Ohm's law, resistance and resistivity
At constant temperature, current is directly proportional to potential difference: V = IR. Resistance depends on length, area and material: R = ρL/A.
Ohm's law
The German physicist Georg Simon Ohm found this law in 1827. Ohm's law states: at a constant temperature, the steady current (I) through a conductor is directly proportional to the potential difference (V) across its ends. So V ∝ I, or V = IR. The constant R is the resistance of the conductor: the opposition it offers to the flow of current. Its SI unit is the ohm (Ω). 1 ohm is the resistance of a conductor that carries 1 A when 1 V is applied across its ends.
How to draw the Ohm's law circuit and graph
- 1Draw a battery, a plug key, an ammeter, a rheostat and the resistor (the wire under test) one after another in a single loop.
- 2Draw a voltmeter across the two ends of the resistor (in parallel).
- 3Label: battery, key (K), ammeter (A), rheostat (Rh), resistor (R), voltmeter (V). Mark the + and − terminals of the meters.
- 4For the graph, take current (I) on the x-axis and potential difference (V) on the y-axis.
- 5★Plot the readings and join them. You get a straight line through the origin. Its slope (V/I) is the resistance R.
★ key step
Resistance depends on four things
| Factor | Effect on resistance | Why |
|---|---|---|
| Length (L) | R ∝ L: a longer wire has more resistance | Electrons hit more atoms on a longer path |
| Area of cross-section (A) | R ∝ 1/A: a thicker wire has less resistance | A wider path lets more electrons pass at once |
| Nature of material | Each material has its own resistivity (ρ) | Silver and copper have low ρ; nichrome has high ρ |
| Temperature | Resistance of a metal rises as it heats up | Hot atoms vibrate more and block electrons more |
Resistivity, conductance and conductivity
Putting length and area together, R ∝ L/A, so R = ρL/A. Here ρ (rho) is the resistivity of the material. Resistivity is the resistance of a conductor of unit length and unit area of cross-section. Its unit is the ohm metre (Ω m). It depends only on the material and its temperature, not on the shape of the wire. Metals like copper have very low resistivity, about 10⁻⁸ Ω m. Rubber and glass have very high resistivity, so they are insulators (materials that do not let charge flow). Conductance (G) is the reciprocal of resistance: G = 1/R. Its unit is ohm⁻¹, also written as mho. Conductivity (σ) is the reciprocal of resistivity: σ = 1/ρ. Its unit is ohm⁻¹ metre⁻¹ (Ω⁻¹ m⁻¹), also written as mho m⁻¹.
Worked example: resistance, resistivity and conductivity
(a) A torch bulb draws a current of 0.3 A from a 3 V battery. Find its resistance. (b) A nichrome wire is 2 m long with an area of cross-section of 2 × 10⁻⁷ m². Its resistance is 11 Ω. Find its resistivity, conductance and conductivity.
- (a) Given V = 3 V, I = 0.3 A. Formula: R = V/I.
- R = 3 V ÷ 0.3 A = 10 Ω.
- (b) Given L = 2 m, A = 2 × 10⁻⁷ m², R = 11 Ω. Formula: ρ = RA/L.
- ρ = (11 Ω × 2 × 10⁻⁷ m²) ÷ 2 m = 1.1 × 10⁻⁶ Ω m.
- Conductance G = 1/R = 1 ÷ 11 Ω ≈ 0.091 Ω⁻¹ (rounded to 3 decimal places).
- Conductivity σ = 1/ρ = 1 ÷ (1.1 × 10⁻⁶ Ω m) ≈ 9.09 × 10⁵ Ω⁻¹ m⁻¹ (rounded to 3 figures).
(a) 10 Ω (b) ρ = 1.1 × 10⁻⁶ Ω m; G ≈ 0.091 Ω⁻¹; σ ≈ 9.09 × 10⁵ Ω⁻¹ m⁻¹
House wiring uses copper, which has very low resistivity, so little energy is lost in the walls. The coil inside an iron box uses nichrome, with high resistivity, so it gets hot.
2 marks: “State Ohm's law.” At constant temperature, the steady current through a conductor is directly proportional to the potential difference across its ends: V = IR.
4 marks: “Define resistivity. A copper wire 10 m long has an area of cross-section of 1 × 10⁻⁶ m² and a resistance of 0.17 Ω. Find its resistivity.” Resistance of a conductor of unit length and unit area. ρ = RA/L = 0.17 × 1 × 10⁻⁶ ÷ 10 = 1.7 × 10⁻⁸ Ω m.
7 marks: “State Ohm's law. Describe an experiment to verify it, with a circuit diagram and a V-I graph.” Law and V = IR; battery, key, ammeter, rheostat and resistor in series, voltmeter in parallel; vary current, record V and I; V/I constant; straight-line graph through the origin.
Resistance grows with Length and falls with Area: R = ρL/A. Long and thin means high resistance.