When the same amount of current, I is passed through different wires under the same conditions, the rates of production of heat are

Analyze the question: The same current I flows through different wires. We want to know how the rate of heat production (which is power) depends on the wire.

Concept: Joule's law of heating says that the rate of heat production is electrical power:

P = I²R

Solution:

1. The current I is the same for every wire, so I² is a constant.
2. That gives P = (constant) × R.
3. So P ∝ R. The rate of heat production is directly proportional to the resistance of each wire.

Checking the options:

• A. P is proportional to I² (not I), and I is fixed anyway. Wrong.
• B. P ∝ R when I is constant. Correct.
• C. P is not proportional to R². Wrong.
• D. P increases with I, so it is not inversely proportional. Wrong.
• E. The inverse relationship P = V²/R holds only when the voltage is constant, which is not the case here. Wrong.

Tip: Look at which quantity is held constant.

• Constant current (series circuit): use P = I²R, so P ∝ R.
• Constant voltage (parallel circuit): use P = V²/R, so P ∝ 1/R.

This distinction is a favourite in JAMB and NECO.