Resistance and resistivity
Introduced| English | Meaning |
|---|---|
| resistance/rɪˈzɪstəns/ | The ratio of potential difference to electric current. |
| Ohm's law/əʊmz lɔː/ | The rule that current is proportional to potential difference across an ohmic conductor when physical conditions such as temperature are constant. |
| conductor/kənˈdʌktə/ | A material through which electric charge or thermal energy can pass readily. |
| filament lamp/ˈfɪləmənt læmp/ | A lamp that produces light by heating a thin conducting filament with an electric current. |
| diode/ˈdaɪəʊd/ | An electronic component that allows current to flow much more readily in one direction than in the other. |
| semiconductor/ˌsemɪkənˈdʌktə/ | A material whose electrical conductivity can be controlled and is generally between that of a conductor and an insulator. |
| lattice/ˈlætɪs/ | A regular, repeating arrangement of particles or points in a solid. |
| resistivity/ˌriːzɪˈstɪvəti/ | A material property relating resistance to length and cross-sectional area at a specified temperature. |
| thermistor/ˈθɜːmɪstə/ | A resistor whose resistance changes strongly with temperature, commonly decreasing as temperature rises. |
Why the kettle element glows
- A kettle's element gets red-hot, but the cable feeding it stays cool.
- The element has a high resistance 电阻 — it turns electrical energy into heat.
- Resistance controls where the energy goes.
Resistance
- Resistance $R = \dfrac{V}{I}$, in ohms ($\Omega$): the p.d. across a component divided by the current in it.
- That is the definition, and it applies to every component. It depends on the conditions — especially the temperature.
Real fixed resistors: the coloured bands code the resistance value
Resistance (Ohm's law)
V = R·I
Ohm's law: voltage is proportional to current — the gradient is the resistance R.
A component has $12\ \text{V}$ across it and $3.0\ \text{A}$ through it. What is its resistance?
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$R = \dfrac{V}{I} = \dfrac{12}{3.0} = 4.0\ \Omega$.
Ohm's law 欧姆定律
- A conductor 导体 obeys Ohm's law when $I \propto V$ (at constant temperature) — so $R$ is constant.
- This is an experimental result, not the definition. $R = \dfrac{V}{I}$ works for any component.
An ohmic conductor at constant temperature has:
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Ohm's law: $I \propto V$, so $R = \dfrac{V}{I}$ is constant and the $I$–$V$ graph is a straight line through the origin.
I–V characteristics
- Metal wire (constant temp): straight line — constant $R$.
- Filament lamp 灯丝灯泡: curves over — heating raises $R$.
- Diode 二极管: passes current one way only (above ~0.7 V).
I-V characteristic of a semiconductor 半导体 diode
Match each component to its $I$–$V$ characteristic.
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Constant $R$ → straight line; heating raises $R$ → flattening curve; a diode blocks reverse current.
A filament lamp's resistance rises at higher voltage because:
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More current heats the filament; in a metal, more lattice vibration scatters electrons, so $R$ increases.
Why the lamp's resistance rises
- The exam chain: a larger current → more heating → higher temperature → the metal ions in the lattice 晶格 vibrate more → the free electrons collide more often → resistance increases.
- Read $R$ off an $I$–$V$ graph as $\dfrac{V}{I}$ at that point — not as the gradient. Only for a straight line through the origin are the two the same.
Put the steps of the explanation for why a filament lamp's resistance rises with current in order.
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Heating → more lattice vibration → more frequent collisions → higher resistance. Each link is a mark in the exam explanation.
Resistivity 电阻率
- $R = \dfrac{\rho L}{A}$ — $\rho$ is the resistivity, a property of the material ($\Omega\cdot\text{m}$).
- Longer wire → more $R$; thicker wire → less $R$.
- To measure it: ammeter in series with the wire, voltmeter across it, a variable supply (or variable resistor) to take several readings; $R$ is the gradient of $V$ against $I$; measure $L$ with a rule and the diameter with a micrometer at several places.
A longer conductor has more resistance
What resistance depends on: R = ρL/A
A longer wire has more resistance; a thicker one (bigger area) has less. Change the length, area and metal.
A wire has resistivity $2.0 \times 10^{-8}\ \Omega\cdot\text{m}$, length $10\ \text{m}$ and area $2.0 \times 10^{-6}\ \text{m}^2$. Find its resistance.
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$R = \dfrac{\rho L}{A} = \dfrac{2.0 \times 10^{-8} \times 10}{2.0 \times 10^{-6}} = 0.10\ \Omega$.
Doubling a wire's length doubles its resistance (same material and area).
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$R = \dfrac{\rho L}{A}$, so $R \propto L$ — twice the length, twice the resistance.
Worked example: resistivity with its uncertainty
A nichrome wire of length $1.20\ \text{m}$ ($\pm 0.5\%$) and diameter $0.40\ \text{mm}$ ($\pm 2.5\%$) has $3.00\ \text{V}$ ($\pm 1\%$) across it when the current is $0.600\ \text{A}$ ($\pm 1\%$). Find the resistivity and its uncertainty.
- Resistance: $R = \dfrac{V}{I} = \dfrac{3.00}{0.600} = 5.00\ \Omega$.
- Area: $A = \dfrac{\pi d^{2}}{4} = \dfrac{\pi (0.40 \times 10^{-3})^{2}}{4} = 1.26 \times 10^{-7}\ \text{m}^{2}$.
- Resistivity: $\rho = \dfrac{RA}{L} = \dfrac{5.00 \times 1.26 \times 10^{-7}}{1.20} = 5.24 \times 10^{-7}\ \Omega\,\text{m}$.
- Percentage uncertainty: add them, counting the diameter twice because it is squared: $1 + 1 + 2(2.5) + 0.5 = 7.5\%$.
- Absolute uncertainty: $0.075 \times 5.24 \times 10^{-7} = 0.4 \times 10^{-7}$, so $\rho = (5.2 \pm 0.4) \times 10^{-7}\ \Omega\,\text{m}$.
- Check: the diameter dominates the uncertainty — which is why it is measured with a micrometer, several times, and averaged.
In a resistivity measurement the percentage uncertainties are: $V$ 2%, $I$ 1%, diameter 3%, length 1%. What is the percentage uncertainty in $\rho = \dfrac{RA}{L}$?
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$\rho \propto \dfrac{V d^{2}}{I L}$, so add $2 + 1 + 2(3) + 1 = 10\%$. The diameter counts twice because it is squared.
$R = \dfrac{V}{I}$ is the definition and works for everything; Ohm's law is the special case where it stays constant. Resistance is not the gradient of an $I$–$V$ curve. The diameter is squared in $A = \dfrac{\pi d^{2}}{4}$, so its percentage uncertainty counts twice — and use the diameter in metres.
Light and heat sensors
- An LDR's resistance falls as light gets brighter (megaohms in the dark, hundreds of ohms in light).
- A thermistor 热敏电阻 (NTC)'s resistance falls as it gets hotter — both are semiconductors.
A thermistor's resistance drops sharply as it gets hotter — the basis of a temperature sensor
Select all the true statements.
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LDRs and NTC thermistors (semiconductors) drop in resistance with more light/heat. A metal does the opposite — its resistance rises with temperature.
You've got it
- resistance $R = \dfrac{V}{I}$ (the definition, at a point — never a gradient); Ohm's law: $I \propto V$ at constant temperature
- filament lamp curves because heating makes the lattice vibrate more and electrons collide more; a diode is one-way
- resistivity $R = \dfrac{\rho L}{A}$, with the diameter's uncertainty counted twice; LDR ↓ with light, thermistor ↓ with heat