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The measuring electric current crosses the layer planes. This geometry motivates the serial-layer paths in the two-current resistor model. A quantitative material calculation can also require interface electrical resistances, spin relaxation and spin accumulation; the circuit below intentionally isolates a smaller set of assumptions.

The motivating observation is that a chemical reaction can produce a signed electrical reading even when its reacting materials occupy separate vessels. UC San Diego's teaching demonstration places a copper strip in $0.1\mathrm{mol}{\mathrm{L}}^{-1}$ $\mathrm{CuS}{\mathrm{O}}_4$ and a zinc strip in $0.1\mathrm{mol}{\mathrm{L}}^{-1}$ $\mathrm{ZnS}{\mathrm{O}}_4$, using two $400\mathrm{mL}$ beakers with about $300\mathrm{mL}$ of chemical solution in each. The strips connect to a voltage meter. Initially there is no ionic connection; the demonstration specifies a zero reading. A filter-paper salt bridge containing aqueous $\mathrm{N}{\mathrm{a}}_2\mathrm{S}{\mathrm{O}}_4$ then connects the chemical solutions. With the meter's positive lead on copper and negative lead on zinc, its reported readings are about $+1.1\mathrm{V}$; exchanging the leads reverses the reading. These are observations reported by the university, not data generated for this article. See UC San Diego, demonstration 9B: apparatus, bridge control and meter polarity.

The controls answer different questions. Exchanging only the meter leads checks the sign convention; it does not reverse the chemical reaction or swap the anode and cathode. Interrupting only the salt bridge tests the need for an ionic return path. Its absence prevents sustained cell electric current, but a very high-input-resistance meter may still show a transient or unstable reading while interfaces store electric charge. The zero reading specified for the university's apparatus is therefore not a universal theorem for every meter. Reconnecting the same salt bridge distinguishes a reversible interruption from a disconnected wire or exhausted chemical solution. This interpretation follows from electric charge conservation in a model allowing interfacial electric charge storage; see IUPAC Gold Book: integral capacitance of an electrode for the capacitance associated with electrode charge.

To distinguish terminal electric potential difference from delivered power, compare the nearly open-circuit reading with readings across a known resistor load. In the nearly open-circuit condition, a meter with input electrical resistance much larger than the cell's effective internal electrical resistance draws little electric current. With a load, measure its terminal electric potential difference $V$ and infer $I=V/R$ using Ohm's law; track the readings over time and repeat the nearly open-circuit reading afterwards. Keep the temperature, chemical solution compositions, immersed electrode areas, bridge geometry and electrical contacts comparable. This is a model-based extension of the documented demonstration, not a claim that UC San Diego published a load sweep. A finite-current drop can reflect ionic electrical resistance, changes of interfacial reaction rates, or depletion near an electrode; a single voltage reading cannot identify which dominates. The interfacial and composition contributions are distinguished in IUPAC Gold Book: overpotential and ^[ref].

For a deliberately simplified calculation, model the cell by a constant electromotive force $\mathcal{E}$ in series with a constant internal electrical resistance $r$, and a passive load $R>0$. Take $I$ positive from the cathode through the load to the anode. ThenThe nearly open-circuit limit $R\to \infty$ gives $V\to \mathcal{E}$ but $I\to 0$ and $P_{\mathrm{load}}\to 0$. A straight $V$-versus-$I$ line is a prediction of constant $r$, not a general law for electrochemistry. See Ling, Sanny and Moebs, University Physics II, §10.1: internal resistance, terminal voltage and load.

For the chosen values, $I=5.50\mathrm{mA}$ and $V=0.550\mathrm{V}$. Holding these toy parameters fixed for $600\mathrm{s}$ gives transported electric charge $Q=It=3.30\mathrm{C}$ and an extent of reaction $\xi =Q/(2F)\simeq 17.1\mu \mathrm{mol}$, assuming all the electric current serves the two specified half-reactions. The same $\xi$ counts zinc consumed and copper deposited, rather than equal masses. The energy budget isThe load's electrical work also becomes Joule heating when the load is a resistor. Here $\mathcal{E}Q$ accounts for the reaction's available reversible electrical work; a full cell thermal budget would also require its thermodynamic entropy contribution. Real measurements need uncertainty estimates, a specified meter input electrical resistance and calibration, and allowance for changing composition and unwanted chemical reactions. The toy value $r=100\Omega$ is not a measured property of the university's cell.

For a capacitor represented by ideal capacitance $C$ and series electrical resistance $R_s$, the complex impedance is $R_s-i/(\omega C)$. The ratio of its dissipative part to the magnitude of its reactive part is $\tan \delta =\omega C{R}_s$. This dimensionless ratio characterizes loss at a specified frequency; equivalent parallel parameters generally differ from the series parameters.

