Resistors in Parallel

A parallel circuit gives charge more than one route between the same two points. Every branch begins and ends at the same potentials, so each resistor receives the full branch voltage.

At a junction, the incoming current separates among the available paths and later recombines.

Two or more resistors are in parallel when each is connected across the same pair of junctions. This shared endpoint structure, rather than how the drawing looks, determines whether components are parallel.

Why branch voltages are equal

Potential difference depends only on the potential at the two endpoints. Because every parallel resistor connects between the same junction A and junction B, each experiences VA−VB.

Why current divides

Charge is conserved at a junction. The rate of charge entering must equal the total rate leaving through all branches.

Since every branch has the same voltage, Ohm's law gives Ii=V/Ri. A branch with smaller resistance carries more current.

Deriving equivalent resistance

Cancelling the common nonzero voltage produces the reciprocal rule.

Why equivalent resistance becomes smaller

Adding a parallel branch creates another route for charge. For the same applied voltage, the total current increases. Since Req=VI, a larger total current means a smaller equivalent resistance.

The reciprocal sum is greater than 1/Rmin, so Req must be less than Rmin. This provides a quick check on every parallel-resistance calculation.

Two-resistor shortcut

The product-over-sum form applies only to two resistors. The reciprocal rule works for any number.

Identical resistors

Conductance viewpoint

Conductance G=1R measures ease of current flow. Parallel branches add conducting paths, so their conductances add directly.

Worked example

A 6 Ω resistor and a 3 Ω resistor are connected in parallel across 12 V.

The lower-resistance 3 Ω branch carries the larger current. Also, 2 A+4 A=6 A, confirming charge conservation at the junction.

Parallel circuits in practice

Household devices are connected in parallel so each receives the supply voltage and can operate independently. Switching off or disconnecting one branch does not normally break the others.

Exam-ready reasoning

• Identify common endpoint pairs to confirm a parallel connection.
• State that branch voltages are equal.
• Use current conservation at junctions.
• Apply Ohm's law to derive the reciprocal-resistance rule.
• Check that Req is below the smallest branch resistance.
• Use the two-resistor shortcut only when exactly two resistors are involved.