Electrical heating (Joule effect): When a steady current I passes through a conductor of resistance R, the rate of heat generation (power) in the conductor equals:

Introduction / Context:Electrical resistance converts electrical energy into thermal energy, a phenomenon exploited in heaters and observed as I^2R losses in power systems. This problem asks for the correct expression for heat generation rate in a resistive element under steady current.

Given Data / Assumptions:

• Steady direct current I (no time-varying fields).
• Ohmic conductor with resistance R (temperature effects neglected).
• Power converted to heat equals electrical input power for a pure resistor.

Concept / Approach:From circuit theory, power P dissipated in a resistor can be written as P = V * I. Using Ohm's law, V = I * R, so P = I * (I R) = I^2 R. Equivalently, P = V^2 / R if voltage is known. The dimensional consistency and widespread engineering use of I^2 R solidify this as the correct form.

Step-by-Step Solution:Start with P = V * I (definition of electrical power).Use Ohm's law: V = I * R.Substitute: P = I * (I * R) = I^2 * R.Therefore, heat generation rate equals I^2 R.

Verification / Alternative check:Energy balance over time: Q = ∫ P dt = I^2 R * t for constant I, matching calorimetric measurements in resistive heating.

Why Other Options Are Wrong:I R and I R^2 have incorrect units for power and are inconsistent with Ohm's law.

I^2 R^2 overstates dependence on R and is dimensionally incorrect for power.

Common Pitfalls:

• Mixing up P = I^2 R and P = V^2 / R; both are correct, but the given data favor I^2 R.
• Ignoring temperature rise changing R; here R is treated as constant.

Final Answer:I^2 R