Question

A close analogy exists between the flow of energy by heat because of a temperature difference (see Section 20.7$)$ 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: $$ \begin{array}{cc}{\text { Charge conduction }} & {\text { Thermal conduction }} \\ {\frac{d q}{d t}=\sigma A\left|\frac{d V}{d x}\right|} & {\frac{d Q}{d t}=k A\left|\frac{d T}{d x}\right|}\end{array} $$ In the analogous thermal conduction equation on the right, the rate of energy flow $d Q / d t$ (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 20.7$)$ 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:
$$
\begin{array}{cc}{\text { Charge conduction }} & {\text { Thermal conduction }} \\ {\frac{d q}{d t}=\sigma A\left|\frac{d V}{d x}\right|} & {\frac{d Q}{d t}=k A\left|\frac{d T}{d x}\right|}\end{array}
$$
In the analogous thermal conduction equation on the right, the rate of energy flow $d Q / d t$ (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.

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