State Fourier's law for one-dimensional steady conduction in a rod.

• A. q = -k A (dT/dx)
• B. Q = h A (T_s - T∞)
• C. Q = k A ΔT / L
• D. P = F v

Answer: (A)

Heat flow in a rod in one dimension is driven by how temperature changes along the rod. Fourier's law says the rate of heat transfer per unit cross-sectional area is proportional to the negative of the temperature gradient: q'' = -k dT/dx, where k is the material’s thermal conductivity. To get the total heat transfer rate through the cross-section, multiply by the area A: Q = -k A dT/dx. If you see q = -k A dT/dx, that’s the same relation written for the total rate rather than the per-unit-area flux.

The negative sign is important: heat flows from regions of higher temperature to lower temperature, opposite the increasing temperature direction. This form is the fundamental description of 1D steady conduction in a rod. Other formulas shown are for different situations (convection, a particular simplifying case, or mechanics) and don’t capture the general conduction behavior in the rod.