This question asks about Fourier’s Law, a fundamental principle in heat transfer that describes the rate of heat conduction through a material. Specifically, it focuses on how the amount of heat flow (denoted as Q) through a body in unit time is determined.

Fourier’s Law Explained

Fourier’s Law of Heat Conduction states that the rate of heat transfer through a material is proportional to the negative temperature gradient (i.e., the rate of change of temperature with distance) and the area of the material perpendicular to that gradient. The amount of heat flow (Q) per unit time is mathematically expressed as:

\(Q = -kA\frac{{dT}}{{dx}}\)

Where:

• Q represents the amount of heat transferred per unit time (e.g., Watts).
• k is the thermal conductivity of the material, a measure of its ability to conduct heat.
• A is the cross-sectional area through which the heat is flowing.
• \(\frac{{dT}}{{dx}}\) is the temperature gradient, representing the change in temperature (dT) over distance (dx).

The negative sign indicates that heat flows in the direction of decreasing temperature.

Analyzing the Options for Heat Flow (Q)

Let’s examine the given options in the context of Fourier’s Law:

• Option 1: \(kA\frac{{dT}}{{dx}}\)

  This expression matches the magnitude part of Fourier’s Law. It correctly includes thermal conductivity (k), area (A), and the temperature gradient (\(\frac{{dT}}{{dx}}\)). While the common formulation includes a negative sign, this option represents the correct relationship between the variables for heat flow rate.

• Option 2: \(kA\frac{{d{T^2}}}{{d{x^2}}}\)

  This option includes the second derivative of temperature with respect to distance (\(\frac{{d{T^2}}}{{d{x^2}}}\)), which is not part of the basic Fourier’s Law for heat flow rate.

• Option 3: \(k\frac{{dx}}{{dT}}\)

  This expression is the inverse of the temperature gradient and lacks the area term (A). It does not represent Fourier’s Law.

• Option 4: \(kA\frac{{dx}}{{dT}}\)

  Similar to option 3, this uses the inverse temperature gradient (\(\frac{{dx}}{{dT}}\)) instead of the correct temperature gradient (\(\frac{{dT}}{{dx}}\)).

• Option 5: (No expression provided)

  This option is incomplete.

Conclusion on Heat Flow Formula

Based on the analysis, the expression that correctly represents the amount of heat flow (Q) through a body in unit time according to Fourier’s Law, considering the provided options, is \(kA\frac{{dT}}{{dx}}\).