Simple Circuits

A simple circuit is governed by two conservation statements. Charge conservation makes the current identical at every cross-section of an unbranched loop, so identical series bulbs glow equally and the return wire carries what the outgoing wire carries. Energy conservation makes the potential drops around the loop sum to the source emf. An ideal battery clamps the voltage across its terminals and lets the network's $R_{\text{eq}}$ set the current through $I = \varepsilon/R_{\text{eq}}$, while an ideal wire has $R = 0$ and therefore drops zero volts, so every point on one unbroken wire is a single node at a single potential.

Three errors dominate. Draining current as it passes through each element, so the second bulb is expected to be dimmer, when what is spent crossing a bulb is energy per charge and not charge itself. Treating the battery as a constant-current pump whose output is fixed by its rating, when adding a series resistor lowers the current and adding a parallel branch raises it. And distributing the source voltage over the connecting wires as well as the resistors, or reading the two ends of a plain wire as different potentials, when $IR$ with $R = 0$ leaves nothing to distribute.