Kirchhoff's laws
Two rules for any circuit
- A tangle of wires can look impossible to solve.
- But just two rules — Kirchhoff's laws — crack any of them.
- Each is really a conservation law in disguise.
First law: junctions
- At any junction, current in = current out.
- It comes from conservation of charge — charge cannot pile up at a point.
$5.0\ \text{A}$ flows into a junction and splits into two branches; one carries $2.0\ \text{A}$. What does the other carry?
Current in = current out: $5.0 = 2.0 + I$, so $I = 3.0\ \text{A}$.
Second law: loops
- Around any closed loop, total e.m.f. = total p.d. across the components.
- It comes from conservation of energy — each coulomb gives back what it gained.
Match each Kirchhoff law to the conservation law behind it.
Charge cannot build up at a junction; energy a charge gains from sources equals what it gives to components round a loop.
Resistors in series
- Same current through each; the p.d.s add.
- $R_{\text{series}} = R_1 + R_2 + \ldots$ — always larger than each one.
A $3.0\ \Omega$ and a $6.0\ \Omega$ resistor are in series. What is the total resistance?
$R = R_1 + R_2 = 3.0 + 6.0 = 9.0\ \Omega$.
Resistors in parallel
- Same p.d. across each; the currents add.
- $\dfrac{1}{R_{\text{parallel}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \ldots$ — always smaller than the smallest one.
A $3.0\ \Omega$ and a $6.0\ \Omega$ resistor are in parallel. What is the total resistance?
$\dfrac{1}{R} = \dfrac{1}{3.0} + \dfrac{1}{6.0} = \dfrac{1}{2.0}$, so $R = 2.0\ \Omega$ — smaller than either.
A parallel combination is always smaller than the smallest resistor in it.
Yes — adding a parallel path gives the current another route, lowering the overall resistance.
Solving a circuit
- Label every current with a direction.
- Apply the junction rule and the loop rule, plus $V = IR$, then solve together.
Two equal resistors $R$ in parallel give a combined resistance of:
$\dfrac{1}{R_{\text{tot}}} = \dfrac{1}{R} + \dfrac{1}{R} = \dfrac{2}{R}$, so $R_{\text{tot}} = \tfrac{1}{2}R$.
You've got it
- first law (junctions): current in = current out — conservation of charge
- second law (loops): total e.m.f. = total p.d. — conservation of energy
- series: $R = R_1 + R_2$; parallel: $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$