A student asked me the following insightful question:
Suppose we have a situation like this:
The lines are wires. The circle is a light bulb. Current runs from left to right. Because the wire has negligible resistance, essentially all the current runs through the top section of wire, skipping around the light bulb, and the bulb doesn’t light.
But what if we look only at this red region?
At the junction, we see current coming in from the left, and deciding to go up rather than continue on to the right. But inside the box, both regions are just made of wire. How does the current know to go up rather than continue straight through?
The answer is that, at the junction, the electric field points mostly up, and very little to the right. But why is that?
Consider two resistors in series. Each one has a voltage drop proportional to its resistance. If the resistors are the same length, the electric field strength inside the resistors is then proportional to the resistance. If one resistor has very high resistance compared to the other, the electric field is much stronger in that resistor.
Go back to the bottom path of our picture. We essentially have three resistors – a wire, a light bulb, and another wire – in series. The light bulb has significant resistance, while the wires do not. Therefore the electric field is much stronger in the light bulb than it is in the wires. It’s weak everywhere, since the bulb isn’t lit. But still, as weak as it is in the bulb, it’s much weaker still in the surrounding wire, by an amount that is proportional to the ratio of the bulb’s resistance to the wire’s resistance.
On the other hand, in the top path, the electric field is the same strength everywhere because it is all just a wire.
Both paths have the same voltage drop because they are in parallel. If they are the same length (they aren’t drawn the same length, but it is easy to imagine), the average electric field strength in them must be the same.
So the average electric field strength is the same in bottom and top. But the electric field in the bottom is localized almost entirely in the light bulb. That means that right at the junction, the electric field is much weaker in the wire heading right than in the wire heading up. Hence, the current almost entirely goes up, making a decision about where to go based only on the local electric field.