Ohm's law from the drift picture
Ohm's law: for a conductor at constant temperature and physical state, the current is proportional to the potential difference: V = IR.
Why? From drift: E = V/l and vd = eEτ/m. Put these in I = n e A vd:
- I = n e A · (e τ / m) · (V / l)
- So V = (m l / n e² τ A) · I
- The bracket is fixed for a given wire at fixed temperature: this is R.
R = m l /(n e² τ A). Unit: ohm (Ω) = V/A.
Vector form: j = σE
Divide V = IR by l and use R = ρl/A: V/l = (I/A) ρ, so E = ρ j, or j = σE. This form works point by point inside a material and does not care about the shape of the wire.
Resistivity and conductivity
R depends on shape: R = ρ l / A. Double l → double R. Double A → half R.
Resistivity ρ = resistance of a wire of the material with unit length and unit area. From drift: ρ = m /(n e² τ). Unit: Ω m. It depends only on the material (n, τ) and temperature, not on size.
Conductivity σ = 1/ρ, unit S/m (siemens per metre) or Ω⁻¹m⁻¹.
- Metals (conductors): ρ ≈ 10⁻⁸ to 10⁻⁶ Ω m (copper 1.7 × 10⁻⁸)
- Semiconductors: ρ ≈ 10⁻⁵ to 10⁶ Ω m (silicon, germanium)
- Insulators: ρ ≈ 10¹¹ to 10¹⁹ Ω m (glass, rubber)
Colour code for carbon resistors (just remember the idea): the first two bands are digits, the third is a power of ten, the fourth is tolerance (gold 5%, silver 10%). A common memory aid for the digit order 0–9: black, brown, red, orange, yellow, green, blue, violet, grey, white.
Linear and non-linear V–I graphs (limits of Ohm's law)
Ohmic devices (metal wires, resistors at steady temperature) give a straight line through the origin. Slope of the I–V line = 1/R.
Non-ohmic behaviour appears in three ways:
- V stops being proportional to I: a filament bulb heats up, its R rises, and the graph bends towards the V axis.
- Current depends on the sign of V: a diode passes current easily one way and hardly at all the other way.
- More than one V for the same I: materials like gallium arsenide (GaAs) give a graph that rises, falls and rises again.
For such devices, we use the dynamic resistance ΔV/ΔI at a point instead of one R value.
How temperature changes resistivity
Metals: hotter ions vibrate more, collisions come faster, τ falls, so ρ = m/(ne²τ) rises. Over a moderate range:
ρT = ρ0 [1 + α (T − T0)], and the same for R: RT = R0[1 + α(T − T0)].
α is the temperature coefficient of resistivity, unit K⁻¹ (or °C⁻¹). Copper α ≈ 3.9 × 10⁻³ K⁻¹.
Alloys (nichrome, manganin, constantan): ρ is high and changes very little with T. So they are used in heaters and in standard resistors.
Semiconductors: heating frees more charge carriers, n rises fast, so ρ falls. α is negative.
Try it
In the 3D: pick copper, set T = 20 °C, note R. Predict R at 220 °C, then move the slider. Switch to silicon and repeat. Did R go up or down?
At home (with an adult): ask to see a pencil lead (graphite). A longer lead has higher resistance, a thick one lower. Compare a used-up short lead with a new long one using a phone torch circuit toy if you have one.
Board exam focus
Derive R = ml/(ne²τA) or ρ = m/(ne²τ) (3 marks); draw V–I graphs of ohmic and non-ohmic devices; numericals on stretching a wire (R ∝ l² at constant volume), colour codes and α.
Key formulas and definitions
- V = I R
- R = m l /(n e² τ A) = ρ l / A
- ρ = m /(n e² τ), σ = 1/ρ = n e² τ / m
- j = σ E
- ρT = ρ0 [1 + α (T − T0)]
- Stretched wire (same volume): R ∝ l²
Worked examples
1. A 2 m wire of area 1 × 10⁻⁶ m² has resistance 34 Ω. Find its resistivity.
Step 1: ρ = R A / l. Step 2: ρ = 34 × 1 × 10⁻⁶ / 2. Answer: ρ = 1.7 × 10⁻⁵ Ω m.
2. Find the resistance of 10 m of copper wire of area 0.5 mm². (ρ = 1.7 × 10⁻⁸ Ω m)
Step 1: A = 0.5 × 10⁻⁶ m². Step 2: R = ρl/A = 1.7 × 10⁻⁸ × 10 / 0.5 × 10⁻⁶. Answer: R = 0.34 Ω.
3. A wire of resistance 5 Ω is stretched to twice its length (volume unchanged). New resistance?
Step 1: l → 2l, and volume fixed means A → A/2. Step 2: R = ρl/A → ρ(2l)/(A/2) = 4ρl/A. Answer: 4 × 5 = 20 Ω.
4. A copper coil has R = 10 Ω at 20 °C. Find R at 120 °C. (α = 4 × 10⁻³ °C⁻¹)
Step 1: RT = R0[1 + α(T − T0)]. Step 2: = 10[1 + 4 × 10⁻³ × 100] = 10 × 1.4. Answer: 14 Ω.
5. A platinum thermometer reads 5.0 Ω at 0 °C and 5.8 Ω in hot water. α = 4 × 10⁻³ °C⁻¹. Temperature of the water?
Step 1: RT/R0 = 1 + αT → 5.8/5 = 1.16. Step 2: αT = 0.16. Step 3: T = 0.16 / 0.004 = 40 °C.
6. Find ρ for copper using n = 8.5 × 10²⁸ m⁻³, τ = 2.5 × 10⁻¹⁴ s, m = 9.1 × 10⁻³¹ kg.
Step 1: ρ = m/(ne²τ). Step 2: ne²τ = 8.5 × 10²⁸ × (1.6 × 10⁻¹⁹)² × 2.5 × 10⁻¹⁴ = 5.44 × 10⁻²³. Step 3: ρ = 9.1 × 10⁻³¹ / 5.44 × 10⁻²³ ≈ 1.7 × 10⁻⁸ Ω m. This matches the measured value for copper.
7. A carbon resistor has bands yellow, violet, red, gold. Find its value.
Step 1: yellow = 4, violet = 7 → 47. Step 2: red multiplier = 10². Step 3: 47 × 100 = 4700 Ω = 4.7 kΩ. Step 4: gold = ±5%.
Common mistakes
- Saying resistivity changes when the wire is cut or stretched. Only R changes; ρ is a property of the material.
- Forgetting that stretching also reduces the area: at constant volume R ∝ l², not ∝ l.
- Calling every device with V and I 'ohmic'. Bulbs, diodes and GaAs are non-ohmic.
- Using °C and K wrongly in α: a change of 1 °C equals a change of 1 K, so ΔT is the same in both.