A metal rod of cross sectional area 1 ·0 cm^2 is being heated at one end. At one time, the temp

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A metal rod of cross sectional area 1 ·0 cm^2 is being...

A metal rod of cross sectional area $1 \cdot 0 \mathrm{~cm}^{2}$ is being heated at one end. At one time, the temperature gradient is $5-0^{\circ} \mathrm{C} \mathrm{cm}^{-1}$ at cross section $A$ and is $2 \cdot 5^{\circ} \mathrm{Ccm}^{-1}$ at cross section $B$. Calculate the rate at which the temperature is increasing in the part $A B$ of the rod. The heat capacity of the part $A B=0-40 \mathrm{~J}^{\circ} \mathrm{C}^{-1}$, thermal conductivity of the material of the $\operatorname{rod}=200 \mathrm{~W} \mathrm{~m}^{-1}{ }^{-1} \mathrm{C}^{-1}$. Neglect any loss of heat to the atmosphere.

Key Concepts

Fourier's Law of Heat Conduction

Fourier's Law is a fundamental principle that states the rate of heat transfer through a material is proportional to the negative gradient of temperatures and to the area through which the heat flows. It is expressed as q = -kA (dT/dx), where k is the thermal conductivity, A is the cross-sectional area, and dT/dx is the spatial temperature gradient. This concept is essential in determining the amount of heat transmitted along a rod or any conductive material.

Temperature Gradient

The temperature gradient represents the rate at which temperature changes with respect to distance within a material. It is typically measured in degrees per unit length (e.g., °C/cm or °C/m). In heat conduction problems, differences in temperature gradients at different cross-sections drive the flow of heat from hotter to cooler regions, which is crucial in setting up the conduction equation.

Thermal Conductivity

Thermal conductivity is a material-specific property indicating how effectively a material can conduct heat. It is measured in units such as W m?¹ °C?¹. A higher thermal conductivity means the material can transfer heat more efficiently. In conduction problems, it appears as a proportionality constant and directly affects the magnitude of heat transfer through the material.

Heat Capacity

Heat capacity is a measure of the amount of heat required to change the temperature of an object by a certain amount. It is expressed in joules per degree Celsius (J/°C). In the context of a conduction problem, the heat capacity of a section of the rod determines how quickly its temperature rises in response to a given amount of heat being supplied or removed, as given by the relation dT/dt = Power/Heat Capacity.

Energy Conservation in Conduction

The principle of energy conservation in the context of heat conduction states that the net heat conducted into a section of a material will result in a change in the internal energy of that section. This change in internal energy is quantified by the heat capacity and the resulting change in temperature over time. By equating the net heat flux (derived from Fourier's Law) with the rate of change of thermal energy, one can determine the rate at which the temperature of the material increases or decreases.

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