Joule's Laws of Heating
Heat ∝ i², ∝ t, ∝ R.
Lesson at a glanceShow the key points
- H = i²Rt (33.1).
- H ∝ i², H ∝ t, H ∝ R: Joule's laws.
- Joule's calorimeter: resistor in oil; temperature rise measures heat.
- Appliance ratings: R = V²/P; in series the larger R takes more power.
Every toaster, kettle and fuse relies on one fact: current through a resistor heats it, by an amount Joule pinned down in three simple laws.
Joule’s laws of heating
With potential difference V across a resistor and current i through it, the field does work W = V(it) = i²Rt on the free electrons in time t. Collisions hand this energy to the lattice, raising the resistor’s thermal energy. We call it the “heat produced”, H (strictly it isn’t heat, since no temperature difference drives it, but the resistor warms and then passes heat to its surroundings).
Read as three laws:
- For a given resistor and time, H ∝ i².
- For a given resistor and current, H ∝ t.
- For a given current and time, H ∝ R.
An 18 V battery (no internal resistance) drives a 3 Ω resistor in series with a parallel pair of 12 Ω and 4 Ω. Find the heat produced in each resistor in one minute.
Verifying the laws
Joule’s calorimeter is a copper calorimeter of kerosene oil with an insulating lid, a stirrer and a thermometer, with a resistor coil dipped in the oil. A battery, rheostat, key and ammeter complete the circuit. The heat produced shows up as a temperature rise of the oil, so Δθ ∝ H.
H ∝ i²
Pass current i₁ for time t and note the rise Δθ₁. Let it cool, change the rheostat, pass i₂ for the same t: you find Δθ₁/Δθ₂ = i₁²/i₂².
H ∝ t
Keep a fixed current and read the thermometer at regular intervals: the temperature rises by equal steps in equal times.
H ∝ R
Put two calorimeters with equal oil and different coils R₁, R₂ in series (so the same current flows through both). The rises satisfy Δθ₁/Δθ₂ = R₁/R₂.
In a Joule calorimeter, 1.0 A for 5 minutes raises the oil by 3.0 °C. What rise would 2.0 A for the same time give?
Ratings and real batteries
A heater is rated 1000 W, 230 V. (a) Find its resistance. (b) What power does it take at 207 V? (c) A 60 W, 230 V bulb is put in series with it on 230 V. Find the power in each.
A 1.5 V dry cell connected straight across an ammeter (of negligible resistance) drives 25 A. How much heat is produced inside the cell in 10 s?
- 1H = i²Rt (33.1): H ∝ i², H ∝ t, H ∝ R (Joule's laws).
- 2Verified with Joule's calorimeter: oil temperature rise ∝ heat.
- 3P = V²/R for ratings; in series the bigger resistance takes more power.
Check yourself
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