= Electrical calorimetry of a resistor
{title2=$E_{\rm el}=\int VI\,dt$}
An electrical <calorimetry> experiment asks whether a measured electrical <work> input accounts for a measured warming. The University of Iowa's demonstration uses an insulated flask containing <water>, an immersed heater, a <thermometer>, electrical meters and a timer. Its procedure compares the duration, electrical readings and <temperature> rise. The following low-<power> measurement design and synthetic readout develop that idea; they are not measurements reported by the demonstration. <5F15.10 - Electrical Equivalent of Heat by University of Iowa Department of Physics and Astronomy>[Equipment and Procedure: insulated flask, immersed heater, thermometry, electrical meters and timed warming]{ref}
Use a covered <calorimeter> with weighed <water>, an immersed <resistor>, a calibrated <thermometer> and gentle stirring. Put an <ammeter> in series with the heater branch and a high-input-<electrical resistance> <voltmeter> across the immersed <resistor>'s terminals. Record $V(t)$, $I(t)$, <temperature> $T(t)$ and room <temperature> $T_a(t)$ before, during and after switching the heater on. Measure the <electric potential difference> at the heater, rather than at the supply: lead and contact <electrical resistance> outside the receiving boundary can consume <power> that never reaches the <water>. Account for <voltmeter loading> if the <ammeter> also includes the sensing branch's <electric current>.<University Physics Volume 2 by Samuel J. Ling and others/10.4 Electrical Measuring Instruments>[Measuring Current with an Ammeter and Measuring Voltage with a Voltmeter, Figures 10.35–10.36: series current measurement, parallel voltage measurement and instrument resistance]{ref}
The electrical readout gives operating <electrical resistance> $R(t)=V(t)/I(t)$ when $I(t)\ne0$, and <power> $P(t)=V(t)I(t)$. To test <Ohm's law>, obtain several pairs at the same <temperature>; one ratio alone cannot establish a linear response. For small <electrical resistance>, use <four-terminal sensing> to exclude lead drops and reverse <electric current> to test stable offsets from the <thermoelectric effect>. Use brief low-<power> resistance measurements if the intended result is the unheated value: the test itself causes <Joule heating>.
For a covered assembly at approximately constant <pressure>, with negligible evaporation, negligible endpoint mechanical <energy> changes and no <phase transition>, take
$$
C_{\rm eff}=m_wc_w+C_{\rm cal},\qquad
E_{\rm el}=\int_{t_0}^{t_1}VI\,dt
=C_{\rm eff}(T_1-T_0)+Q_{\rm loss,out}-W_{\rm stir,on}.
$$
Here $c_w$ is the <water>'s <specific heat capacity at constant pressure>, $C_{\rm cal}$ is the participating vessel, heater and sensor <heat capacity>, $Q_{\rm loss,out}$ is signed outward <heat> leakage, and $W_{\rm stir,on}$ is stirring <work> entering the assembly. The storage term approximates its <enthalpy> change; for a rigid fixed-<volume (thermodynamics)> receiver use the corresponding <heat capacity at constant volume>. Calibrate $C_{\rm eff}$ independently if the purpose is to test electrical input against thermal storage. Using the same heating run to define $C_{\rm eff}=E_{\rm el}/\Delta T$ and then claiming it verifies the equality would be circular.
\Image[/joule-heating-electrical-calorimetry-readout.png]
{title=An immersed <resistor> connects electrical readings to a corrected <calorimeter> warming}
{id=joule-heating-electrical-calorimetry-readout}
{description=Original apparatus schematic and synthetic <temperature> trace, not Joule's original apparatus or a measured dataset. Separate sensing leads indicate the heater-terminal <electric potential difference>; the <ammeter> is in the heater branch. Shading marks $120\,\mathrm s$ of input at $5\,\mathrm W$. The dashed trace suppresses room <heat> leakage but keeps the same <heat capacity>.}
For the synthetic trace, stipulate $V=5\,\mathrm V$, $I=1\,\mathrm A$, $R=5\,\Omega$, $C_{\rm eff}=500\,\mathrm{J/K}$, room <temperature> $20\,{}^\circ\mathrm C$, and loss coefficient $G=0.5\,\mathrm{W/K}$, with negligible stirring input. The heater runs for $120\,\mathrm s$, beginning at room <temperature>. The <calorimeter heat-loss correction> model is
$$
C_{\rm eff}\frac{dT}{dt}=VI-G(T-T_a),\qquad
\Delta T=\frac{VI}{G}\left(1-e^{-G\Delta t/C_{\rm eff}}\right).
$$
It predicts $E_{\rm el}=600\,\mathrm J$, $\Delta T\approx1.13080\,\mathrm K$, storage $565.398\,\mathrm J$ and outward loss $34.602\,\mathrm J$. Neglecting leakage undercounts deposited <energy> by about $5.77\%$; assuming no loss predicts $1.20\,\mathrm K$. After switching off, the excess <temperature> undergoes <exponential decay> with <time constant> $C_{\rm eff}/G=1000\,\mathrm s$. A cooling slope can estimate $G$ only with known $C_{\rm eff}$ and a credible single-<temperature> model.
Controls separate causes of a <temperature> rise. Check the electrical meters' zeros and <thermometer calibration>; keep <water> <mass>, immersion, lid and stirring fixed between runs. Record a heater-off stirred baseline to estimate sensor drift and stirring input. Repeat cooling runs over the same <temperature> range to estimate environmental <heat transfer>; a baseline taken only at room <temperature> cannot measure warming-dependent leakage. Vary <electric current> while keeping $R$ and initial thermal conditions comparable: corrected deposited <energy> should scale as $I^2$ for equal durations. An independent <calorimeter calibration> or known reference <heat capacity> makes this a quantitative test rather than a qualitative observation.
Finite sensor response, incomplete <thermal equilibrium>, changing $R(T)$ or <heat capacity>, evaporation, and imperfect loss corrections limit the inference. A final <temperature> alone cannot distinguish smaller electrical input from larger <heat> leakage or a larger participating <heat capacity>. Agreement tests the stated assembly and its <energy> accounting; it does not show that all electrical <work> in arbitrary devices becomes <heat>.
Historically, the Royal Society's archive description of <James Prescott Joule>'s manuscript dated 13 October 1840 records a coil of wire in a jar of <water>, a <thermometer> and a <galvanometer>, and summarizes his inference that warming depends on <electrical resistance> and the square of <electric current> intensity. This electrical experiment precedes <Joule's paddle-wheel experiment>; the input here is electrical <work>, whereas the paddle supplies mechanical <work>. The catalogue, rather than the inaccessible manuscript images, supports this account. <Archive catalogue record AP-23-34: On the production of heat by voltaic electricity by The Royal Society Archives>[Royal Society Archives, AP/23/34, Description and Date: apparatus, resistance dependence and duplicate-ratio conclusion; manuscript dated 13 October 1840]{ref}
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