Relating voltage in a circuit to distance

Question

I'm having trouble understanding voltage in relation to distance and the electric field in a circuit.

A volt is a joule per coulomb, and a joule is a newton meter. Say you were to measure the voltage across a loop with a voltmeter. If you were to put both the prongs at one terminal of the battery, and then slowly move one of the prongs around the loop until you reached the other side of the battery, I would expect the voltage to increase in proportion to the distance (measured along the length of the wire) between prongs. In my mind this is analogous to someone going up an elevator in a tall building. As the elevator moves up, the person's potential energy increases proportionally. Likewise, as you move the prong back to the starting side of the battery, I would expect the voltage to decrease along with the distance between prongs. This would be like if the person fell from the top of the building and their potential energy was now being converted into kinetic energy.

I did some PHET circuit simulation and the voltage behaved as I just described when it was a fiery loop with no resistors. When a resistor is in, voltage only drops at the resistor instead of along the length of the wire. The wires on either side of the resistor act as extensions of the battery terminals. How are these voltage levels created, and how can I still relate the distance component of voltage to a circuit like this?

Answer 1

The voltage drop over a segment of a circuit is proportional to its resistance. Therefore, if you have a uniform wire, the amount of voltage drop will be proportional to distance as you thought.

The opposite case is where the circuit consists of one section of very low resistance (a wire) and one section of very high resistance (a resistor). In this case, since the voltage drop is proportional to the resistance, there will be a much greater drop over the resistor than the wire. Often the approximation is made that the resistance of the wire is infinitely smaller than the resistor, so that there is actually no voltage drop over the wire, and all the voltage drop is across the resistor. If you are able to measure the voltage at some point within the resistor, you should be able to see a spatial dependence.