Developing a Mental Model of Resistance

The purpose of this lesson is to help students develop a mental model of how resistance affects a circuit and to understand what variables contribute to resistance. Resistance is an impediment to the flow of electrical current. The electrons in the filament of the bulb are jostled and move very vigorously, but their movement is random and therefore it inhibits forward flow. With enough "push" or voltage, electrons do flow through the filament material, but due to all the random movement, they heat up, and this heat gives off light.

Students often forget to think about resistance when conceptualizing a simple circuit because the role of resistance is passive. While a filament's resistance impedes current flow and is the reason the bulb lights, most learners focus on the active role of electrons in making the bulb light. However, it is important not to use language that tries to make the passive effect active. For instance, some teachers talk about resistance as a push in the opposite direction. One problem with this is that it could lead students to incorrectly think that the push is there when there is no current flow. Instead, help students focus on resistance as an impediment to flow. Students also tend to think of resistance as slowing down the current. Many analogies inadvertently support this "speed" notion. This can lead to confusions. For instance, the more particles heat up, the faster—or better put, more vigorously—they move. However, faster or more vigorously moving particles result in greater impediment to flow. For these reasons and others, it is important to get students to focus on the impediment as limiting the amount of current that can flow rather than the mistaken idea that it has to do with speed of flow.

Some Variables that Contribute to Resistance

The variables of material type, diameter, and the temperature of the filament wire all contribute to the amount of resistance. These variables work to create an impediment to the flow of electrons, thereby affecting the current in the circuit.

• The type of material of the filament wire. The type of material affects the amount of resistance, as described in the previous lesson. In an insulator, nearly all of the electrons are tightly held by individual atoms or are shared by pairs of atoms. Since no electrons are free to move from atom to atom, insulators do not conduct electricity. In a conductor, one or two electrons from each atom are free to move from atom to atom. The movement of these free electrons from areas of higher electron concentration to areas of lower electron concentration is what we call current.

  Some materials, like Nichrome, include both free electrons and electrons that are tightly held in covalent bonds. The electrons in the covalent bonds repel the free electrons and act as obstacles to the movement of the free electrons. As the free electrons move through the material they bump into the electrons and atoms in the covalent bonds. This makes it more difficult for electrical current to flow and all the jostling causes the material to heat up. Materials like Nichrome are said to have higher resistance than better conductors like pure copper. Even good conductors include minor obstacles to the movement of free electrons, which is why even a good conductor will heat up when a large current passes through it.

• The diameter (width or thickness) of the filament wire. The diameter of the wire also affects the amount of resistance. The conductivity of the wire is proportional to the area of the cross-section. This means that the more cross-sectional area a wire has, the more easily it will conduct the forward flow of electrons; and the less cross-sectional area it has, the more it will resist the forward flow of electrons. So as diameter (and cross-sectional area) is increased or decreased, a change takes place. (For example, compare two wires, one with a 1 inch diameter and one with a 2 inch diameter. The 2 inch wire actually has four times the cross-sectional area of the 1 inch wire. For round wires, the cross-sectional area is proportional to the square of the radius.) A way to envision it (see Comparison of Conductivity of Material in Relation to Size: Square Pipe Example chart) is to think of a pipe and imagine that it is square instead of round. Draw a square that is 1 inch on each side and then expand it to a square that is 2 inches on each side. You will see that the area of the 2 inch pipe (and therefore the flow that it can carry) is 4 times that of the 1inch pipe. The 1 inch square pipe fits in 1/4 of the 2 inch square pipe.

• The temperature of the filament wire. The temperature of the wire also matters. As electrons flow throughout the wires when the circuit is complete, all the atoms in the wire become more energetic. This excited state of the atoms is reflected in a higher temperature (that is, temperature is a measure of a higher energy state). The resistance materials increase as the temperature increases. For instance, when current flows through the filament, it gets hotter, which means that it becomes increasingly thermally energetic, making it harder for current to flow. Think of it this way: heat is kinetic energy. This means that the hotter something is, the more vigorously its atoms are moving. The greater the random motion of the atoms (and of the electrons as part of atoms), the harder it is to sustain the flow, because as electrons move forward, they collide with other fast moving atoms and get knocked around. Eventually the filament gets so hot that it glows white.

