Resistance and resistivity. Not quite the same word, not quite the same thing. Make sure you know which is which…

Everything has Resistance

1. If you put a potential difference across an electrical component, a current will flow.

2. How much current you get for a particular p.d. depends on the resistance of the component.

3. You can think of a component’s resistance as a measure of how difficult it is to get a current to flow through it.

Mathematically, resistance is: R = V/I

The equation really defines what is meant by resistance.

4. Resistance is measured in ohms (Ω).

A component has a resistance of 1 Ω if a potential difference of 1 V makes a current of 1 A flow through it.

Three Things Determine Resistance

If you think about a nice, simple electrical component, like a length of wire, its resistance depends on:

1. Length (l). The longer the wire the more difficult it is to make a current flow.

2. Area (A). The wider the wire the easier it is to make a current flow.

3. Resistivity (p). This depends on the material the wire’s made from, as the structure of the material may make it easy or difficult for charge to flow. In general, resistivity depends on environmental factors as well, like temperature. p is the Greek letter rho, the symbol for resistivity.

The resistivity of a material is defined as the resistance of a 1 m length with a 1 m² cross-sectional area, so p = RA/I. Resistivity is measured in ohm metres (Ωm).

In your exams, you’ll be given this equation in the form:

Typical values for the resistivity of conductors are really small.

However, if you calculate a resistance for a conductor and end up with something really small (e.g. 1 × 10^-7 Ω), go back and check that you’ve converted your area into m² .

The resistivity of a material is related to the number density or charge carriers, (and their mean drift velocity, which often varies with temperature). The higher the number of charge carriers, (and the higher their mean drift velocity), the higher the current at a given p.d. (as I = nqvA), and so the lower the resistance and therefore the lower the material’s sensitivity. The number density of charge carriers varies greatly between different materials, which means there can be a huge variation in their resistivities.

For an Ohmic Conductor, R is a Constant

A chap called Ohm did most of the early work on resistance. He developed a rule to predict how the current would change as the applied potential difference increased, for certain types of conductor.

The rule is now called Ohm’s law and the conductors that obey it (mostly metals) are called ohmic conductors.

Provided the temperature is constant, the current through an ohmic conductor is directly proportional to the potential difference across it (that’s I ∝ V).

1. Doubling the p.d. doubles the current.

2. What this means is that the resistance is constant.

3. Often external factors, such as temperature will have a significant effect on resistance, so you need to remember that Ohm’s law is only true for ohmic conductors at constant temperature.

To Find the Resistivity of a Wire You Need to Find its Resistance

Before you start, you need to know the cross-sectional area of your test wire.

Assume that the wire is cylindrical, and so the cross-section is circular.

Then you can find its cross-sectional area using: area of a circle = πr²

Use a micrometer to measure the diameter of the test wire for at least three different points along the wire. Take an average value of the diameter and divide by 2 go get the radius (make sure this is in m). Plug it into the equation for cross-sectional area and… ta da. Now you can get your teeth into the electricity bit…

1. The test wire should be clamped to a ruler and connected to the rest of the circuit at the point where the ruler reads zero.

2. Attach the flying lead to the test wire — the lead is just a wire with a crocodile clip at the end to allow connection to any point along the test wire.

3. Record the length of the test wire connected in the circuit, the voltmeter reading and the ammeter reading.

4. Use your readings to calculate the resistance of the length of wire, using - R=V/I

5. Repeat for several different lengths within a sensible range, e.g. at 0.10 m intervals from 0.10m to 1.00m.

6. Plot your results on a graph of resistance against length, and draw a line of best fit.

The gradient of the line of best fit is equal to R/I = p/A. So multiply the gradient of the line of best fit by the cross-sectional area of the wire to find the resistivity of the wire material.

7. The resistivity of a material depends on its temperature, so you can only find the resistivity of a material at a certain temperature. Current flowing in the test wire can cause its temperature to increase, so failing to keep the wire at a constant temperature could invalidate your results. Try to keep the temperature of the test wire constant by e.g. only having small currents flow through the wire.

Practice Questions

Q1 State the equation that links the resistance of a wire to its resistivity.

Q2 The resistivity of a piece of glass is 1 × 10¹¹ Ωm at 20 ℃. The resistivity of aluminium at 20 ℃ is around 3 × 10^-8 Ωm. Explain this difference in terms of charge carriers.

Q3 What is Ohm’s law?

Q4 Describe an experiment to find the resistivity of a metal.