Compound Direct Current Circuits
Introduced| English |
|---|
| series/ˈsɪəriːz/ |
| parallel/ˈpærəlel/ |
| equivalent resistance/ɪˈkwɪvələnt rɪˈzɪstəns/ |
One bulb dies — does the whole string go dark?
- Old fairy lights: one bulb fails and the entire string goes out.
- Newer ones: a bulb fails but the rest stay lit.
- The difference is series 串联 versus parallel 并联 wiring.
- Real circuits mix both, so we need rules to combine resistors.
Series: resistances add
- In series, components sit one after another on a single path.
- The same current flows through each; their voltages add up.
- The total resistance is just the sum: $R = R_1 + R_2 + \cdots$.
- Adding a series resistor always increases the total resistance.
Two resistors in series have a total resistance of:
Series resistances add: $R = R_1 + R_2$.
In a series circuit, what is the same through every component?
Series shares one current; voltages add.
Parallel: reciprocals add
- In parallel, components share the same two ends.
- Each gets the same voltage; their currents add up.
- The reciprocals add: $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots$.
- The total is smaller than any single branch — more paths, less resistance.
Series vs parallel
Wire bulbs in parallel and compare the current and brightness with a series wiring.
Two $4\ \Omega$ resistors in parallel give what total resistance (in Ω)?
$1/R = 1/4 + 1/4 = 1/2$, so $R = 2\ \Omega$.
The total resistance of a parallel combination is smaller than any single branch.
More paths for current means less total resistance.
Reduce a network step by step
- Find any pure series or pure parallel group and replace it with one resistor.
- Repeat, shrinking the circuit until a single resistor remains.
- That final value is the equivalent resistance 等效电阻.
- Then use $V = IR$ to find the total current from the battery.
The single resistor that replaces a whole network is its ____ resistance.
That value is the equivalent resistance.
What each wiring means
- Series: one break kills the whole loop (old fairy lights).
- Parallel: each branch is independent (house wiring, new lights).
- Parallel also draws more total current — every branch adds its own.
- Homes use parallel so one switch doesn't turn off everything.
Select all true statements about series and parallel.
Series adds, parallel adds reciprocals, and they share current/voltage respectively. Parallel is NOT a sum.
Two $6\ \Omega$ resistors: find the total in series, then in parallel.
- Series: $R = 6 + 6 = 12\ \Omega$.
- Parallel: $\dfrac{1}{R} = \dfrac{1}{6} + \dfrac{1}{6} = \dfrac{2}{6}$, so $R = 3\ \Omega$.
Parallel resistance is not the sum. Add the reciprocals, then flip: the total is always less than the smallest branch. Writing $R = R_1 + R_2$ for parallel is the most common circuit error.
In series, resistances add ($R = R_1 + R_2$) and share the current; in parallel, reciprocals add ($1/R = 1/R_1 + 1/R_2$) and share the voltage, giving a smaller total. Reduce a network step by step to its equivalent resistance.