# Ways to think about current -- a toolbox of models

#### Prerequisites

The phenomenon of electric current is quite complex, even at the macroscopic level. We need to call on a number of levels of reasoning. These lead to a set of guiding principles (Kirchhoff's principles) that can interact in complex and subtle ways.

Trying to figure out "what's really happening" — to develop a sense of the true underlying mechanism responsible — is not helpful. (In the case of batteries and wires, the "true mechanism" deeply involves quantum physics.)

Instead, we develop a set of analogies — models of the system — to help us think about what's going on. Each of these models correctly represent one or more aspects of the macroscopic phenomenon and, as we learn to blend them, they help us develop some intuition.

This situation is actually quite common in physics (and in the other scientific disciplines as well). The "true" phenomenon is something very far out of our everyday experience. As a result, the best we can do is to take things we know, work by analogy, and eventually build a mental blend of the variety of viewpoints that works for us. The Indian legend of the Blind Men and the Elephant illustrates the point well. The picture at the right is from a Jain temple. (See also the 19th century poem by John Godfrey Saxe.)

The four models we use explicitly here are:

1. The rope model
2. The nail-board model
3. The water-flow model
4. The air-flow model

Each model represents a facet of the complex phenomenon we are trying to build intuition for. Other models are also sometimes useful (for example, a traffic-flow model) and you may come up with additional models of your own. These particular models are intended to show systems that represent the following characteristics of charge current flow:

• Because of the strong energy advantage to an equal balance of positive and negative charges, the moving charge tends to maintain its volume and move like an incompressible fluid.
• The electric forces that build up as a result of the compression of the moving charges at the front end of a resistive region creates an E field that drives the charges through the resistance and leads to a drop in potential ("electric pressure").
• The charge flow, while conserving volume and moving like an incompressible fluid, can divide and recombine, but the total is conserved.
• The resistance arises from irregularities and randomness in the resistor that "gets in the way" of the smooth flow of the charges.
• More resistance results in less flow.
• For a given applied pressure (voltage difference), the amount of flow depends on what's connected to it. More paths can result in more flow.

### 1. The rope model

The first tricky idea we have to deal with is that the electrons in a wire are not sent from a battery into an empty pipe (the wire). The wire is already full of electrons so a push from the battery starts them all moving together. A good model for this is a loop of rope.

The simplest electric circuit is a loop with a battery and a resistor. Since like charges repel each other so strongly but are balanced by the background of the opposite charges, there can't be any buildup of excess charge anywhere in the circuit (unless we make a special arrangement — see motivating capacitance). So moving charges push other movable charges in front of them. The electrons move like links in a chain or rope (or an incompressible fluid).

Imagine a loop of rope. A battery is like a person holding the rope and pulling one side of it, causing a high tension on one side (due to the rope being pulled taut) and a low tension on the other (due to the rope being compressed and pushed towards the resistance). A resistor is like a second person squeezing the rope, having it pulled through her hands. The friction of the rope generates heat. The more people squeezing, the slower the rope goes, even if the battery person pulls with the same tension. The tension (the pull) doesn't determine the flow by itself. It depends on how many people are squeezing the rope.

This shows a number of the aspects:

1. More resistance implies less flow, since the pull (tension) is always the same;
2. the flow around the loop is the same everywhere.

### 2. The nail-board model

Another hard idea is that a resistive region requires a potential drop to create an electric field to force the charges through a resistance at a constant speed.

Since electric potential energy is very like gravitational potential energy, the gravitational PE, $gh$, is a good analogy for electric potential, $V$. We can use a drop in height to model a drop in electric potential.

For our second model, we consider ping-pong balls as stand-ins for our electrons rolling on a wooden track — the wires.

The balls begin in a flat region (no g field in the direction of allowed motion) where there is no resistance. They can move at a constant velocity without need for a driving force.

The balls then enter a region with resistance — nails that are driven into the board that cause the balls to bump around and change direction, on the average, slowing down. To keep them going at a constant velocity, we need a force to speed them up through the nails. We get this by tipping the board — introducing a gravitational force (and an associated change in gravitational potential). If the nails provide an effective velocity dependent drag, $-bv$, the component of the gravitational force down the track, $mg\sin{θ}$, will balance the drag and keep the ping-pong balls going at a constant speed (on the average). When they get to the bottom to a region with no nail, they can now continue at a constant speed.

But to make a loop — to get them back up to the top — someone or something will have to put in energy (since they gain PE as they go up) to bring them back up to the top (like a battery).

This model illustrates the balance of drag against driving force and the role of random interference in creating resistance. It also shows the energy balance well and that PE is lost going through the resistor so energy needs to be put in to keep the thing going.

### 3. Air flow

Some aspects of electric current are well described by a model using air flow as an analogy. In this case, air pressure is analogous to electric potential. A pressure drop produces a wind — a flow of air. This example is a nice one since you can actually feel the difference between the voltage drop (the pressure difference —how hard you are blowing) and the current flow (how long it takes you to empty your lungs). I highly recommend that you actually do this; don't just read it!

Consider the example of the seven small straws shown in the figure at the right. Suppose we scotch tape three together next to each other as shown at the left ("in parallel") and three together one after the other as shown at the right ("in series"). Now consider blowing as hard as you can through each set of straws.

If you take a deep breath and blow as hard as you can, you are creating a high pressure in your mouth, higher than the ambient air pressure. As a result, the pressure on the end of the straws in your mouth is higher than the pressure at the end of the straws open to the air. This pressure difference will drive air through the straws.

If you try this with both configurations, blowing as hard as you can both times so you create the same pressure differential, you will find that you can empty your lungs a LOT faster with the parallel straws; the air flows through them at a faster rate, even though the pressure differential is the same. This makes a lot of sense since each of the straws in parallel can carry air equally. You expect they would carry as much air as three single straws, draining the air in your lungs three times as fast as for a single straw. And the straws connected in series will offer more resistance and flow less quickly than the three in parallel — or the single straw.

This model does not provide for conservation (air is compressible) or loops, but shows nicely that the same pressure difference (voltage drop) can lead to different currents drawn from the pressure source (battery).

### 4. The water-flow model

In many ways, since electron current moves like an incompressible fluid, water flow is the most useful of the analogies. Water is practically incompressible, its flow is conserved, it has a pressure that is analogous to voltage, and it can divide and recombine. The equation for fluid flow in a pipe (the Hagen-Poiseuille equation) is closely analogous to Ohm's law.

Of course water doesn't repel itself and there is no alternative kind of charge that cancels it.

But the gravitational height and potential provides an excellent and useful analogy for thinking about electric potential, and the conservation of the amount of water flowing at a junction (rate at which fluid enters a junction = rate at which fluid leaves) provides an excellent analogy for Kirchhoff's flow rule.

While none of these models are perfect, each reminds us of some aspect of the flow of electric charge in wires. Building up a better mental model requires borrowing pieces from many models and thinking about what is actually happening in a situation where charge is flowing

Joe Redish 2/27/12

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