resistors are connected in parallel, they provide multiple independent paths for electric current to flow between two points in a circuit. Unlike a series arrangement — where all current is forced through a single route — a parallel combination allows the total current from the source to split across each branch. This is exactly how the sockets in your home are wired: every appliance receives the full mains voltage, and each can be switched on or off without disrupting the others. Understanding parallel resistor networks is therefore fundamental to analysing almost every practical electrical circuit. The most important consequence of connecting resistors in parallel is that the equivalent (combined) resistance of the network is always less than the smallest individual resistor in the group. This seems counterintuitive at first — surely adding more components should increase resistance? — but think of it like opening extra lanes on a motorway. Each new lane reduces the overall congestion, making it easier for traffic (current) to flow. Mathematically, the equivalent resistance is found using the reciprocal formula, which comes directly from applying Kirchhoff's current law alongside Ohm's law. Parallel resistor networks appear in power distribution systems, electronic filters, and sensor circuits. Mastering the reciprocal formula and understanding why parallel combinations reduce resistance will equip you to design and analyse circuits confidently. This article derives the formula from first principles, walks through a worked numerical example, and connects the idea to other core concepts in current electricity.
- Shared Voltage Across Each Branch
- Applying Ohm's Law to Each Branch
- Applying Kirchhoff's Current Law
- Defining the Equivalent Resistance
- Why R_eq Is Always Smaller
Shared Voltage Across Each Branch
In a parallel network, both ends of every resistor are connected to the same two nodes. This means every resistor experiences exactly the same potential difference (voltage) V across it. This is the defining feature of a parallel connection, and it contrasts directly with a series circuit where voltage is shared out.
Applying Ohm's Law to Each Branch
Because each resistor R₁, R₂, R₃, … has the same voltage V across it, the current through each branch follows Ohm's law independently: I₁ = V/R₁, I₂ = V/R₂, I₃ = V/R₃, and so on. A smaller resistance carries a larger share of the current — the branches with the easiest path attract the most traffic.
Applying Kirchhoff's Current Law
Kirchhoff's Current Law (KCL) states that the total current entering a junction equals the total current leaving it. The current supplied by the source therefore equals the sum of all branch currents: I_total = I₁ + I₂ + I₃ + … Substituting the Ohm's law expressions gives I_total = V/R₁ + V/R₂ + V/R₃ + … = V(1/R₁ + 1/R₂ + 1/R₃ + …).
Defining the Equivalent Resistance
The equivalent resistance R_eq is the single resistance that would draw the same total current from the same voltage: I_total = V/R_eq. Equating the two expressions for I_total and dividing both sides by V yields the parallel resistor formula: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + … Always take the reciprocal of your calculated sum at the end to find R_eq itself.
Why R_eq Is Always Smaller
Because you are summing positive reciprocals, 1/R_eq is always larger than any individual reciprocal 1/Rₙ. Taking the reciprocal of a larger number gives a smaller number, so R_eq is always less than the smallest resistor in the network. For the special case of two equal resistors R in parallel, R_eq = R/2 — a handy check. Adding more parallel branches always decreases R_eq further.
The Key Equation
- R_{\text{eq}} — Equivalent resistance (Ohm (Ω)): The single resistance that replaces the entire parallel network and draws the same total current from the source.
- R_1, R_2, R_3 — Individual branch resistances (Ohm (Ω)): The resistance of each separate resistor connected in parallel between the two common nodes.
- \frac{1}{R} — Conductance (reciprocal resistance) (Siemens (S)): The reciprocal of resistance, representing how easily current flows. In parallel circuits, conductances add directly, which is why the formula uses reciprocals.
Worked Example
Problem. Three resistors of 6 Ω, 12 Ω, and 4 Ω are connected in parallel across a 12 V battery. Find (a) the equivalent resistance of the network and (b) the total current supplied by the battery.
Answer. Equivalent resistance = 2 Ω; total current = 6 A
Origins · 1 minGeorg Simon Ohm and how this was discovered
Ohm formulated the relationship between voltage, current, and resistance (Ohm's Law, 1827), which is the cornerstone of all resistor network analysis. The formula for equivalent resistance in parallel is derived directly by combining Ohm's Law with Kirchhoff's Current Law. In recognition of his foundational work, the unit of electrical resistance is named the ohm (Ω) in his honour. Ohm's groundbreaking 1827 paper was initially ridiculed by German scientists, and he lost his teaching position as a result. It took nearly two decades — and recognition from British scientists — before his work was widely accepted in Germany.
Related Concepts
- Resistors in Series — Series combinations are the natural counterpart to parallel ones; together they cover all fundamental resistor network configurations and are often combined in the same circuit.
- Ohm's Law — Ohm's Law (V = IR) is applied to each branch of a parallel network to find individual branch currents, making it the essential starting point for this derivation.
- Kirchhoff's Current Law — The parallel resistance formula is derived by applying KCL at the junction nodes — the total current equals the sum of all branch currents.
- Potential Divider — Potential dividers use series and parallel resistor combinations to produce specific output voltages, building directly on the analysis of resistor networks.