11.5 Compound Direct Current Circuits

Elements are in series when they sit one after another on a single path with no branches between them. The same current passes through each. Their potential differences add up to the total across the group, and their resistances add: $R_{eq} = R_1 + R_2 + \cdots$.

Adding a resistor in series always raises the equivalent resistance. The equivalent resistance is bigger than any single resistor in the group.

Elements are in parallel when both their ends connect to the same two points, so each sits on its own branch. They all have the same potential difference across them. The currents in the branches add up to the current entering the group, and the reciprocals add: $\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots$.

Adding a branch in parallel always lowers the equivalent resistance, because it gives the charge another path. The equivalent resistance is smaller than the smallest branch. For just two resistors, $R_{eq} = \dfrac{R_1R_2}{R_1 + R_2}$, and n identical resistors R in parallel give $\dfrac{R}{n}$.

To simplify a compound circuit, find the innermost group that's purely series or purely parallel, replace it with its equivalent, redraw, and repeat. Then work back out to find each current and potential difference.

A real battery behaves like an ideal emf $\mathcal{E}$ in series with a small internal resistance r. When current I flows out of it, some potential is lost inside, so the potential difference you measure across its terminals, the terminal voltage, is $\Delta V = \mathcal{E} - Ir$.

With no current, the terminal voltage equals the emf. The more current you draw, the lower it gets. That's why car headlights dim while the starter motor draws a big current. The current from the battery is $I = \dfrac{\mathcal{E}}{R_{ext} + r}$, where $R_{ext}$ is the equivalent resistance of everything outside the battery.

Wires are usually treated as having no resistance, because a good conducting wire's resistance is tiny next to the other elements. That only works when the circuit has other elements with resistance. A bare wire straight across a battery is a short circuit (11.2), and then the wire's own resistance and the battery's internal resistance are what limit the current.

An ammeter measures current, so the current must pass through it: connect it in series. An ideal ammeter has zero resistance, so it doesn't change the current.

A voltmeter measures the potential difference between two points: connect it in parallel, across the element. An ideal voltmeter has infinite resistance, so it draws no current.

Real meters aren't ideal. An ammeter's small resistance slightly lowers the current it measures. A voltmeter with a resistance comparable to the element it's across draws noticeable current and lowers the reading. A good meter has a resistance far from the element's: much smaller for an ammeter, much larger for a voltmeter.