9.3 Resistance and resistivity
- Syllabus
- 9702–2028–2029
- Topic
- 9.3
- Level
- AS
Learning objectives
9.3.1Resistance• define resistance9.3.2V = IR• recall and use V = IR9.3.3The I–V characteristics of a metallic conductor at constant temperature, a• sketch the I–V characteristics of a metallic conductor at constant temperature, a semiconductor diode and a filament lamp9.3.4That the resistance of a filament lamp increases• explain that the resistance of a filament lamp increases as current increases because its temperature increases9.3.5Ohm’s law• state Ohm’s law9.3.6R = ρL / A• recall and use R = ρL / A9.3.7The resistance of a light-dependent resistor (LDR) decreases• understand that the resistance of a light-dependent resistor (LDR) decreases as the light intensity increases9.3.8The resistance of a thermistor decreases as the temperature increases (it• understand that the resistance of a thermistor decreases as the temperature increases (it will be assumed that thermistors have a negative temperature coefficient)
Resistance measures opposition to current through the ratio of voltage to current
Resistance is R=V/I at an operating point, measured in ohms. It describes how much potential difference is needed for a given current.
For a non-ohmic device the ratio can change with voltage, current or temperature, so call it operating-point resistance when appropriate.
A component carrying 0.50 A at 4.0 V has resistance 8.0 Ω at that point.
Resistance is not “used up” by a component and is not necessarily constant for every material or device.
Use V = IR for one operating point
V=IR
V is potential difference in volts, I is current in amperes and R = V/I is resistance in ohms at the stated operating point. Rearrangement gives I = V/R or R = V/I.
A component has 6.0 V across it and carries 0.25 A. Its resistance at that operating point is R = 6.0/0.25 = 24 Ω. If that resistance is constant, 3.0 V would produce 0.125 A.
From an I-V graph, choose the required point and calculate R = V/I using its coordinates. The gradient of an I-versus-V graph is 1/R only when the graph is a straight line through the origin.
Using V = IR at one point does not by itself show that a component obeys Ohm's law. Ohm's law additionally requires I to remain directly proportional to V under constant physical conditions.
Sketch three I-V characteristics from their physical behaviour
For an I-V characteristic, place potential difference V on the horizontal axis and current I on the vertical axis. Include positive and negative values unless the question restricts the range.
| Component | Required I-V sketch features | What the shape means |
|---|---|---|
| metallic conductor at constant temperature | straight line through origin; same line in quadrants I and III | constant gradient = 1/R, so I ∝ V |
| filament lamp | smooth curve through origin, symmetric in quadrants I and III; becomes less steep as | V |
| semiconductor diode | almost zero current for reverse V; almost zero forward current until turn-on, then a steep rise | conducts strongly in one direction only after sufficient forward V |
Sketch from the origin outward: decide whether the component is symmetric under voltage reversal; decide whether resistance stays constant, increases, or falls; then convert that into I-V steepness. On these axes, greater resistance means smaller I/V and therefore a less-steep line from the origin to the operating point.
Do not copy a V-I shape onto I-V axes without reflecting what the gradient represents. A diode is asymmetric; a filament lamp is symmetric but curved; only the constant-temperature metallic conductor is a straight line through the origin.
Heating makes a filament lamp's resistance rise
Increasing current increases electrical power transferred to the filament. Its temperature rises, lattice ions vibrate more strongly, and conduction electrons undergo more frequent scattering. Their drift is hindered, so the filament's resistance increases.
As voltage magnitude increases, current still increases, but less than proportionally. Therefore V/I grows and an I-V characteristic becomes less steep away from the origin. The curve is approximately symmetric for positive and negative voltage because either current direction heats the filament.
If current decreases, heating falls, the filament cools and its resistance decreases. The changing resistance—not a change in electron charge—is why a filament lamp does not have a straight constant-temperature metal characteristic during normal operation.
A filament lamp remains conducting and has a resistance at every operating point. It is non-ohmic because temperature and therefore V/I change as current changes.
Ohm’s law is a condition-dependent model, not a universal rule for every component
A component obeys Ohm’s law only when its V–I relation is proportional under constant conditions, giving constant resistance.
Test proportionality experimentally and identify variables such as temperature. Use a non-linear model or a local resistance when the graph curves.
A fixed resistor can be approximately ohmic over its rated range, while a filament lamp fails the test as it warms.
“Resistance” existing for a device does not prove that V/I stays constant across all voltages.
Use resistivity and true cross-sectional area to find wire resistance
R=ρL/A
| Symbol | Meaning | SI unit |
|---|---|---|
| R | resistance of the uniform conductor | Ω |
| ρ | resistivity of its material at the stated temperature | Ω m |
| L | conductor length along current flow | m |
| A | cross-sectional area perpendicular to current | m² |
A wire has ρ = 1.7 × 10⁻⁸ Ω m, L = 2.0 m and diameter 0.50 mm. Radius r = 0.25 × 10⁻³ m, so A = πr² = 1.96 × 10⁻⁷ m². Hence R = (1.7 × 10⁻⁸ × 2.0)/(1.96 × 10⁻⁷) = 0.17 Ω (2 s.f.).
For unchanged material and temperature, R ∝ L/A. Doubling length doubles R. Doubling diameter makes area four times larger, so R becomes one quarter. Rearrangement gives ρ = RA/L when resistivity is required.
Do not substitute diameter or radius directly for A. Resistivity belongs to the material and depends on physical conditions such as temperature; resistance also depends on the specimen's length and area.
An LDR has lower resistance when incident light intensity increases
A light-dependent resistor’s resistance decreases as light intensity increases, because illumination creates more mobile charge carriers in its semiconductor material.
Use it as a variable sensor: the circuit output depends on how the LDR is arranged with other resistors and the supply.
In a potential divider, brighter light lowers the LDR resistance and changes the share of supply voltage across the other component.
An LDR responds to light intensity, not simply to elapsed time or temperature; the direction of voltage change depends on circuit placement.
An NTC thermistor's resistance falls as temperature rises
For the negative-temperature-coefficient (NTC) thermistors assumed in this course, increasing temperature decreases resistance; decreasing temperature increases resistance.
| Temperature change | NTC resistance change |
|---|---|
| increases | decreases |
| decreases | increases |
A qualitative graph of resistance R against temperature T slopes downward throughout the stated range and is usually curved rather than linear. It starts from the stated resistance at the initial temperature and remains above R = 0.
With a fixed supply directly across the thermistor, heating lowers R and increases current. In a series circuit or potential divider, first update the thermistor resistance, then use the actual topology to determine current, power or the measured output voltage.
Do not apply the positive temperature behaviour of a metal filament to an NTC thermistor. Lower thermistor resistance alone does not determine whether an unspecified output voltage rises or falls; the measured points and component placement matter.