The New Jersey Institute of Technology laboratory uses a silicon 1N4148 diode. For its forward current-voltage characteristic, the supply operates in current-source mode; a benchtop voltmeter independently reads the diode electric potential difference. Small electric current steps resolve the steep branch. For reverse measurements, a sensitive ammeter sits in series; the handout suggests a $10$–$100\mathrm{k\Omega }$ series resistor. Reversing the applied polarity supplies the comparison. Meter placement must avoid counting the voltmeter's own electric current, and the zero reading may have an offset. The same diode is then measured with an Agilent U2722A source measure unit for graphical comparison.^[ref]

The prescribed analysis uses a logarithmic scale for forward electric current to infer $I_S$ and $n$ from the straight region. Large-electric current departures can indicate series electrical resistance. The handout cautions that the accessible range may not determine it precisely; meter loading also matters. This is a measurement procedure, without a published set of device readings.^[ref]

The reason for the logarithmic scale follows directly from the Shockley diode equation. If $I_D\gg I_S$ and series electrical resistance is negligible,where $I_0>0$ is a fixed reference electric current making the logarithm dimensionless. The slope is $1/(n{V}_T)$, so extracting $n$ needs the junction temperature. Fitting only an appropriate interval matters: a straight-looking subset cannot establish the model at all electric currents.

A direct current is an electric current that flows in one direction; its magnitude may vary with time. A constant or steady electric current additionally has a value independent of time. Thus a direct current need not be constant. The steady direct current model of a voltage divider assumes a constant input electric potential difference and fixed electrical resistances.

An eddy current is a circulating electric current induced within an electrical conductor by changing magnetic flux, as described by Faraday's law of induction. Its magnetic field opposes the inducing change according to Lenz's law; electrical resistance converts electrical energy into heat.

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. ^[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.^[ref]

The electrical readout gives operating electrical resistance $R(t)=V(t)/I(t)$ when $I(t)\ne 0$, 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, takeHere $c_w$ is the water's specific heat capacity at constant pressure, $C_{\mathrm{cal}}$ is the participating vessel, heater and sensor heat capacity, $Q_{\mathrm{loss},\mathrm{out}}$ is signed outward heat leakage, and $W_{\mathrm{stir},\mathrm{on}}$ is stirring work entering the assembly. The storage term approximates its enthalpy change; for a rigid fixed-volume receiver use the corresponding heat capacity at constant volume. Calibrate $C_{\mathrm{eff}}$ independently if the purpose is to test electrical input against thermal storage. Using the same heating run to define $C_{\mathrm{eff}}=E_{\mathrm{el}}/\Delta T$ and then claiming it verifies the equality would be circular.

For the synthetic trace, stipulate $V=5\mathrm{V}$, $I=1\mathrm{A}$, $R=5\Omega$, $C_{\mathrm{eff}}=500J/K$, room temperature $20{}^{\circ }\mathrm{C}$, and loss coefficient $G=0.5W/K$, with negligible stirring input. The heater runs for $120\mathrm{s}$, beginning at room temperature. The calorimeter heat-loss correction model isIt predicts $E_{\mathrm{el}}=600\mathrm{J}$, $\Delta T\approx 1.13080\mathrm{K}$, 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 $C_{\mathrm{eff}}/G=1000\mathrm{s}$. A cooling slope can estimate $G$ only with known $C_{\mathrm{eff}}$ 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. ^[ref]

An electrical circuit connects components that carry electric current and sustain electric potential difference differences. Its dynamics may couple electrical resistance, inductance and capacitance.

For an ohmic element with finite nonzero electrical resistance $R$, its electrical conductance is $G=1/R$: an electric potential difference $\Delta V$ drives electric current $I=G\Delta V$. Greater positive electrical conductance means a greater electric current for the same positive electric potential difference. Electrical conductivity is instead a bulk material coefficient relating current density to electric field.

Electrical conductivity $\sigma$ measures the local response of current density to electric field. In the rest frame of an isotropic Ohmic conductor, the local form of Ohm's law isLarger positive electrical conductivity gives larger current density at the same electric field. It is a bulk material coefficient, distinct from the geometry-dependent electrical conductance of a whole component and from a sheet's surface conductivity. Electrical resistivity is ${\rho }_{\mathrm{el}}=1/\sigma$ in this scalar model. ^[ref].

