Question

Which of the following Law's are based on gradient:
(A) Stefan-Boltzmann law
(B) Fourier's law
(C) Newton's law of cooling
(D) Fick's law

Choose the correct answer from the options given below:

• A. (A), (B) and (C) only.
• B. (B), (C) and (D) only.
• C. (A), (C) and (D) only.
• D. (A), (B) and (D) only.

Hint:

Ask which law depends on temperature to the fourth power instead of on a driving-force gradient or difference.

Answer 1

The Correct Option is B

Step 1: A gradient-based transport law is one that writes a flux as a constant times the spatial rate of change, or gradient, of some driving property, in the general form flux equals minus a coefficient times the gradient of the potential.
Step 2: Fourier's law of heat conduction is written as \(q = -k\dfrac{dT}{dx}\), heat flux tied directly to the temperature gradient across the material. This is a textbook gradient law, so (B) qualifies.
Step 3: Fick's law of diffusion is written as \(J = -D\dfrac{dC}{dx}\), mass flux tied to the concentration gradient. This is the mass-transfer counterpart of Fourier's law, so (D) qualifies too.
Step 4: Newton's law of cooling, \(q = hA(T_s - T_\infty)\), is usually written using a temperature difference across a thin boundary layer rather than a continuous derivative, but it belongs to the same family of linear rate laws, flux proportional to a driving potential, used for convective heat transfer, and it is grouped together with Fourier's and Fick's laws as a driving-force based rate law.
Step 5: The Stefan-Boltzmann law, \(E = \sigma T^4\), describes radiant heat emission and depends on the fourth power of absolute temperature, with no gradient or simple linear difference anywhere in it. It stands apart from the gradient-type laws entirely.
Step 6: So among the four, Fourier's law, Newton's law of cooling and Fick's law belong together as gradient or driving-force based rate laws, while the Stefan-Boltzmann law does not.