No Voltage Drop Inside a Wire: Examining Electric Potential & Kinetic Energy

[maccha]

My textbook says that electrons in a wire begin with electric potential energy- they are then accelerated by the electric field and their potential is converted to kinetic. Once inside a resistor, they collide with molecules and their kinetic energy is converted to thermal energy. However, it also says that there is no voltage drop inside an ideal wire- only inside a resistor. If their potential energy is being converted to kinetic energy inside the wire, how can there be no voltage drop?

 

[collinsmark]

My textbook says that electrons in a wire begin with electric potential energy- they are then accelerated by the electric field and their potential is converted to kinetic. Once inside a resistor, they collide with molecules and their kinetic energy is converted to thermal energy. However, it also says that there is no voltage drop inside an ideal wire- only inside a resistor. If their potential energy is being converted to kinetic energy inside the wire, how can there be no voltage drop?

The confusion stems from the implication that the bulk of the change from potential to kinetic energy happens in the wire (not the resistor). That's somewhat misleading. The truth is that the potential on one side of the wire is nearly identical to the potential on the other side of the wire. And in the case of modeling circuits with ideal components (using lumped parameter theory), the potential is identical.

It takes a comparatively tiny amount of energy to move charge through a conductor. And when you model charge as a bunch of electrons, the kinetic and potential energies are essentially the same on each side of the wire. The main reason is because it is so easy for electrons to freely move around in a conducting wire.

The bulk of the Potential --> Kinetic --> Thermal energy transitions happen all within the resistor, on a sub-microscopic scale. But this process is not one simple step, it is a submicroscopic process that happens a little bit, repeatedly, many times over.

Allow me to present an analogy. Don't take this analogy too seriously, as it's not particularly accurate. But I'll present it anyway to introduce the concept.

Imagine a large, nearly frictionless, horizontal surface on the top of a steep hill, and another one at the bottom of the hill. In between is the hill itself, and it is steep, rough and rocky. The slope and roughness of each surface are proportional to the surface's resistance. But since the top and bottom plateaus are practically frictionless, they remain slippery and nearly horizontal.

Now imagine thousands of people, each on a sled, covering all parts of the hill and the plateaus. And all the sleds are loosely connected to each other with bungee-cords. Imagine yourself as one of these sledders on top of the hill, on the top plateau.

While on the top plateau, you find yourself gradually moving to one side. But you are not accelerating, but rather moving at a nice, comfortable constant velocity from one side of the plateau to the other. Those sledders in your vacinity are moving with you, since all the sleds are attached to each other with the bungee cords. It's very relaxing, and you drink a cup of tea, and chat about the weather with those sledders around you.

Eventually you reach the edge of the hill-top, and the terror begins. You and all the sledders in your vicinity suddenly fall on rough boulders on the hill, bounce back up a bit, fall on each other, and tumble some more before smacking into more rocks and more bodies. Scrapes and bruises abound. The process continues for some time. There are skulls crashing into rocks, limbs crashing into skulls, and the mayhem continues over and over again. Oh, the humanity!

Thankfully after awhile, you reach the bottom of the hill. Like the top plateau, the bottom is also smooth and relaxing. You are able to relax and talk about the mutual experience with the other sledders.

So again, don't take that analogy too seriously (it has its share of flaws). But in the analogy, the sledders represent electrons. The smooth plateaus represent the conducting wires and the steep hill represents the resistor. The bungee cords represent the Coulomb force (or if you wish, change the analogy to make the sledders negatively charged and the ground positively charged), such they are all kept more-or-less uniformly distributed about all the surfaces. Like real electrons in a DC circuit, the average acceleration of the electrons is zero. On average they travel at a constant velocity. However, there are periods of brief acceleration in the analogy, particularly when a sledder bounces off a rock or into another sledder, and falls down the hill a bit more, little by little, banging his noggin on all sorts of things on the way. This is where the Potential --> Kinetic --> Thermal energy transitions take place. And as in the analogy, it's not one single event; rather it is many, many small transitions.

[Edit: Oh, and in this analogy, the height difference between the peak and valley of the hill represents the potential of the battery. The number of total sledders crossing a given point, per unit time, represents the current. If you were to increase the resistor's resistance, it means (in the analogy) increasing the roughness of the hill. Sledders would travel slower due to the increased difficulty in getting down the hill so the "current" would decrease. On the other hand if you were to decrease the resistance, the sledders would move faster, thus the current would increase. And like how kinetic energy increases with the square of the velocity, sum and severity of bruises per unit time (representing power) increases with the square of the current.]

 

[maccha]

Thank you so much for that explanation! I think I understand now!