The local and whole-wire forms fit together particularly neatly. For uniform current in a straight homogeneous wire, with length $L$, wire cross-sectional area $A$, constant electrical conductivity and terminal electric potential difference $V$, the axial electric field has magnitude $E=V/L$. ThusFor the defined electrical resistance example $\sigma =5\times 10^5(\Omega \mathrm{m}{)}^{-1}$, $L=1\mathrm{m}$, $A=10^{-6}{\mathrm{m}}^2$ and $V=5\mathrm{V}$, the local values are $E=5V/m$ and $j=2.5\times 10^{6}A/{\mathrm{m}}^2$; integrating gives $I=2.5\mathrm{A}$. One material law therefore supplies both a spatial description and a terminal relation.

The scalar form requires isotropy. In an anisotropic material the response can instead be a tensor, $J_i={\sum }_j{\sigma }_{ij}{E}_j$, and current density need not be parallel to electric field; uniaxial conductivity tensor is one existing example. These statements concern the material's rest frame. A moving electrical conductor instead uses the moving-conductor Ohm law.

An electrical insulator has sufficiently small transport of electric charge for its operating conditions. Insulation is a practical requirement involving electric potential difference, temperature, geometry and duration, rather than a universal claim of zero electrical conductivity. Electric polarization can be substantial in an electrical insulator; dielectric breakdown can end the insulating regime.

For an idealized uniform slab with small constant electrical conductivity $\sigma$, the Ohmic conductor model predicts leakage electrical resistance $R=d/(\sigma A)$ in parallel with capacitance $C=ϵA/d$. An isolated capacitor then hasThis is a derived model of slow discharge, not an assertion that every leakage mechanism obeys Ohm's law. Its time scale matches charge relaxation. The fixed-charge comparison in dielectric capacitor response needs an observation time short compared with $RC$, but long enough for electric polarization to settle. During a maintained voltage, leakage consumes power $V^2/R$ as Joule heating; delayed electric polarization can add dielectric loss under changing electric fields.^[ref]

Large relative permittivity alone does not rule out either mechanism.^[ref]

Electrical reactance $X$ is the signed imaginary part of electrical impedance $Z=R+\mathrm{i}X$, measured in ohms. It describes the part of the terminal electric potential difference in phase quadrature with the electric current in a sinusoidal steady state. In the $e^{\mathrm{i}\omega t}$ convention, an ideal inductor has $X_L=\omega L>0$ and an ideal capacitor has $X_C=-1/(\omega C)<0$ for $\omega >0$. A large $|X|$ reduces electric current at fixed applied voltage, but an ideal purely reactive component returns the energy it receives over each cycle. This is why electrical reactance and electrical resistance have the same units yet describe different physical behavior. ^[ref].

In a series RLC circuit, electrical impedances add because the same electric current passes through every component. For ideal positive inductance $L$, capacitance $C$, and total electrical resistance $R_{\Sigma }>0$,Here $\varphi$ is voltage wave phase minus current wave phase. Below ${\omega }_0=1/\sqrt{LC}$, $X<0$ and electric current leads terminal voltage; above it, $X>0$ and electric current lags. At ${\omega }_0$, the two signed electrical reactances cancel, leaving $Z=R_{\Sigma }$ and the maximum electric current at fixed terminal voltage. This is the resonant series RLC response. Electrical resistance does not cancel: the toy circuit gives $f_0={\omega }_0/(2\pi )=1591.55\mathrm{Hz}$ and $I_{\mathrm{rms}}=9.091\mathrm{mA}$ there. These numbers follow directly from the stipulated component values. ^[ref].

Cancellation concerns terminal voltage, not the disappearance of stored energy. In the toy circuit at resonance, the individual inductor and capacitor voltages are each $0.9091\mathrm{V}$ in root mean square magnitude, with opposite wave phases, while their sum is zero. Their peak stored magnetic energy and capacitor energy are each $0.8264\mu \mathrm{J}$, attained at different times. With smaller positive $R_{\Sigma }$, either component voltage can exceed the applied voltage. This ability to select a frequency while sustaining internal energy exchange connects electrical reactance to tuning and frequency response.

For an ideal inductor or capacitor, the absorbed instantaneous power equals the rate of change of stored energy:Positive power stores energy; negative power returns it. Over a periodic steady state, each stored energy returns to its initial value, so its cycle-averaged absorbed power is zero. An ideal resistor instead has $p_R=R{i}^2\ge 0$, transferring energy into Joule heating. More generally, the average terminal power of this series RLC circuit isZero average power in an ideal reactive component does not imply zero instantaneous power, zero electric current, or zero stored energy. Real coils and capacitors have losses, so zero ideal reactive loss is a model assumption rather than a claim about every component. ^[ref].

The motivating observation is a pair of sinusoids on an oscilloscope: changing frequency changes both their relative amplitudes and their phase difference. Gary Tuttle's EE 201 laboratory uses resistor voltage as the current-phase reference, compares predictions with readings at several frequencies, and explicitly asks students to measure component values and account for coil electrical resistance. The following bench arrangement applies that method to a series RLC circuit; its component values and illustrated traces are our own toy example. ^[ref].

