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This topic is the foundation for every circuit question on the papers. Examiners test it through quick multiple-choice items (units, ratios, I–V graph shapes), short calculations with I=ΔQ/Δt, R=V/I and R=ρL/A, the resistivity practical (a favourite for data analysis and 6-mark method answers), and “explain in terms of charge carriers” questions about lamps, thermistors and superconductors.
By the end you’ll be able to
Define current, potential difference and resistance, and use I = ΔQ/Δt, V = W/Q and R = V/I
Use I = nAvq to compare drift velocities in metals, semiconductors and insulators
Sketch and explain I–V characteristics of a resistor, filament lamp, diode/LED and thermistor
Apply R = ρL/A and describe the resistivity required practical with a micrometer
Explain how resistance of metals and NTC thermistors varies with temperature, and describe superconductivity and critical temperature
What the exam asks
Definitions of current, potential difference and resistance, with units and base units.
Charge as the area under a current–time graph, and counting electrons with N=Q/e.
The mean drift velocity equation I=nAvq (all boards except AQA).
Sketching and explaining I–V characteristics: ohmic resistor, filament lamp, diode or LED, NTC thermistor.
Resistivity calculations, especially ratio questions and diameter-to-area conversions.
The resistivity practical: micrometer technique, the against graph, the intercept, and uncertainties.
vii.Check your understanding
3 questions on current, potential difference, resistance and resistivity. Every option is explained once you answer.
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PromptCard 1 of 3
Define potential difference.
R
L
How resistance changes with temperature in metals and thermistors, and superconductivity (AQA, Eduqas/WJEC).
Core ideas
Current, charge and potential difference
Current is the rate of flow of charge:
I=ΔtΔQ
One ampere is one coulomb per second. Conventional current flows from + to −; electrons drift the other way. Charge is quantised in multiples of e=1.60×10−19 C, so the number of electrons carrying a charge Q is N=Q/e.
The area under an I–t graph is the charge that has flowed.
The gradient of a Q–t graph is the current.
Potential difference is the energy transferred from electrical to other forms per unit charge between two points:
V=QW,1V=1J C−1
Drift velocity: I=nAvq (not AQA)
n is the number density of charge carriers (m⁻³), A the cross-sectional area, v the mean drift velocity and q the charge on each carrier.
Material
n / m⁻³ (order of magnitude)
Consequence
Metal (copper)
1028–1029
Tiny drift velocity (fractions of a mm s⁻¹)
Semiconductor (silicon)
1016–1023, rising with temperature
Much faster drift for the same current
Insulator
Close to zero
Negligible current
For the same current in series, a thinner section of the same wire has a larger drift velocity (v∝1/A). A lamp still lights almost instantly because the electric field is set up around the whole circuit at close to the speed of light, so free electrons everywhere, including those in the filament, start drifting together.
Resistance and Ohm’s law
Resistance is defined as R=V/I, measured in ohms (Ω = V A⁻¹). Always use the ratioV/I at the point of interest, never the gradient of the I–V graph.
Ohm’s law: for a metallic conductor at constant temperature, the current is directly proportional to the potential difference. A component that obeys it is ohmic.
I–V characteristics
Component
Shape of the I–V graph
Why
Ohmic resistor (constant temperature)
Straight line through the origin in both quadrants
R is constant
Filament lamp
Curve through the origin that gets less steep as ∣V∣ increases; symmetric
Filament heats up, so R rises
Semiconductor diode
Almost no current until about 0.6 V (silicon), then rises steeply; tiny current in reverse
Conducts in one direction only
LED
Like a diode, but the threshold is higher (about 1.5–3 V, depending on colour)
Photon energy is about eVthreshold
NTC thermistor (self-heating)
Curve that gets steeper as ∣V∣ increases
Warms up and releases more charge carriers, so R falls
To record a full characteristic, supply the component from a potential divider (so V can start from 0), use an ammeter in series and a voltmeter in parallel, then reverse the connections for negative values. Meters are treated as ideal: an ammeter has zero resistance and a voltmeter has infinite resistance.
Resistivity
R=AρL⟺ρ=LRA
Resistivity ρ is a property of the material (at a given temperature), with unit Ω m. Resistance belongs to a particular object. With A=πd2/4:
R∝d2L
Material
ρ / Ω m at room temperature
Copper
1.7×10−8
Aluminium
2.8×10−8
Constantan
about 4.9×10−7
Nichrome
about 1.1×10−6
Silicon (pure)
about 103
Glass or PVC
1010 or more
Temperature and resistance
Metals: as temperature rises, the positive ions vibrate with larger amplitude. Free electrons collide with them more often, so their drift velocity for a given pd falls and resistance rises. The number of charge carriers stays essentially constant.
NTC thermistors (semiconductors): as temperature rises, many more charge carriers are released, so n increases sharply. This outweighs the extra lattice vibration, so resistance falls, steeply at first and then more gradually (a non-linear, roughly exponential curve).
LDRs: more light releases more charge carriers, so resistance falls as light intensity rises.
Superconductivity (AQA, Eduqas/WJEC)
A superconductor has zero resistivity at and below its critical temperatureTc. The drop is abrupt, not gradual. With no resistance there is no pd across it and no heating, so a current can circulate indefinitely. Most superconductors need cooling to a few kelvin; the best-known “high-temperature” ceramics work at roughly 90–130 K, so liquid nitrogen (77 K) can be used. Uses: very strong magnetic fields (MRI scanners, particle accelerators, maglev trains) and power cables with no resistive loss.
The resistivity practical
Measure the wire’s diameter with a micrometer at several points and in perpendicular directions; check the zero reading and take the mean.
Tape the wire to a metre rule. Attach one crocodile clip at zero and move the other to lengths L from about 0.10 m to 0.90 m.
Measure R with an ohmmeter or from V/I. Keep the current small and switch off between readings so the wire does not heat up.
Plot R (y-axis) against L (x-axis). Then gradient=ρ/A, so ρ=gradient×A.
A non-zero intercept comes from the resistance of the leads and clip contacts (a systematic error). It does not change the gradient, which is why the graph method beats a single reading.
Worked examples
Exam technique
Ratio questions: write the proportionality first, for example R∝L/d2, then scale each factor. It takes seconds and avoids rounding errors.
Convert before substituting: mm to m (×10−3), mm² to m² (×10−6), mA to A, minutes to seconds.
“Explain in terms of charge carriers”: name the carriers, say what happens to n and to their collisions with the lattice, then link to R. For Edexcel, OCR and Cambridge, quote I=nAvq.
Reading an I–V graph:R=V/I at the point. If the axes are swapped (V against I), the gradient still does not give R for a curve.
6-mark resistivity method: cover instruments and measurements, the graph and how ρ comes from the gradient, and precautions (repeat diameters, small current, switch off between readings, intercept from contact resistance).
Common mistakes
Quick recap
I=ΔQ/Δt (area under I–t = charge), V=W/Q, R=V/I at the point, not the gradient.
I=nAvq: tiny v in metals, larger v in thinner wires and in semiconductors (not AQA).
Lamp: I–V curve flattens because R rises. Thermistor: curve steepens because R falls. Diode: threshold about 0.6 V, one direction only.
R=ρL/A with A=πd2/4, so ; is in Ω m.
Metals: more lattice vibration, more collisions, higher R. Semiconductors: more carriers, lower R.
Superconductor: zero resistivity at and below Tc; used for strong magnets and loss-free cables.
Resistivity practical: R against L, ρ=gradient×A; the intercept is contact resistance.