Ohm's law, resistance and resistivity
At constant temperature, V = Ri: current is proportional to potential difference. Resistance grows with length and falls with thickness: R = ρl/A, with ρ in ohm metre.
Ohm's law and resistance
Georg Simon Ohm found that the current through a metal wire is directly proportional to the potential difference across its ends. This holds provided its temperature stays the same. So V ∝ i, or V = Ri. The constant R is the resistance of the wire. Resistance is the property of a conductor that opposes the flow of current. Its SI unit is the ohm (Ω). A wire has a resistance of 1 Ω if a potential difference of 1 V drives a current of 1 A through it. A wire made to have a chosen resistance is a resistor. Resistance is unwanted in power lines, where it wastes energy as heat. It is useful in bulbs and heaters, where we want light and heat.
Activity (practical): V-I graph for a resistor
You need fresh 1.5 V cells, a resistor or a piece of wire, an ammeter, a voltmeter, a plug key, connecting wires and sandpaper.
- 1Note the range and least count (smallest reading) of the ammeter and the voltmeter.
- 2Join one 1.5 V cell, a plug key, the ammeter and the resistor (or wire) in series. Join the voltmeter across the resistor. Keep the key open while you connect.
- 3Clean the wire ends with sandpaper. Check that both meters read zero with the key open.
- 4Close the key. Record the current i and the potential difference V.
- 5Repeat with two cells and then three cells in series. Record i and V each time, and work out V/i.
- 6Plot the graph and find R.V/i is nearly the same every time, and the graph is a straight line through the origin. This verifies Ohm's law. Take out the key between readings so the wire does not heat up.
How to draw the circuit and the V-I graph
- Circuit: draw the cells in a row with + and − marked. Join a plug key, an ammeter (A in a circle) and a zigzag resistor R in one loop.
- Draw the voltmeter (V in a circle) across R only, in parallel. Mark the + terminal of each meter towards the + end of the battery.
- Graph: your NIOS lab manual puts V on the x-axis and i on the y-axis. Plot the points and draw one straight line through the origin.
- With i on the y-axis, the slope is i/V, so R = 1 ÷ slope. If V is on the y-axis instead, the slope itself is R.
- So in an i-V graph, the steeper line belongs to the smaller resistance.
What resistance depends on: resistivity
- Length (l): a longer wire has more resistance: R ∝ l. Double the length, double the resistance.
- Area of cross-section (A): a thicker wire has less resistance: R ∝ 1/A. Double the area, half the resistance.
- Material: copper has less resistance than iron of the same length and thickness. Nichrome has much more.
- Temperature: the resistance of a metal wire increases as it gets hotter.
- Putting length and area together: R = ρl/A. Here ρ (rho) is the resistivity of the material.
- Resistivity is the resistance of a piece 1 m long and 1 m² in cross-section. Its SI unit is the ohm metre (Ω m). It depends on the material and temperature, not on the size of the wire.
Using V = Ri
(a) An electric kettle draws a current of 10 A from the 230 V mains. Find its resistance. (b) What current flows through a 30 Ω resistor when the potential difference across it is 12 V?
- (a) Given: V = 230 V (the mains in India; your NIOS lesson and older books say 220 V), i = 10 A. Formula: R = V/i.
- R = 230 V ÷ 10 A = 23 Ω.
- (b) Given: R = 30 Ω, V = 12 V. Formula: i = V/R.
- i = 12 V ÷ 30 Ω = 0.4 A.
(a) 23 Ω; (b) 0.4 A
Using resistivity
A copper wire is 100 m long and has a cross-section of 2 × 10⁻⁶ m². The resistivity of copper is 1.7 × 10⁻⁸ Ω m. Find its resistance. What would it be if the wire were twice as long?
- Given: l = 100 m, A = 2 × 10⁻⁶ m², ρ = 1.7 × 10⁻⁸ Ω m. Formula: R = ρl/A.
- R = (1.7 × 10⁻⁸ Ω m × 100 m) ÷ (2 × 10⁻⁶ m²) = (1.7 × 10⁻⁶) ÷ (2 × 10⁻⁶) Ω.
- R = 0.85 Ω.
- R ∝ l, so a wire twice as long has R = 2 × 0.85 Ω = 1.7 Ω.
0.85 Ω; 1.7 Ω for twice the length
House wiring uses thick copper wire, so little energy is lost as heat. The coil of an electric iron is made of nichrome, which has a high resistivity. Old fan regulators slowed the fan by adding resistance to the circuit.
1 marks: “In R = ρl/A, what does ρ stand for, and what is its SI unit?” ρ is the resistivity of the material. Its SI unit is the ohm metre (Ω m).
2 marks: “A wire is replaced by one of the same metal that is twice as long and has twice the area of cross-section. How does the resistance change? Give a reason.” It stays the same. R = ρl/A: doubling l doubles R, and doubling A halves it, so the two changes cancel.
3 marks: “State Ohm's law. Describe how you would draw a V-I graph for a resistor and find its resistance from it.” At constant temperature, current is proportional to p.d. (V = Ri). Join cells, key, ammeter and resistor in series, voltmeter across the resistor. Take readings for 1, 2 and 3 cells. Plot the points: a straight line through the origin. R = V/i (or 1 ÷ slope of the i-V graph).
Long and thin: high resistance. Short and thick: low resistance. In R = ρl/A, length sits on top and area below.