Trace the path with a finger

Charge circulates around a closed conducting loop. It passes through an element and continues on, so every working element needs two connections that lead somewhere.

For a bulb that means both the tip and the metal side have to sit in a path that returns to the other battery terminal. Run a single lead from the positive terminal to the base and nothing happens, because the charge that arrives has no way to continue.

The working procedure is literal: put a finger on one terminal and trace. Through every element, along every wire, back to the other terminal. If the trip cannot be completed, the current in that path is zero, whatever the diagram looks like. The same test catches a schematic with one side left dangling, which students often still assign a current to.

Connect a plain wire across a bulb. The wire has essentially no resistance, so its two ends are at the same potential, and those ends are the bulb's two ends. So

$$\Delta V_{\text{bulb}} = 0 \quad \Longrightarrow \quad I_{\text{bulb}} = \frac{\Delta V}{R} = 0.$$

The bulb goes out, not dim. Current still arrives at the junction, and all of it takes the resistanceless route.

The other half of the result is easy to miss and is the reason shorts are dangerous. Removing the bulb's resistance from the loop lowers $R_{\text{eq}}$, often drastically, so the current the battery delivers rises. A short across the battery itself is the extreme case: nothing limits the current but the internal resistance of the battery and the wires, and both get hot fast.

The habit worth building: before deciding whether anything flows through an element, compare the potentials at its two terminals.

An open switch or a burned-out bulb creates a gap. That gap stops the current in every loop that passes through it, and leaves every other closed loop alone.

The chain picture, in which a circuit is one continuous line and any break kills it, works perfectly for a single series loop, which is why it survives into circuits where it is wrong. Once there is more than one path it has to go.

The method is to redraw:

1. Erase the broken branch entirely.
2. Look at what is left and find every path that still closes.
3. Solve that circuit from scratch, because $R_{\text{eq}}$ has changed and so has everything downstream of it.

Two bulbs in parallel: one burns out, the other stays lit. Two bulbs in series: one burns out, both go dark. The topology decides it, not the number of bulbs.

The two arrangements differ in exactly one respect, and everything else follows from it.

1. Series. One path. The same current passes through every element, and the source potential difference is divided among them.
2. Parallel. Several paths between the same two points. Every branch has the same potential difference across it, and the current divides among them.

That difference explains why household wiring is parallel. Every outlet gets the full supply voltage regardless of what else is plugged in, and switching off one appliance does not open the path for the others. Old series Christmas lights are the counterexample everybody has met: one bulb fails and the whole string goes dark, because there was only ever one loop.