A cubical box of volume 216 cm^3 is made up of 0.1 cm thick wood. The inside is heated electri

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A cubical box of volume 216 cm^3 is made up of 0.1 cm...

A cubical box of volume $216 \mathrm{~cm}^{3}$ is made up of $0.1 \mathrm{~cm}$ thick wood. The inside is heated electrically by a $100 \mathrm{~W}$ heater. It is found that the temperature difference between the inside and the outside surface is $5^{\circ} \mathrm{C}$ in steady state. Assuming that the entire electrical energy spent appears as heat, find the thermal conductivity of the material of the box.

Key Concepts

Thermal Conductivity

Thermal conductivity is a material property that quantifies the ability of a substance to conduct heat. It determines how effectively thermal energy is transmitted through the material due to a temperature gradient. High thermal conductivity implies that the material is a good conductor of heat, while low thermal conductivity indicates that the material acts as an insulator.

Fourier's Law of Heat Conduction

Fourier's law is the fundamental principle governing steady-state heat conduction. It states that the rate of heat transfer through a material is proportional to the negative gradient of temperature and the area through which the heat flows, and is directly proportional to the material's thermal conductivity. The law is mathematically represented by Q = kA(?T/?x), linking the heat energy transfer rate to the material properties, temperature difference, and physical dimensions.

Steady-State Conduction

Steady-state conduction refers to a condition where the temperature distribution within a material does not change over time, meaning that the heat entering any region of the material is equal to the heat leaving it. This allows for a straightforward analysis of heat flow because it implies that the energy generation and losses are balanced, which is key to applying Fourier's law accurately.

Geometric Considerations in Conduction

In heat conduction problems, the geometry of the material or system plays a critical role in determining the rate of heat transfer. Parameters such as the surface area through which heat is conducted and the thickness or length of the material along the direction of heat flow are essential. These geometric factors directly influence the overall heat transfer rate, as seen in the conduction equation where the area and distance variables are included in the relationship.

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