Discussion
The free ends of an electric circuit comprising a cell and switch are connected one by one to wires A, B, C and D. Which of these wires would produce least amount of heat ?
| Wire | length (cm) | Area of Cross-section (m2) |
|---|---|---|
| A | 10 | 10 × 10-8 |
| B | 10 | 8 × 10-8 |
| C | 12 | 6 × 10-8 |
| D | 5 | 10 × 10-8 |
Answer: A
The least heat is produced by wire C.
With the same cell, the voltage V is fixed, so the heat produced in a given time is \(H = \frac{V^2 t}{R}\): the higher the resistance, the less the heat. Resistance is \(R = \rho \frac{l}{A}\), so for wires of the same material it is proportional to length divided by area.
Taking l/A in units of cm per \(10^{-8}\ \text{m}^2\):
\[ \begin{aligned} A &: 10/10 = 1 \\ B &: 10/8 = 1.25 \\ C &: 12/6 = 2 \\ D &: 5/10 = 0.5 \end{aligned} \]
C has the highest resistance and so draws the least current and produces the least heat. D, the shortest and thickest, produces the most. Choosing D comes from using \(H = I^2Rt\) as if the current were the same in every wire, which is not so when each wire is connected to the same cell.
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