Kirchhoff's laws
| English | Chinese | Pinyin |
|---|---|---|
| Kirchhoff's laws | 基尔霍夫定律 | jī ěr huò fū dìng lǜ |
| junction | 节点 | jié diǎn |
| conservation of charge | 电荷守恒 | diàn hè shǒu héng |
| conservation of energy | 能量守恒 | néng liàng shǒu héng |
| series | 串联 | chuàn lián |
| parallel | 并联 | bìng lián |
Two rules for any circuit
- A tangle of wires can look impossible to solve.
- But just two rules — Kirchhoff's laws 基尔霍夫定律 — crack any of them.
- Each is really a conservation law in disguise.
First law: junctions 节点
- At any junction, current in = current out.
- It comes from conservation of charge 电荷守恒 — charge cannot pile up at a point.
Circuit for measuring the e.m.f. and internal resistance of a cell
Series & parallel circuits
Switch between series and parallel and add bulbs. In series they share the voltage and one break kills them all; in parallel each gets the full voltage and a break only loses its branch.
$5.0\ \text{A}$ 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 能量守恒 — each coulomb gives back what it gained.
Two resistors in series 串联 and their single equivalent resistor
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 并联
- 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.
Two resistors in parallel and their single equivalent resistor
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: $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$