A close analogy exists between the flow of energy by heat...

A close analogy exists between the flow of energy by heat because of a temperature difference (see Section 19.6 ) and the flow of electric charge because of a potential difference. In a metal, energy $d Q$ and electrical charge $d q$ are both transported by free electrons. Consequently, a good electrical conductor is usually a good thermal conductor as well. Consider a thin conducting slab of thickness $d x,$ area $A$, and electrical conductivity $\sigma$, with a potential difference $d V$ between opposite faces. (a) Show that the current $I=d q / d t$ is given by the equation on the left: Charge conduction $$ \frac{d q}{d t}=\sigma A\left|\frac{d V}{d x}\right| $$ Thermal conduction $$ \frac{d Q}{d t}=k A\left|\frac{d T}{d x}\right| $$ In the analogous thermal conduction equation on the right (Eq. 19.17 ), the rate $d Q / d t$ of energy flow by heat (in SI units of joules per second) is due to a temperature gradient $d T / d x$ in a material of thermal conductivity $k$. (b) State analogous rules relating the direction of the electric current to the change in potential and relating the direction of energy flow to the change in temperature.

A close analogy exists between the flow of energy by heat because of a temperature difference (see Section 19.6 ) and the flow of electric charge because of a potential difference. In a metal, energy $d Q$ and electrical charge $d q$ are both transported by free electrons. Consequently, a good electrical conductor is usually a good thermal conductor as well. Consider a thin conducting slab of thickness $d x,$ area $A$, and electrical conductivity $\sigma$, with a potential difference $d V$ between opposite faces. (a) Show that the current $I=d q / d t$ is given by the equation on the left:
Charge conduction
$$
\frac{d q}{d t}=\sigma A\left|\frac{d V}{d x}\right|
$$
Thermal conduction
$$
\frac{d Q}{d t}=k A\left|\frac{d T}{d x}\right|
$$
In the analogous thermal conduction equation on the right (Eq. 19.17 ), the rate $d Q / d t$ of energy flow by heat (in SI units of joules per second) is due to a temperature gradient $d T / d x$ in a material of thermal conductivity $k$. (b) State analogous rules relating the direction of the electric current to the change in potential and relating the direction of energy flow to the change in temperature.

Key Concepts

Electric Conduction

Electric conduction describes the movement of electric charge, typically through free electrons in a conductor, in response to an electric field. The differential form of Ohm’s law relates current density to the electric field by the material's conductivity, with the electric field expressed as the spatial rate of change of the electric potential. This concept explains that charge flows from regions of higher potential to lower potential.

Thermal Conduction

Thermal conduction is the process by which thermal energy is transferred through a material due to a temperature gradient. Governed by Fourier’s law, this phenomenon states that the rate of heat flow through a material is proportional to both the temperature gradient and the material's thermal conductivity. In this context, heat flows from regions of higher temperature to lower temperature.

Analogy in Driving Forces

Both electric and thermal conduction rely on gradients of their respective potentials as driving forces. In electric conduction, the gradient of electric potential drives the movement of charges, resulting in an electric current, whereas in thermal conduction, the temperature gradient acts as the motivator for the flow of heat energy. This analogy highlights that in both cases, energy flows from regions of high potential (electrical or thermal) to regions of low potential.

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