Kirchhoff's Loop Rule

The loop rule states that the potential changes around any closed walk sum to zero, which is energy conservation per unit charge. Signs are assigned from the traversal direction: a resistor crossed along its current arrow contributes $-IR$ and crossed against it contributes $+IR$, while a battery contributes $+\varepsilon$ when entered at its negative plate and $-\varepsilon$ when entered at its positive plate, in both cases without reference to which way current flows. A real source carries internal resistance $r$, so its terminal voltage is $\varepsilon - Ir$ while discharging and $\varepsilon + Ir$ while being charged.

Three errors dominate. Assigning signs by component type, making every battery positive and every resistor negative, which breaks as soon as two sources oppose or a guessed current runs backward. Using the EMF as the terminal voltage under load, so a measured battery reading below its label looks like a defective battery rather than $\varepsilon - Ir$. And reasoning locally after a change: closing a switch or swapping a resistor alters $R_{\text{eq}}$, the source current and every node potential, so the entire circuit must be re-solved before any element is called brighter or dimmer.