Connect a low-voltage sine generator to a series chain ending in a known sensing resistor $R_s$ at the common return. Use a resistor whose unwanted inductance is negligible over the chosen frequency range. An oscilloscope displays voltage against time: channel A reads the loaded voltage across the whole chain, and channel B reads the voltage $v_s$ across $R_s$. Both channel ground connections and the generator return meet at that same node. Ordinary bench oscilloscope grounds are electrically common; attaching one to an intermediate circuit node would change the circuit. A separate floating component voltage needs a suitable differential measurement, rather than moving a ground clip. ^[ref].

After the free transient decays, record frequency $f$, period $T=1/f$, both voltage amplitudes, and matched rising zero-crossing times $t_A,t_B$. Check that the traces are sinusoidal. Define $\Delta t=t_B-t_A$ on the branch $|\Delta t|<T/2$. With negligible probe loading and negligible sensing-resistor electrical reactance, $v_s=R_{s}i$, so channel B is the electric current reference, and in the reactance phase conventionA later current crossing gives $\Delta t>0$, hence positive electrical reactance; an earlier crossing gives negative electrical reactance. Use the same amplitude convention for both channels. An amplitude-only ratio recovers $|Z|$, so it cannot distinguish the two signs: timing supplies the missing information. The formulas follow from $Z=|Z|e^{\mathrm{i}\varphi }$ and the chosen terminal orientation.

Begin with a known resistor replacing the reactive chain: its expected phase difference is approximately zero, testing channel polarity and relative delay. Measure the same signal with both channels to establish the instrumental delay baseline, then exchange probes to check that a phase offset does not belong to one channel. Next test an inductor and a capacitor separately with $R_s$: after subtracting $R_s$ from the inferred real part, expect approximately $X/f=2\pi L$ and $fX=-1/(2\pi C)$, respectively. Finally combine $L,C,R,R_s$ and sweep through the predicted resonance. These are proposed controls and consequences of the model, not reported observations.

For the toy series RLC circuit, the prediction at $500\mathrm{Hz}$ is $X=-286.894\Omega$, $|Z|=307.259\Omega$, $\varphi =-69.02^{\circ }$, and $V_{s,\mathrm{rms}}=32.55\mathrm{mV}$. At $5000\mathrm{Hz}$, $X=+282.328\Omega$ and $\varphi =+68.71^{\circ }$. Near $1591.55\mathrm{Hz}$ the model predicts zero phase difference and $V_{s,\mathrm{rms}}=90.91\mathrm{mV}$. Keep actual loaded $V_{A,\mathrm{rms}}$ fixed or normalize by it at each frequency; the generator's display alone does not establish that voltage. The $10\Omega$ sensing resistor is part of the $110\Omega$ total and therefore changes the predicted peak electric current.

Real components add unintended electrical impedance: coil wire has electrical resistance, nearby turns add capacitance, and capacitor leads add inductance. These effects can produce a component's own resonance, beyond which the ideal $\omega L$ or $-1/(\omega C)$ approximation fails. Work in a range where that approximation is adequate, measure coil wire electrical resistance, and compare the inferred $R_{\mathrm{chain}}$ with the sensing resistor, intended resistor, and component losses. A measured nonzero real part is evidence of loss in the assembly, not negative or positive electrical reactance becoming heat. ^[ref]; ^[ref].

The probes also add electrical resistance and capacitance to the measured nodes; their loading becomes more important as frequency increases. Use compensated probes with known input electrical impedance and short ground connections, and repeat with a smaller drive voltage to test whether the inferred $X$ depends on amplitude. ^[ref]. Record actual voltage and phase readings separately from calculated predictions, with component tolerances and timing uncertainty. A phase error $\delta \varphi$ alone produces approximately $\delta X=|Z|\cos \varphi \delta \varphi$, so near zero electrical reactance, a small delay can imitate a small residual $X$. Agreement within stated uncertainties tests the model; the smooth toy curves establish no experimental accuracy.

Electrical resistance measures the electric potential difference needed to drive a given electric current through a component. For an Ohmic conductor under fixed material conditions, $R=V/I$ is constant and Ohm's law gives $V=IR$. The SI unit is the ohm, $1\Omega =1V/A$. At a fixed positive electric potential difference, increasing electrical resistance reduces electric current. Electrical conductance $G=1/R$ expresses the same response reciprocally. ^[ref].

