4.4: Energy Dissipation

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Let me conclude this brief chapter with an ultra-short discussion of energy dissipation in conductors. In contrast to the electrostatics situations in insulators (vacuum or dielectrics), at dc conduction, the electrostatic energy \(\ U\) is “dissipated” (i.e. transferred to heat) at a certain rate \(\ \mathscr{P} \equiv-dU/dt\), with the dimensionality of power.¹⁷ This so-called energy dissipation may be evaluated by calculating the power of the electric field’s work on a single moving charge:

\[\ \mathscr{P}_{1}=\mathbf{F} \cdot \mathbf{v}=q \mathbf{E} \cdot \mathbf{v}.\tag{4.37}\]

After the summation over all charges, Eq. (37) gives us the average dissipation power. If the charge density \(\ n\) is uniform, multiplying by it both parts of this relation, and taking into account that \(\ q n \mathbf{v}=\mathbf{j}\), for the energy dissipation in a unit volume we get the differential form of the Joule law¹⁸

\[\ \text{General Joule law}\quad\quad\quad\quad\mathscr{I} \equiv \frac{\mathscr{P}}{V}=\frac{\mathscr{P}_{1} N}{V}=\mathscr{P}_{1} n=q \mathbf{E} \cdot \mathbf{v} n=\mathbf{E} \cdot \mathbf{j}.\tag{4.38}\]

In the case of the Ohmic conductivity (8), this expression may be also rewritten in two other forms:

\[\ \text{Joule law for Ohmic conductivity}\quad\quad\quad\quad\mathscr{I}=\sigma E^{2}=\frac{j^{2}}{\sigma}.\tag{4.39}\]

With our electrostatics background, it is also straightforward (and hence left for the reader’s exercise) to prove that the dc current distribution in a uniform Ohmic conductor, at a fixed voltage distribution along its borders, corresponds to the minimum of the total dissipation in the sample,

\[\ \mathscr{P} \equiv \int_{V} \mathscr{I} d^{3} r=\sigma \int_{V} E^{2} d^{3} r.\tag{4.40}\]

Reference

¹⁷ Since this electric field and hence the electrostatic energy are time-independent, this means that the energy is replenished at the same rate from the current source(s).

¹⁸ Named after James Prescott Joule, who quantified this effect in 1841.