Fourier’s law in one dimension — sign convention Relate the heat-flux density Q (W/m^2), thermal conductivity K (W/m·°C), and temperature gradient dT/dx (°C per metre) for one-dimensional steady conduction, using the proper sign convention.

Introduction / Context:
Fourier’s law is the cornerstone of heat conduction analysis. The sign convention indicates that heat flows from higher to lower temperature, opposite to the direction of increasing temperature.

Given Data / Assumptions:

• One-dimensional steady conduction along x.
• Q is defined positive in the +x direction.
• K is positive for physical materials.

Concept / Approach:
The physical statement “heat flows down the temperature gradient” is captured mathematically by a negative sign. If temperature decreases with x (dT/dx < 0), then Q should be positive in +x, which requires Q = -K dT/dx.

Step-by-Step Solution:
Write Fourier’s law: Q = - K * (dT/dx).Check sign with a linear profile: T(x) = T0 - ax with a > 0 ⇒ dT/dx = -a.Then Q = -K(-a) = K a > 0, i.e., heat flows toward +x where T is lower.

Verification / Alternative check:
Dimensional consistency: K has units W/m·°C, dT/dx has °C/m, product gives W/m^2, which matches Q.

Why Other Options Are Wrong:
Option (a) misses the sign; (c), (d), and (e) are algebraically or dimensionally incorrect as written.

Common Pitfalls:
Dropping the minus sign; mixing up heat rate (W) with heat flux (W/m^2); confusing K’s units.

Final Answer:
Q = - K (dT/dx)