Note to Teacher: In the process of experimentation, students may discover that the length of the filament wire also matters. If students use a ½ inch length of #32 Nichrome wire, it will glow. If they use a 1 inch length of #32 Nichrome wire, it will not. This is hard to explain without understanding a potential difference model and attending to the circuit as an entire system. This lesson does not attempt to explain filament length.

The important contribution of resistance to current flow can be seen when one creates a short circuit. A short circuit (i.e. a battery and a wire, but no bulb) has no resistors or devices that add significant resistance into the circuit. Therefore, there is nothing to minimize the amount of current, and the wires heat up very quickly. This can be counter-intuitive. Above we said that increased resistance (in the filament) results in heat. Here, the opposite case is being made. How can both be true? They heat up for slightly different reasons. Whereas the filament heats up because of the random bouncing around of electrons and the resistance to forward flow, the circuit wires in a short circuit heat up because there is nothing to minimize the amount of flow. The more atoms that pass through a given area, the more jostling and bouncing that can occur—this movement is kinetic energy, or heat energy.

The Challenge of Isolating the Role of the Filament

The explanation of the filament that follows offers information that is beyond the scope of what this module covers, but addresses some puzzles that could arise for students. It is offered here for teachers as background, but is not needed for the lesson.

Like most science concepts, as soon as you delve deeply into electricity/electrical circuits, the concepts become complex. This lesson introduces the concept of resistance at a level that students should be able to grasp given the background provided thus far. However, a deep understanding of why filaments glow requires students to reason about the circuit as a complex system. As students begin to deeply probe the model presented here, they may discover some puzzles. This is because it isn't strictly possible to examine just the filament wire without attending to the entire system. You can't actually isolate the role of the filament in the circuit. However, students also need to develop some understanding of how the filament acts as a limiting factor in the broader system. The resolution to these puzzles engages students in complex concepts that are beyond the scope of this module. However, the issues are elaborated further here for those teachers who would like to understand it for background information or would like to consider presenting aspects of the ideas to their students.

The experiments in this lesson are designed to help students understand the variables involved in resistance. However, it is important for teachers to realize that the experiment is actually an imperfect one in a couple of ways. The primary reason is that a battery is an imperfect voltage source. In the case of a circuit with a copper filament wire, the wire's resistance is so low that the battery, and not the wire, limits the rate of current flow. Electrons can flow from the negative terminal to the positive terminal through the wire faster than the chemical reactions in the battery can move the charge. The battery cannot maintain the full level of electrical imbalance (voltage difference) between the terminals. As a result the current through the wire drops until it matches the rate at which the battery can move charge internally. If you measure the voltage between the terminals of the battery when you do the filament experiments, you will find a much lower voltage than 1.5V (the voltage of the batteries used here and in most classroom experiments) while the current flows.

By contrast, the Nichrome filament wire glows because it has enough resistance to be the rate-limiting factor in the system, meaning that the battery maintains a full 1.5V difference between the terminals. The greater potential difference distributed through the Nichrome wire means electrons flow more energetically through the wire than they do through copper, thus jostling the atoms of the Nichrome more vigorously.

Because of the different voltages, the comparison between the copper and Nichrome wires is not completely parallel. Some classrooms use a rheostat, a device that enables the teacher to adjust the amount of voltage coming out of outlets in the science lab. This makes it possible to make a more direct comparison. However, we DO NOT recommend this. Students can easily forget that the voltage has been adjusted to be different from that in their home and might engage in unsafe behavior with electrical outlets at home.

While the lesson introduces three contributing variables (material type, diameter and temperature of the filament wire) the story of how a light bulb lights is actually quite complex—more complex than students will readily be able to reason about or are asked to in this module. Therefore, they may discover puzzles when trying to apply the three variables. For instance, students might notice that electrical bulbs have different brightness levels.¹ They might also notice that bulbs with high intensity also have filaments that are thicker and low resistance, and bulbs with low intensity have filaments that are thinner and high resistance. This appears to contradict what was learned in the lesson. The contradiction can be resolved, but it quickly engages students in a complex systems problem beyond the scope of this module.