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Resistance

What's the physical difference between a 100-watt lightbulb and a 200-watt one? They both plug into a 110-volt outlet, so according to the equation P=IV, the only way to explain the double power of the 200-watt bulb is that it must pull in, or “draw,” twice as much current. By analogy, a fire hose and a garden hose might be served by pumps that give the same pressure (voltage), but more water will flow through the fire hose, because there's simply more water in the hose that can flow. Likewise, a wide, deep river could flow down the same slope as a tiny creek, but the number of liters of water flowing through the big river is greater. If you look at the filaments of a 100-watt bulb and a 200-watt bulb, you'll see that the 200-watt bulb's filament is thicker. In the charged-particle model of electricity, we expect that the thicker filament will contain more charged particles that are available to flow. We say that the thicker filament has a lower electrical resistance than the thinner one.

fat-and-skinny-pipes

 A fat pipe has less resistance than a skinny pipe.

Although it's harder to pump water rapidly through a garden hose than through a fire hose, we could always compensate by using a higher-pressure pump. Similarly, the amount of current that will flow through a lightbulb depends not just on its resistance but also on how much of a voltage difference is applied across it. For many substances, including the tungsten metal that lightbulb filaments are made of, we find that the amount of current that flows is proportional to the voltage difference applied to it, so that the ratio of voltage to current stays the same. We then use this ratio as a numerical definition of resistance,

\[R = \frac{V}{I}\]

which is known as Ohm's law. The units of resistance are ohms, symbolized with an uppercase Greek letter Omega, Ω. Physically, when a current flows through a resistance, the result is to transform electrical energy into heat. In a lightbulb filament, for example, the heat is what causes the bulb to glow.

ohm

Georg Simon Ohm (1787-1854).

Ohm's law states that many substances, including many solids and some liquids, display this kind of behavior, at least for voltages that are not too large. The fact that Ohm's law is called a “law” should not be taken to mean that all materials obey it, or that it has the same fundamental importance as the conservation laws, for example. Materials are called ohmic or nonohmic, depending on whether they obey Ohm's law.

On an intuitive level, we can understand the idea of resistance by making the sounds “hhhhhh” and “ffffff.” To make air flow out of your mouth, you use your diaphragm to compress the air in your chest. The pressure difference between your chest and the air outside your mouth is analogous to a voltage difference. When you make the “h” sound, you form your mouth and throat in a way that allows air to flow easily. The large flow of air is like a large current. Dividing by a large current in the definition of resistance means that we get a small resistance. We say that the small resistance of your mouth and throat allows a large current to flow. When you make the “f” sound, you increase the resistance and cause a smaller current to flow. In this mechanical analogy, resistance is like friction: the air rubs against your lips. Mechanical friction converts mechanical forms of energy to heat, as when you rub your hands together. Electrical friction --- resistance --- converts electrical energy to heat.

If objects of the same size and shape made from two different ohmic materials have different resistances, we can say that one material is more resistive than the other, or equivalently that it is less conductive. Materials, such as metals, that are very conductive are said to be good conductors. Those that are extremely poor conductors, for example wood or rubber, are classified as insulators. There is no sharp distinction between the two classes of materials. Some, such as silicon, lie midway between the two extremes, and are called semiconductors.

Applications

Superconductors

All materials display some variation in resistance according to temperature (a fact that is used in thermostats to make a thermometer that can be easily interfaced to an electric circuit). More spectacularly, most metals have been found to exhibit a sudden change to zero resistance when cooled to a certain critical temperature. They are then said to be superconductors. A current flowing through a superconductor doesn't create any heat at all.

Theoretically, superconductors should make a great many exciting devices possible, for example coiled-wire magnets that could be used to levitate trains. In practice, the critical temperatures of all metals are very low, and the resulting need for extreme refrigeration has made their use uneconomical except for such specialized applications as particle accelerators for physics research.

But scientists have recently made the surprising discovery that certain ceramics are superconductors at less extreme temperatures. The technological barrier is now in finding practical methods for making wire out of these brittle materials. Wall Street is currently investing billions of dollars in developing superconducting devices for cellular phone relay stations based on these materials. In 2001, the city of Copenhagen replaced a short section of its electrical power trunks with superconducing cables, and they are now in operation and supplying power to customers.

There is currently no satisfactory theory of superconductivity in general, although superconductivity in metals is understood fairly well. Unfortunately I have yet to find a fundamental explanation of superconductivity in metals that works at the introductory level.

