Class Content > Electric Currents > Quantifying electric current

Prerequisites:

• Electric Currents
• Quantifying electric current  
• Resistive electric flow: Ohm's law

The phenomenon of electric current is quite complex, even at the macroscopic level. There are a number of levels of reasoning that need to be called on and a set of guiding principles (Kirchoff's rules) that can interact in complex and subtle ways. Trying to call on "what's really happening" -- a sense of the true underlying mechanism responsible -- is not helpful in the case of batteries and wires as the "true mechanism" deeply involves quantum physics. Instead, we will develop a set of analogies -- models of the system -- to help us think of what's going on. Each of these models will correctly represent one or more aspects of the macroscopic phenomenon and help us develop some intuition.

The situation is actually quite common in physics (and in the other scientific disciplines as well). The "true" phenomenon is something very much 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. The 19th century poem on this topic, "The Blind Men and the Elephant," by John Godfrey Saxe illustrates the point well.

The four models we will use explicitly 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 possibly useful (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 and 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.

1. The rope model

Let's consider the simplest possible electric circuit: a simple 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 charge anywhere in the circuit (unless we make a special arrangement -- see capacitance). So moving charges push other movable charges in front of them. The electrons move like links in a chain or rope.

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.

This shows a number of the aspects: (1) More resistance implies less flow; (2) the flow around the loop is the same everywhere.

2. The nail-board model

For this model we consider ping-pong balls as stand-ins for our electrons rolling on a wooden track -- the wires. In this model, gravity stands in for the electric field.   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. 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 them going at a constant speed (on the average). When it gets to the bottom to a region with no nail, it 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.   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, 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 picture of a water tank on top of a hill, sluices that carry it at the same level and then run it down through a water wheel, with a pump to carry it back uphill is very like a battery (tank + pump), wires (sluices), and a water wheel (bulb or resistance) that can do useful work.

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.

Follow ons:

• Electrical energy and power
• Kirchhoff's principles

Joe Redish 2/27/12