For a straight uniform wire at rest, with constant electrical conductivity $\sigma >0$, length $L$ and wire cross-sectional area $A$, the uniform Ohmic conductor model givesThe electrical resistivity ${\rho }_{\mathrm{el}}$ characterizes the material; electrical resistance also includes geometry. Doubling length doubles $R$, while doubling wire cross-sectional area halves $R$, with material and temperature fixed. The uniform-field assumption includes insulated sides and equipotential end faces, as explained in uniform current in a straight homogeneous wire. ^[ref].

As a defined wire example, take $L=1\mathrm{m}$, $A=1\mathrm{m}{\mathrm{m}}^2$ and $\sigma =5\times 10^5(\Omega \mathrm{m}{)}^{-1}$. Then $R=2\Omega$; applying electric potential difference $5\mathrm{V}$ drives electric current $2.5\mathrm{A}$. With the same length and material, a second wire of wire cross-sectional area $2\mathrm{m}{\mathrm{m}}^2$ has $R=1\Omega$ and carries $5\mathrm{A}$ at that same electric potential difference. Fixing electric potential difference here differs from fixing electric current in the current density comparison: stating what is held fixed is essential.

An explicit lumped model makes the cancellation understandable. Treat the receiver as having uniform temperature $T$, heat capacity $C$, heater power $P_h$ deposited in the receiver, net parasitic input $P_{\mathrm{par}}$, and outward heat transfer $Q_{\mathrm{out}}(T,T_b)$ to a reference at temperature $T_b$. With incident radiant flux $P_{\mathrm{in}}$ and radiometric absorptance $a$, conservation of energy givesThe dark state has no admitted beam. Compare settled dark and light states, denoted $d$ and $l$; each has $dT/dt=0$. Subtracting their balances giveswhere $\Delta Q_{\mathrm{out}}=Q_{\mathrm{out},l}-Q_{\mathrm{out},d}$ and $\Delta P_{\mathrm{par}}=P_{\mathrm{par},l}-P_{\mathrm{par},d}$. At identical $T$ and $T_b$, with unchanged surroundings and parasitic heating, both correction differences vanish. Thus $P_{\mathrm{abs}}=\Delta P_h$, and $P_{\mathrm{in}}=\Delta P_h/a$. Notice the signs: extra stray thermal radiation admitted with the beam also reduces heater power, so it must be subtracted from that reduction before attributing the result to the intended beam.

Measure the heater's own terminal voltage $V_h$ and electric current $I_h$: for steady electrical heating, $P_h=V_h{I}_h=I_h^2{R}_h$, where $R_h$ is its electrical resistance. A supply voltage that includes wire voltage drops counts Joule heating in the leads too; only the portion deposited in the receiver belongs in this balance. Changes in that deposited lead heat belong in $\Delta P_{\mathrm{par}}$. Optical-electrical heating non-equivalence adds a further instrument-specific correction when the uniform-temperature model is insufficient.^[ref]

An electrical synapse passes ionic electric current directly between neurons through gap junctions, whose paired channels connect the two cytoplasms. It does not require synaptic vesicle release and postsynaptic neurotransmitter binding. Transmission across the junction is very fast; the receiving cell membrane's capacitance and other conductances still determine its voltage response. Electrical coupling can coordinate the timing of activity across neurons. See Purves and colleagues: electrical synapses and gap junctions.

For a passive junction with constant conductance $g_j$, the inward electric current into cell 2 iswhere each $V_k$ is the cell's membrane potential, measured against a common extracellular reference. Taking $g_j=2\mathrm{nS}$, $V_1=-55\mathrm{mV}$ and $V_2=-65\mathrm{mV}$ gives $20\mathrm{pA}$ into cell 2 and the equal opposite electric current into cell 1. Exchanging the voltages reverses the electric current. Both depolarization and hyperpolarization can therefore be transmitted; an electrical synapse is not inherently an excitatory connection.

Many electrical synapses conduct in both directions, but this does not guarantee equal voltage responses: the cells can have different effective electrical resistance and capacitance. Some junctions rectify, meaning they conduct preferentially in one direction, so the constant-$g_j$ formula is insufficient. A concrete molecular example is the original study ^[ref]. Bidirectionality is thus a common property with important qualifications, rather than part of the definition.

In electrostatics, the electric potential $\Phi$ tells us the interaction potential energy per unit test electric charge in the electric field of prescribed sources. Its most directly useful quantity is an electric potential difference: how much work per unit electric charge an external agent must supply to move that test electric charge slowly between two points. This replaces the bookkeeping of a vector force along a path with the difference of two scalar values. The test electric charge must be small enough not to rearrange the sources appreciably.

