Electric potential difference → Electrical work

Matrix context

D = disequilibrium row · C = coupling family column. A cell reads the state of the atlas for that pair.

D.06 × C.01 · Electric potential difference × Thermoelectricfamily: ThermoelectricD.06 × C.02 · Electric potential difference × Thermionicfamily: Thermionic

Typed bounds on this route

Every limit the atlas holds for this route, as constraint records: reached through bounded_by claims on the route's own entities. Hard bounds (an upper bound or a formula) decide the thermodynamic-bound check above; benchmarks, constitutive relations and resource limits are listed and never decisive. No prose limit exists anywhere (pass 42).

• Carnot limit · formula-bound · hard, applies to this route
  η ≤ 1 − T_c/T_h for any heat engine between reservoirs at T_h and T_c
• Space-charge limit · formula-bound · hard, applies to this route
  J ≤ (4ε₀/9)√(2e/m) V^{3/2}/d² (Child–Langmuir)

Steps

1. 01
   Electric potential difference —drives→ Ettingshausen effectestablishedclaim record

   electrical → thermal

   • a current in a conductor with a perpendicular magnetic field

   solid · conductor required · magnetic field required

   refs [1] [2]
2. 02
   Ettingshausen effect —produces→ Temperature gradientestablishedclaim record

   thermal → thermal

   refs [2] [3]
3. 03
   Temperature gradient —drives→ Thermionic emissionestablishedclaim record

   thermal → electrical · loss to radiative

   • cathode hot enough that kT is a useful fraction of the work function (typically > 1500 K for metals, lower for low-work-function coatings)
   • a cooler anode across a vacuum or caesium-vapour gap

   vacuum required · above ~700 K · solid

   refs [4] [5]
4. 04
   Thermionic emission —produces→ Charge carriersestablishedclaim record

   electrical → electrical · loss to thermal

   • emitted electrons collected at the anode at a potential difference set by the work-function difference

   refs [5]
5. 05
   Charge carriers —converts into→ Electrical workestablishedclaim record

   electrical → electrical · loss to thermal

   • carriers driven through an external load at the potential difference the source maintains

   refs [6]

Evidence for the complete composition · 0

No source on record for this exact composition. The sources below support the individual steps, not the route as a whole.

Evidence for the constituent relations · 6

1. [1]
   A. von Ettingshausen, W. Nernst. Ueber das Auftreten electromotorischer Kräfte in Metallplatten, welche von einem Wärmestrome durchflossen werden und sich im magnetischen Felde befinden (1886)

   Annalen der Physik 265, 343–347 · doi:10.1002/andp.18862651010 · crossref ✓ 2026-09-19 · record
2. [2]
   H. B. Callen. The application of Onsager's reciprocal relations to thermoelectric, thermomagnetic, and galvanomagnetic effects (1948)

   Physical Review 73, 1349 · doi:10.1103/PhysRev.73.1349 · crossref ✓ 2026-09-19 · record
3. [3]
   H. J. Goldsmid. Introduction to Thermoelectricity (2nd ed.) (2016)

   Springer Series in Materials Science 121 · doi:10.1007/978-3-662-49256-7 · crossref ✓ 2026-09-19 · record
4. [4]
   O. W. Richardson. On the negative radiation from hot platinum (1901)

   Proceedings of the Cambridge Philosophical Society 11, 286–295 · paper, no DOI · record
5. [5]
   G. N. Hatsopoulos, E. P. Gyftopoulos. Thermionic Energy Conversion, Vol. 1: Processes and Devices (1973)

   MIT Press · book, no DOI · record
6. [6]
   N. W. Ashcroft, N. D. Mermin. Solid State Physics (1976)

   Holt, Rinehart and Winston · book, no DOI · record