Constant voltage throughout a conductor

The idea of a superconductor leads us to the question of how we should expect an object to behave if it is made of a very good conductor. Superconductors are an extreme case, but often a metal wire can be thought of as a perfect conductor, for example if the parts of the circuit other than the wire are made of much less conductive materials. What happens if the resistance equals zero in the equation

\[R = \frac{V}{I}?\]

The result of dividing two numbers can only be zero if the number on top equals zero. This tells us that if we pick any two points in a perfect conductor, the voltage difference between them must be zero. In other words, the entire conductor must be at the same voltage. Using the water metaphor, a perfect conductor is like a perfectly calm lake or canal, whose surface is flat. If you take an eyedropper and deposit a drop of water anywhere on the surface, it doesn't flow away, because the water is still. In electrical terms, a charge located anywhere in the interior of a perfect conductor will always feel a total electrical force of zero.

Suppose, for example, that you build up a static charge by scuffing your feet on a carpet, and then you deposit some of that charge onto a doorknob, which is a good conductor. How can all that charge be in the doorknob without creating any electrical force at any point inside it? The only possible answer is that the charge moves around until it has spread itself into just the right configuration. In this configuration, the forces exerted by all the charge on any charged particle within the doorknob exactly cancel out.

We can explain this behavior if we assume that the charge placed on the doorknob eventually settles down into a stable equilibrium. Since the doorknob is a conductor, the charge is free to move through it. If it was free to move and any part of it did experience a nonzero total force from the rest of the charge, then it would move, and we would not have an equilibrium.

It also turns out that charge placed on a conductor, once it reaches its equilibrium configuration, is entirely on the surface, not on the interior. We will not prove this fact formally, but it is intuitively reasonable (see discussion question B).

Short circuits

So far we have been assuming a perfect conductor. What if it's a good conductor, but not a perfect one? Then we can solve for

V=IR .

An ordinary-sized current will make a very small result when we multiply it by the resistance of a good conductor such as a metal wire. The voltage throughout the wire will then be nearly constant. If, on the other hand, the current is extremely large, we can have a significant voltage difference. This is what happens in a short-circuit: a circuit in which a low-resistance pathway connects the two sides of a voltage source. Note that this is much more specific than the popular use of the term to indicate any electrical malfunction at all. If, for example, you short-circuit a 9-volt battery as shown in the figure, you will produce perhaps a thousand amperes of current, leading to a very large value of P=IV. The wire gets hot!

The voltmeter

A voltmeter is nothing more than an ammeter with an additional high-value resistor through which the current is also forced to flow, l/1. Ohm's law relates the current through the resistor is related directly to the voltage difference across it, so the meter can be calibrated in units of volts based on the known value of the resistor. The voltmeter's two probes are touched to the two locations in a circuit between which we wish to measure the voltage difference, l/2. Note how cumbersome this type of drawing is, and how difficult it can be to tell what is connected to what. This is why electrical drawing are usually shown in schematic form. Figure l/3 is a schematic representation of figure l/2.

voltmeter

l / 1. A simplified diagram of how a voltmeter works. 2. Measuring the voltage difference across a lightbulb. 3. The same setup drawn in schematic form. 4. The setup for measuring current is different.

The setups for measuring current and voltage are different. When we're measuring current, we're finding “how much stuff goes through,” so we place the ammeter where all the current is forced to go through it. Voltage, however, is not “stuff that goes through,” it is a measure of electrical energy. If an ammeter is like the meter that measures your water use, a voltmeter is like a measuring stick that tells you how high a waterfall is, so that you can determine how much energy will be released by each kilogram of falling water. We don't want to force the water to go through the measuring stick! The arrangement in figure l/3 is a parallel circuit: one in there are “forks in the road” where some of the current will flow one way and some will flow the other. Figure l/4 is said to be wired in series: all the current will visit all the circuit elements one after the other.

If you inserted a voltmeter incorrectly, in series with the bulb and battery, its large internal resistance would cut the current down so low that the bulb would go out. You would have severely disturbed the behavior of the circuit by trying to measure something about it.

Incorrectly placing an ammeter in parallel is likely to be even more disconcerting. The ammeter has nothing but wire inside it to provide resistance, so given the choice, most of the current will flow through it rather than through the bulb. So much current will flow through the ammeter, in fact, that there is a danger of burning out the battery or the meter or both! For this reason, most ammeters have fuses or circuit breakers inside. Some models will trip their circuit breakers and make an audible alarm in this situation, while others will simply blow a fuse and stop working until you replace it.

Discussion Questions
  • In figure g/4 on page 102, what would happen if you had the ammeter on the left rather than on the right?
  • Imagine a charged doorknob, as described on page 109. Why is it intuitively reasonable to believe that all the charge will end up on the surface of the doorknob, rather than on the interior?

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