For motion from $A$ to $B$, write $\Delta \Phi =\Phi (B)-\Phi (A)$. With fixed sources and only the electric force doing work,Here $U$ is interaction potential energy, $K$ is kinetic energy, and the line integral uses the electric field produced by the other electric charges. For slow transport with unchanged kinetic energy, the external work is $W_{\mathrm{ext}}=\Delta U$, so $\Delta \Phi =W_{\mathrm{ext}}/q$ for nonzero $q$. This external work has the opposite sign to the work done by the electric field. The SI unit is the volt,A volt measures energy per unit electric charge, not energy itself. For example, crossing an electric potential difference of $+5\mathrm{V}$ changes the potential energy of $q=+2\mathrm{nC}$ by $+10\mathrm{nJ}$, but changes that of $q=-2\mathrm{nC}$ by $-10\mathrm{nJ}$. Both test electric charges encounter the same electric potential difference; their opposite signs reverse the energy change.^[ref]

The electrostatic potential is a scalar function of position. The electric field is a conservative vector field, and all three components follow from the negative gradient of the electric potential:The electric field points toward decreasing electric potential. A positive test electric charge feels electric force in that direction; a negative test electric charge feels electric force in the opposite direction, since $\mathbf{F}=q\mathbf{E}$. A freely accelerated negative test electric charge therefore moves toward higher electric potential while lowering its potential energy $U=q\Phi$.^[ref]

Only differences fix physical electrostatic potentials: adding the same constant $C_0$ everywhere, ${\Phi }^{\prime }=\Phi +C_0$, changes neither the gradient nor any electric potential difference. Choosing an electrical reference potential is this elementary gauge transformation. Setting zero at infinity is convenient for finitely many localized point charges; it is not available for an ideal uniform electric field extending to infinity. There we can choose a finite reference point. Assigning zero at ground is a convention, whereas physically grounding an electrical conductor also allows exchange of electric charge to impose a boundary value. Zero electric potential at a point never by itself implies zero electric field.^[ref]

For a fixed point charge $Q$ at the origin in vacuum, choose zero at infinity. Coulomb's law giveswhere ${\epsilon }_0$ is the vacuum permittivity. Use the stipulated approximation $k=8.99\times 10^9\mathrm{N}{\mathrm{m}}^2/{\mathrm{C}}^2$ and $Q=+2\mathrm{nC}$. At $r=0.20\mathrm{m}$,A test electric charge $q=-3\mathrm{nC}$ has $U=-2.697\times 10^{-7}\mathrm{J}$ and radial electric force $F_r=-1.3485\times 10^{-6}\mathrm{N}$, directed inward. Moving it inward from $r_A=0.20\mathrm{m}$ to $r_B=0.10\mathrm{m}$ raises the electric potential from $89.9$ to $179.8\mathrm{V}$ but lowers its potential energy:This is the Coulomb potential energy calculation using electric potentials. The source's singular self-field at $r=0$ is excluded; when finding the force on a source point charge, use only the electric field of the other sources.^[ref]

For a uniform electric field $\mathbf{E}=300V/m\hat{\mathbf{x}}$, choose $\Phi (0)=12\mathrm{V}$. ThenMoving from $x_A=0$ to $x_B=0.04\mathrm{m}$ gives $\Delta \Phi =-12\mathrm{V}$. For $q=+2\mathrm{nC}$ the change in potential energy is $-24\mathrm{nJ}$ and the electric field does $+24\mathrm{nJ}$ of work. For $q=-2\mathrm{nC}$, the same prescribed displacement gives $\Delta U=+24\mathrm{nJ}$: the external agent must supply that work for slow transport. This uniform electric field approximates the interior of a parallel-plate capacitor sufficiently far from the edges; the fringing field of a capacitor limits the approximation.^[ref]

An equipotential is a level set $\Phi =\text{constant}$. On a smooth equipotential with nonzero gradient, every tangent displacement has $d\Phi =\nabla \Phi \cdot d\mathbf{l}=0$, so the electric field is normal to the equipotential. Thus moving a test electric charge along an equipotential requires no net electrical work. Around a point charge, the equipotentials are spheres; the circles in the diagram are their sections, not an assumption of two-dimensional Coulomb's law. A uniform electric field has parallel plane equipotentials. For contours separated by the same small electric potential difference, smaller normal separation means a stronger electric field, approximately $|\mathbf{E}|\approx |\Delta \Phi |/\Delta n$. At a point where $\nabla \Phi =0$, the electric field is zero and has no direction to call normal.^[ref]

The superposition principle adds electric potentials as signed scalars. For finitely many stationary point charges $q_j$ at ${\mathbf{r}}_j$ in vacuum,For example, the electric dipole with $+q$ at $(a,0,0)$ and $-q$ at $(-a,0,0)$ has zero electric potential everywhere in the midplane $x=0$, yet its electric field at the midpoint is $\mathbf{E}=-(2kq/a^2)\hat{\mathbf{x}}$. With $q=2\mathrm{nC}$ and $a=0.10\mathrm{m}$, this is $-3596N/C\hat{\mathbf{x}}$. At $(2a,0,0)$, the same model gives $\Phi =2kq/(3a)\approx 119.867\mathrm{V}$ and $E_x=8kq/(9a^2)\approx 1598.222N/C$. Cancellation of electric potentials is quite different from cancellation of vector electric fields.

