Which expression represents Fourier's law in differential form for a rod?

• A. q = -k A (dT/dx)
• B. Q = k A ΔT / L
• C. dQ/dt = h A (T_s - T∞)
• D. Q = σ A T^4

Answer: (A)

Heat conduction in one dimension relates the flow of heat to the local temperature gradient. For a rod with cross-sectional area A, the rate at which heat passes through that cross-section is proportional to the negative temperature gradient: Q = -k A (dT/dx). The negative sign ensures heat moves from hotter to cooler regions. Here k is the material’s thermal conductivity, and dT/dx is the temperature change per unit length along the rod. The form q'' = -k dT/dx is the heat flux per unit area, and multiplying by A gives the total heat transfer rate through the cross-section. The other expressions describe different situations: a simple linear drop along the rod (lumped form), convective heat transfer to a fluid, or radiation from a surface. So the appropriate Fourier’s law form for a rod is Q = -k A dT/dx.