### Resistance to the flow of electron current

Only very special substances, called superconductors (they're actually special substances placed in special situations, namely extreme cold) offer no resistance to the flow of electric current. All other substances resist the flow current to some degree.

Because of the way that electrons are arranged around their nuclei, metals tend to be excellent conductors. Some, like copper (Cu), silver (Ag) and gold (Au) are better than others, like lead (Pb), but all resist the flow of electric charge (current) to some degree.

Electrical resistance of a material is a property of that material, and its size and temperature. It is a result, like most of physics and all of chemistry, of the way electrons behave when bound to atoms.

If you know a bit about chemistry, the conducting properties of metals arises from how their electrons in d-orbitals arrange and behave in bulk materials.

### Unit of resistance: Ohms

The unit we use to measure and compare electrical resistance in materials and in electrical and electronic circuits is the Ohm, defined in the box and referred to by the Greek letter omega.

Resistance can range from a few pico-Ohms (1 pΩ = 1 × 10-12Ω) in good conductors to many Giga-Ohms (1 GΩ = 1 × 109 Ω) in insulators.

The base units of Ohms (1 Ω = 1 Kg·m2s-3A2) seem a bit complicated, but you generally won't need them except in special situations. For now, just know the rough range of resistance for conductors and insulators. We'll use Ohms a lot to do circuit calculations.

By the way, Ohms is capitalized because it's a person's name: Georg Simon Ohm.

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#### The Greek alphabet

 alpha Α α beta Β β gamma Γ γ delta Δ δ epsilon Ε ε zeta Ζ ζ eta Η η theta Θ θ iota Ι ι kappa Κ κ lambda Λ λ mu Μ μ nu Ν ν xi Ξ ξ omicron Ο ο pi Π π rho Ρ ρ sigma Σ σ tau Τ τ upsilon Υ υ phi Φ φ chi Χ χ psi Ψ ψ omega Ω ω

### Resistivity in materials

The electrical resistance of a material depends on two things:

• The nature of the material and
• It's physical dimensions.

We'll get to size and the effect of dimensions below. The intrinsic property of materials to resist the flow of electric current is called resistivity. The table shows the resistivities of several substances in units of Ohms per meter (Ω/m).

You can see that the things you would expect to conduct electricity, like metals, have low resistivities, and the things you hope don't, insulators like glass and rubber, have very high resistivities.

Conductors have low resistivities, and insulators have high resistivity.

Notice that the difference in resistivity between a good conductor and a good insulator can be many orders of magnitude of resistivity.

Also notice that many materials commonly held to be good conductors, like water, are actually not so good.

Silicon (and other semi-metallic elements around it in the periodic table) can either conduct or resist current in different circumstances, and they are called semiconductors – and there's a whole industry built around them!

### Resistance is proportional to size

The resistance of a material is a property of the identity (atomic & molecular structure) of the material, but also its size and shape. You can think of resistance to the flow of electric current much like resistance to the flow of water in a pipe:

• If the pipe is narrow, the amount of water you will be able to push through will be limited.
• If the pipe is long, the small amount of attraction of the water for the walls will add up and slow it down.

In a wire like the one shown below, the resistance is proportional to the length, and inversely proportional to the cross-sectional area.

The longer the wire the greater the resistance, and thicker wires have less resistance. Mathematically, the resistance proportionality statement is:

The constant of proportionality is ρ (the Greek letter "rho"), the resistivity of the material. Resistivity is an intensive property, and length and area are extensive.

### Choke points

Let's say you have a long and very conductive wire through which you're running an electric current. The wire is a good conductor so the resistance is low, and we would say that the resistance of the whole system (we'll call it a circuit later) is low.

Now interrupt that wire with a small element that has a high resistance – it could just be a thin piece of wire like the fine wires in an incandescent light bulb – and the resistance of the whole system will just be the resistance of that least conductive piece. It's a choke point for the flow of current.

You can think of electricity like water flowing through pipe. You can get a lot of water to flow through a 10 cm diameter pipe, but if there's just one small section that has a diameter of 1 cm, then that section will determine the overall flow. It's the rate-limiting or current-limiting element of the system.

