Resistivity and the potential divider flashcards

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Resistivity
The material property ρ in R = ρl/A; numerically, the resistance of a 1 m length of the material with a cross-sectional area of 1 m². Unit Ω m.
Unit of resistivity
Ω m (ohm metre), from ρ = RA/l: Ω × m² ÷ m.
Uniform wire
A wire of the same material and the same cross-sectional area all along its length, so equal lengths have equal resistance.
Potential divider
Two or more resistors in series across a supply, giving an output p.d. that is a fraction of the supply p.d.
Potential gradient along a wire
The p.d. across the wire divided by its length (V m⁻¹); constant along a uniform wire carrying a steady current.
Thermistor (NTC)
A resistor whose resistance decreases as its temperature increases.
Light-dependent resistor (LDR)
A resistor whose resistance decreases as the light intensity on it increases.
Resistance from resistivity
R = ρl/A (Yes)
Cross-sectional area of a round wire from its diameter d
A = πd²/4 (= πr² with r = d/2). (No — learn it)
Potential divider output across R₂
V_out = R₂ × V_in / (R₁ + R₂) (No — derive it)
Ratio of p.d.s across two resistors in series
V₁/V₂ = R₁/R₂ (same current through both). (No — derive it)
Resistivity from an R–l graph
ρ = gradient × A, since R = (ρ/A) × l. (No — derive it)
Graph of R against l for a uniform wire
Straight line; gradient = ρ/A. Through the origin, or with a small intercept from contact resistance.
Graph of potential against distance along a uniform current-carrying wire
Straight line from 0 V at one end to the full p.d. at the other; gradient = potential gradient.
Thermistor in the top position, output across the fixed resistor: output against temperature
Output rises with temperature (thermistor resistance falls, its share of the supply falls).
Resistivity practical: circuit
Battery, switch and ammeter in series with the test wire on a metre rule; voltmeter across the length l between a fixed clip and a movable clip.
Two wires of the same metal plotted on one R–l graph
The thinner wire has the steeper line: gradient ρ/A is larger when A is smaller.
Why does resistance increase with length and decrease with cross-sectional area?
Longer: charge passes more ions on its way. Larger area: more parallel paths for the charge.
Resistivity practical: why measure the diameter at several places and orientations?
The wire may not be uniform or round; the mean gives a better area and the spread gives the uncertainty.
Resistivity practical: why keep the current small and switch off between readings?
To stop the wire heating; resistivity increases with temperature, so R would be too high and drift.
Why is the gradient method better than a single R reading?
Uses all the results, shows anomalies, and a constant contact resistance only shifts the intercept, not the gradient.
Why does potential change linearly along a uniform wire?
Same current everywhere and equal resistance per unit length, so equal p.d. (IR) across equal lengths.
Why does connecting a voltmeter across R₂ lower the output?
Voltmeter and R₂ are in parallel, so their combined resistance is lower and takes a smaller share of the supply.
How do you set the switching point of a sensor circuit?
Replace the fixed resistor with a variable resistor and adjust it until the output reaches the switching p.d. at the chosen temperature or light level.
Resistance of 20 m of copper wire (ρ = 1.7 × 10⁻⁸ Ω m) of cross-section 2.5 mm²?
R = 1.7 × 10⁻⁸ × 20 ÷ 2.5 × 10⁻⁶ = 0.14 Ω.
A wire has R = 9.0 Ω. Same metal, three times the length and three times the diameter: new R?
× 3 for length, ÷ 9 for area: 9.0 × 3 ÷ 9 = 3.0 Ω.
R–l graph gradient 5.2 Ω m⁻¹; wire diameter 0.30 mm. Resistivity?
A = π(0.15 × 10⁻³)² = 7.07 × 10⁻⁸ m²; ρ = 5.2 × 7.07 × 10⁻⁸ = 3.7 × 10⁻⁷ Ω m.
5.3 kΩ and 4.7 kΩ in series across 10.0 V. P.d. across the 4.7 kΩ?
10.0 × 4.7 ÷ 10.0 = 4.7 V.
R₁ = 1.5 kΩ on a 12 V supply. What R₂ gives V_out = 2.0 V across R₂?
R₁ takes 10 V: R₂/1.5 = 2.0/10 → R₂ = 0.30 kΩ (300 Ω).
3.0 V across a 1.50 m uniform wire. Separation of two points with 0.60 V between them?
Potential gradient 2.0 V m⁻¹; 0.60 ÷ 2.0 = 0.30 m.
Two 10 kΩ resistors on 6.0 V; a 40 kΩ voltmeter across one. Reading?
Parallel: 8.0 kΩ; 6.0 × 8.0 ÷ 18 = 2.7 V (not 3.0 V).
Diameter in A = πr²
Halve the diameter first. Using d as r makes A four times too big.
Converting mm² to m²
1 mm² = 10⁻⁶ m², not 10⁻³ m².
Potential divider: denominator
V_out = R₂V_in/(R₁ + R₂) — the TOTAL resistance, not R₁. Check the p.d.s add up to the supply.
Explaining a sensor circuit
Full chain: surroundings change → sensor resistance changes → its share of the p.d. changes → output changes.
Uncertainty in the diameter
Repeated readings: half the range. Single reading: half the micrometer resolution (0.005 mm).

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