A motivating measurement is conductive-paper electric potential mapping. A documented teaching experiment uses slightly conductive paper on an insulating board, two circular electrodes in highly conducting ink, a steady supply, and a digital voltmeter. Connect the supply to the two electrodes, keep the voltmeter common lead on the negative electrode, and use its other lead as a movable probe. Adjust the supply using the measured electric potential difference between electrodes; the published procedure uses $10\mathrm{V}$. The observable is the voltmeter reading $\Phi (P)-\Phi (\mathrm{COM})$ at the probe position $P$. Locate several points reading $2\mathrm{V}$, record their positions on a separate map, and join them into an equipotential; repeat at other readings. Lift the probe between contacts so it does not damage the paper. This procedure makes the electrical reference potential concrete: every reading compares two contacts.^[ref]

From neighboring readings infer an electric field component by a finite difference, $E_x\approx -[\Phi (x+h)-\Phi (x)]/h$. Reconstruct two in-plane components with displacements along two independent directions. Alternatively, draw the electric field normal to mapped equipotentials, toward lower electric potential. The second diagram illustrates the same measurement principle with bar electrodes spanning a uniform sheet, insulating top and bottom edges, and a synthetic model $\Phi (x)=(50V/m)x$. The model readings $4.0\mathrm{V}$ at $x=0.08\mathrm{m}$ and $5.0\mathrm{V}$ at $x=0.10\mathrm{m}$ imply $E_x=-50V/m$. These values are calculated illustrations, not reported experimental data. The documented circular-electrode experiment similarly estimates the electric field from readings at successive centimetre marks.^[ref]

Useful controls include checking the voltmeter zero, checking that the electric potential difference along each painted electrode is negligible, and repeating reference readings to detect supply drift. Reversing the supply should reverse the signs relative to the same reference contact. Contact resistance, leakage electric currents, and touching the sheet can distort the map. A voltmeter has finite input electrical resistance and draws a small electric current; voltmeter loading changes the quantity being measured if that electrical resistance is insufficiently large relative to the sheet's effective source electrical resistance.^[ref][ref]

The paper carries a steady electric current, so this is a planar analogue rather than a direct measurement of isolated point charges in three-dimensional vacuum. For constant electrical conductivity, Ohm's law and steady charge conservation give $\mathbf{j}=-\sigma \nabla \Phi$ and $\nabla \cdot \mathbf{j}=0$, hence the two-dimensional Laplace equation away from electrodes. Insulating sheet edges impose zero normal electric current, while finite electrodes fix boundary values. This derivation explains both the analogy and its limits: inhomogeneous paper, finite boundaries, and finite electrodes can change the contours. One should not fit the three-dimensional point charge formula $kQ/r$ to a planar map without testing that model. Small probe spacing resolves spatial variation better, but amplifies uncertainty in the inferred finite difference. The published laboratory instructions explicitly ask students to assess whether their proposed field fit is appropriate.^[ref][ref]

The scalar description has a crucial limit. Faraday's law of induction for a stationary closed path $\Gamma$ sayswhere ${\Phi }_B$ is magnetic flux. Nonzero changing magnetic flux gives nonzero circulation, incompatible with a single-valued electric potential alone. More generally, a scalar electric potential and a magnetic vector potential together giveConsequently, in the presence of induction, two voltmeter lead routes between the same contacts can produce different readings if the enclosed magnetic flux changes. This follows by subtracting the two path line integrals; it is a deduction from Faraday's law of induction, not a claimed outcome of the paper experiment. The steady setup and negligible induction justify interpreting its readings as electric potential differences. Merely introducing a scalar electric potential that depends on time cannot supply a nonzero curl of the electric field.^[ref]

A short mathematical justification connects the statements above. Along a smooth path, the chain rule gives $d\Phi =\nabla \Phi \cdot d\mathbf{l}$. Integrating and using $\mathbf{E}=-\nabla \Phi$ yields the endpoint formula by the fundamental theorem for line integrals, proving path independence. Conversely, vanishing circulation around every closed path lets us define $\Phi (P)=-{\int }_{P_0}^P\mathbf{E}\cdot d\mathbf{l}$ independently of the route, with $\Phi (P_0)=0$. Zero curl is a local test; for a continuously differentiable curl-free vector field, a simply connected open region suffices for a global potential. The explicit finite-source formula already gives a global electric potential away from its source points.^[ref]