Later we'll use this idea to create different "local" currents around the same loop of current in circuits.

### A water circuit analogy

We can make a handy analogy between the flow of electric current and the flow of water in a closed circuit of pipes.

The figure on the left shows a closed pipe system which includes a water pump (pumping counterclockwise), a valve for shutting the whole thing down and a constriction in the pipe. If we set the pump at a certain level (volume of water pumped per unit time), then the flow of water through the system is determined by the constriction. The water in the pipe can flow no faster than it does through the narrow part.

In the electric circuit (right), the pump is replaced by a battery, a source of electric potential. It provides the force that pushes and pulls electrons around the circuit, which consists of wire. The valve in the water circuit is replaced by a switch in the electric circuit.

The constriction is replaced in the circuit by the squiggly line, the common symbol for a resistor. A resistor is a section of wire that resists the smooth flow of electrons. Electric current can flow no faster through a circuit than it can through the resistor.

### Circuit resistors

Resistors in electric circuits usually look something like those pictured here. They consist of two lead wires connected to one or more of a variety of materials that don't conduct electric current as well as the wire.

Resistors are constructed to have a fixed resistance. The ones shown have a color code that you can look up. This makes it easy to read the resistance levels when looking at a circuit board.

Below is the common resistor color code table. It lists the color codes for numbers, the multiplier and the tolerance or precision of the resistance.

Two examples of banded resistors are shown below the table. Some resistors have five colored bands and others four. In the five-band case, the first three numbers give the resistance, the fourth band gives the multiplier (e.g. × 10 KΩ) and the fifth the tolerance in plus or minus some percentage of the resistance value.

The second example shows a four-band code. the difference is that only the first two bands are numbers. Everything else is the same.

Not all resistors use the color code; those will simply have a resistance written on them. And there are many kinds of specialty resistors, some with adjustable resistances called potentiometers.

One further important designation is the power capacity of the resistor. Circuits operating at high voltages with high currents can burn up low-power resistors, so one must be careful in choosing the right kind of resistors for the task at hand.

#### Examples

Here are two examples of how to read the resistor code on 5-band and 4-band resistors.

### Incandescent light bulbs are resistive devices

Incandescent light bulbs operate using the choke-point principle above.

Current is directed through a very fine wire (the horizontal section in the center of the bulb on the right) that is resistive because of the choice of material, and because it is very thin.

Because of the high resistance and high current, electrons moving in the wire experience a "friction" that heats the wire. In fact, it's heated so hot that it emits visible light.

You may have seen that metal objects heated in a fire glow red. It's the same principle here except that the wire gets much hotter. The hotter the wire, the more light is emitted in the visible and ultraviolet range.

Source: Wikipedia Commons

### Resistance can be temperature-dependent

Very often, and to varying degrees, the electrical resistance of a material can depend on the temperature of the material.

When a material, such as a wire, is relatively cold, its atoms move more slowly and don't oscillate as far from their equilibrium positions in the metal. The path of a charged particle, like an electron, through the wire might look like this:

But if the substance is hotter, its atoms will oscillate more rapidly, and will travel farther from their equilibrium positions, and thus the probability of a collision with the traveling charge will be greater:

The path of such a charge will be more convoluted, so it will take longer, on average, to travel the same distance in cold material

This charge dependence of resistance can be important in many kinds of electric circuits, so it's worth keeping in mind.

### Ohm's law and the flow of current

In most applications of electricity, what we're really interested in is regulating the flow of electric current to do useful things. We'd like to make the flow of current predictable and to be able to manipulate it.

It turns out the the flow of current through a material is directly proportional to the potential, and it's inversely proportional to the resistance of the material to the flow of current. That's known as Ohm's law, and it's written like this:

Ohm's law is one of the most important relationships in all of the field of electricity and magnetism.

If we want more current, we can either

• increase the potential (voltage) or
• decrease the resistance.

Later you will see electric circuits in which the potential, current and resistance are fixed, and those in which each can depend upon other things, such as frequency of switching current on and off. All of this leads to all of the wonderful electronic devices to which we've become so accustomed.

Ohm's law is usually written on one line as a product, V = IR.

