One of the consequences of adding a resistor in the circuit (with higher electron speed and a higher electric field) is that a large energy transfer occurs across the resistor. In thin wire resistors (sometimes referred to as filaments), this effect is particularly visible. The amount of energy transferred to a filament is sufficient to heat the thin wire to the point where it produces heat and light. This is actually how incandescent light bulbs work.

Let's continue to look at the simple circuit that we were using in the video above (a mechanical battery, wires, and a thin filament). To analyze the energy in our circuit, we can refer back to the Energy Principle. If we take everything to be in our system, including the battery, wires, filament, and the surrounding air or materials, then we know that:

$$\Delta E_{sys}=0$$ If we breakdown what is in our system, this means that $$\Delta E_{bat}+\Delta E_{wires}+\Delta E_{filament}+\Delta E_{surr}=0$$ From this statement of energy conservation, this would tell us that any energy provided by the battery must be used in the wires, heating up the filament, or providing light and heat to the surroundings.

We could also consider what is happening to the energy of a single electron as it makes a complete trip around the circuit. The energy gained by the electron as it is transported across the mechanical battery is dissipated by the collisions the electron has as it moves around the wire, particularly with the positive nuclei in the wire. While this is certainly true, it becomes cumbersome to think about every single electron that is moving around the circuit. Instead, we will often think about energy in circuits in terms of the energy per charge that is moving around the circuit. As we talked about before, electric potential is the energy per charge, which is something that you have been working with over the last few weeks. So rather than talk about changes in energy around the circuit, we will generally be talking about changes in electric potential around the circuit.

This means we can rewrite our energy conservation statement in terms of the energy per charge instead. This is called The Loop Rule or sometimes Kirchhoff's Loop Rule. This Loop Rule says that if you follow a complete, round-trip path in a circuit, the total change in electric potential across all of the circuit elements (i.e., batteries, wires, resistors, capacitors) should be zero.

$$\Delta V_1+\Delta V_2+\Delta V_3+...=0$$

We will refer back to the Loop Rule and the