Resistor Circuits: Ohm's Law, Series and Parallel

Ohm's law is the whole story for one resistor

For an ideal resistor, current and voltage are directly proportional: V = IR. V is the potential difference across the resistor in volts, I is the current through it in amps, and R is its resistance in ohms. Double the voltage and the current doubles too — that linearity is what makes resistor networks solvable with algebra instead of iterative simulation.

V = I * R           (Ohm's law)
P = I * V = I^2 * R = V^2 / R      (power dissipated as heat)

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Series: same current, voltages add

Resistors in series share a single path, so the same current I flows through every one of them. The total voltage across the chain is the sum of the individual drops (Kirchhoff's voltage law: the sum of voltage drops around any closed loop is zero), so the equivalent resistance is simply the sum: R_total = R1 + R2 + ... + Rn. Adding a resistor in series always increases total resistance and decreases current for a fixed supply voltage.

Parallel: same voltage, currents add

Resistors in parallel share both endpoints, so each one sees the same voltage across it. The current splits between the branches (Kirchhoff's current law: current into a node equals current out), and the equivalent resistance follows the reciprocal rule 1/R_total = 1/R1 + 1/R2 + ... + 1/Rn. Adding a resistor in parallel always decreases total resistance, because it opens up an additional path for current.

series:    R_total = R1 + R2 + ... + Rn
parallel:  1/R_total = 1/R1 + 1/R2 + ... + 1/Rn
special case, two resistors in parallel:  R_total = (R1*R2) / (R1+R2)

Kirchhoff's laws for anything more complex

Real circuits are rarely pure series or pure parallel. Kirchhoff current law (KCL) says the sum of currents entering a node equals the sum leaving it. Kirchhoff voltage law (KVL) says the sum of voltage rises and drops around any closed loop is zero. Writing one KCL equation per independent node and one KVL equation per independent loop gives a system of linear equations that solves for every unknown current and voltage in an arbitrarily complex resistive network.

Power dissipation and why it matters for design

Every resistor converts electrical energy to heat at a rate P = I²R = V²/R. Circuit designers pick series resistors to limit current (protecting an LED, for instance) and parallel combinations to lower an effective resistance or split current between multiple load paths. A low-resistance parallel path can silently carry most of the current, and most of the power, if the parallel resistances are mismatched.

Frequently asked questions

Why does adding a resistor in parallel lower the total resistance?

Because a parallel branch opens up an additional path for current to flow, so for the same applied voltage more total current can flow through the network — and total resistance is voltage divided by that larger total current, so it goes down, even though you added a component.

What is the difference between Kirchhoff's current and voltage laws?

KCL applies at a node: the total current flowing in must equal the total current flowing out, because charge cannot accumulate at a junction. KVL applies around a closed loop: the sum of all voltage rises and drops going around the loop is zero, because you return to the same starting potential.

How do you find the current through one resistor in a mixed network?

First reduce the network step by step using the series and parallel formulas to find an equivalent resistance, then find the total current from the source using Ohm's law, then use the voltage-divider or current-divider relationships to recover the current or voltage at the specific resistor you care about.

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