#### Ohm's law

Ohm's law: Potential is current multiplied by resistance.

V   =   IR

### Example 1

Calculate the resistance of 10 meters of AWG-14 wire made of the following materials, with the given resistivities (all at T = 298 K). AWG stands for "American Wire Gauge." 14-gauge wire has a cross-sectional area of 2.5 mm2:

1. Copper,   ρ = 1.75 × 10-8 Ωm
2. Barium,   ρ = 3.4 × 10-7 Ωm
3. Lithium,   ρ = 9.47 × 10-8 Ωm

Solution: This is a good opportunity to use a spreadsheet. They're perfect for doing repetitive calculations.

We'll put the resistivities on the first column and calculate the resistance of each wire in the second. We'll need to convert 2.5 mm2 to square meters:

$$2.5 \, mm^2 \left( \frac{1 \, m}{1000 \; mm} \right)^2 = 2.5 \times 10^{-6} \, m^2$$

Now the second column of our spreadsheet will be the resistance,

$$R = \frac{\rho \cdot L}{A}$$

Here are the results. You can look up resistivities of substances in a number of places. They are also temperature-dependent.

### Example 2

The Taser™ company claims that their stunning device delivers a voltage of approximately 1200 V across a human body with as much as 30 mA (0.03 A) of electric current. Using these data, calculate the approximate resistance of a human body between the electrodes of the Taser.

Solution: This problem suggests a rearrangement of Ohm's law:

$$V = IR \; \longrightarrow \; R = \frac{V}{I}$$

The resistance is then:

$$R = \frac{1200 \, V}{0.03 \, A} = 40,000 \; \Omega$$

When writing resistances, it's customary to keep the numerical part at three significant digits or less, so we express this resistance in kilo-ohms, 40 KΩ

### Example 3

A 14.4 V battery-operated drill draws 3.0 A of current in use. Calculate the internal resistance of the drill.

Solution : We work this problem in the same way as example 2, R = V/I gives a resistance of 4.8 Ω for the drill.

Small resistances like this are common for motor-driven tools like a drill.

### Practice problems

 1 A certain sump pump (pumps water from low spaces like basements) is specified to plug into a 110 V outlet and draw a current of 10.8 A. Calculate the "internal" resistance of the pump. Internal resistance means that for the purposes of the discussion, we're treating the sump pump like a simple resistor. Solution Use Ohm's law, V = IR. Rearrange to R = V/I = 110V/10.8A = 10.2 Ω 2 A human body can survive large voltage differences if the current is low. A TASER is a device used to immobilize a person by passing a current of about 3 mA (0.003 A) at 1200 V through a person. Using these values, calculate the internal resistance of a person. Solution Use Ohm's law, V = IR. Rearrange to R = V/I = 1200V/0.003A = 400,000 Ω or 400 KΩ 3 A defibrillator can restore a normal heartbeat to a defibrillating heart (one that's quivering randomly and can't pump blood). Using the resistance of a human body from problem 2, and assuming that at least 20 mA (0.020 A) of current must pass through the heart in order to electrically "convert" the rhythm, what minimum voltage must be applied by the defibrillator? Solution Use Ohm's law, V = IR. V = 0.020A · 400,000 Ω = 8000 V or 8 KV 4 Use the data in the table above to calculate the resistance of a piece of platinum wire 1 cm long and 1 mm in diameter. Do the same calculation with a piece of hard rubber of the same dimensions. Solution Platinum (Pt): use   $R = \frac{\rho\cdot L}{A}$, where $\rho = 1.06 \times 10^{-6} \; \Omega/m$, $L = 0.01 \, m$   and   $A = \pi(0.5 \times 10^{-3})^2 \; m^2$. So   $R = 0.04 \Omega.$   This is a low resistance. Pt is a good electrical conductor Hard rubber: Use $R = \frac{\rho\cdot L}{A}$, where $\rho = 1 \times 10^{13} \; \Omega/m$, $L = 0.01 \, m$   and   $A = \pi(0.5 \times 10^{-3})^2 \; m^2$. So   $R = 4 \times 10^{17} \Omega.$   This is a high resistance. Hard rubber is a poor electrical conductor and a good insulator.

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