Electrolysis uses an external source of electrical work to drive redox chemical reactions at the electrodes of an electrolytic cell. Its characteristic application is forcing a chemical reaction in a direction that is nonspontaneous under the specified conditions: a discharging galvanic cell supplies electrical work, whereas electrolysis consumes electrical work. This makes electrolysis useful for producing otherwise difficult-to-obtain chemical substances, charging a rechargeable battery, and depositing a metal coating on a conducting object.^[ref]

Always identify the anode by oxidation and the cathode by reduction. In ordinary driven two-terminal operation, the source withdraws electrons from the positive anode and delivers electrons to the negative cathode. In a discharging galvanic cell, the anode is negative and the cathode positive. Thus terminal signs change with operating mode, while the reaction definitions remain valid. Through the electrolyte, ions transport electric charge; they need not all be consumed at an electrode. The electrolyte and electrolytic cell articles explain this transport and its conventional current direction.

For a net chemical equation transferring $z$ electrons per reaction event, the molar reaction Gibbs free energy obeyswhere $F$ is the Faraday constant and $E_{\mathrm{chem}}$ is the reversible cell electromotive force for the reaction in the written direction. A nonspontaneous direction has ${\Delta }_{r}G>0$ and $E_{\mathrm{chem}}<0$ at fixed temperature and pressure. The ideal opposing voltage magnitude is ${\Delta }_{r}G/(zF)$; sustaining a finite electric current generally needs additional drive because of electrical resistance and finite electrode reaction rates. This thermodynamic voltage concerns the actual composition, rather than necessarily standard conditions.^[ref] Not every use of electrolysis increases bulk chemical Gibbs free energy: transferring copper between equivalent surfaces can have zero bulk ${\Delta }_{r}G$ while the source still supplies dissipated electrical work. See the existing electrolytic cell example.

A particularly revealing experiment measures copper transfer instead of merely looking for a coating. Two separated copper electrodes stand in aqueous $\mathrm{CuS}{\mathrm{O}}_4$ electrolyte in a vessel, with insulated leads to a regulated direct-current power supply, a series ammeter and a switch. A balance reads each electrode's mass before and after a timed run. The Institute of Physics describes copper cells with ammeters, adjustable current and removable cathodes; its comparison uses three similar cells at $1\mathrm{A}$ for ten minutes, $0.5\mathrm{A}$ for ten minutes, and $0.5\mathrm{A}$ for twenty minutes. The specified observation to test is that deposited copper mass is proportional to electric current times time. These are documented experimental settings, not measurements generated here.^[ref]

The ideal half-reactions areEach deposited copper chemical atom requires two electrons, so $z=2$. The anode loses copper and the cathode gains it; ideally their mass changes have equal magnitudes, and the bulk copper cation inventory stays constant. Local concentrations near the electrodes can nevertheless differ. The sulfate anions participate in ionic transport without appearing in these net half-reactions; electronic electric current in the wires becomes ionic electric current in the electrolyte through the two interfaces.^[ref]

Clean, rinse, dry and identify the electrodes before their initial weighing. After interrupting the electric current, rinse away adhering electrolyte, dry again and weigh without rubbing off the deposit. KIT's university practical explicitly calls for distilled-water rinsing, careful drying and weighing both electrodes; it warns that loose copper grains can be lost during handling.^[ref] Useful additional controls are repeated drying and weighing until the reading stabilizes, and a matched unpowered immersion followed by the same rinse/dry cycle. These controls test retained liquid, residual dissolved salts, handling loss and chemical changes unrelated to passed electric charge; they are proposed checks, not additional observations reported by KIT. Keep immersed electrode area, spacing, electrolyte composition and temperature comparable across runs, and keep the electrodes from touching.

Record the electric current as a function of time, together with the exact switch-on and switch-off times. Taking $I(t)\ge 0$ in the chosen deposition direction, the transported electric charge isFor constant electric current, $Q=I(t_1-t_0)$; with sampled readings, use numerical integration rather than assuming that one ammeter reading remained constant. The two equal-charge Institute of Physics settings each give $600\mathrm{C}$, while its shorter $0.5\mathrm{A}$ setting gives $300\mathrm{C}$. With $M_{\mathrm{Cu}}=63.546\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, the mass per amount of substance tabulated by KIT, and $F\approx 96485.33212\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$, ideal Faraday's laws of electrolysis predictThese are calculated predictions for complete conversion of the measured electric charge into retained copper, not experimental results. For a stable process, doubling the electric charge doubles the deposited mass; equal electric charge gives equal deposited mass even when the electric current and duration differ.

The balance reading is a test of the whole model. Competing reduction of water or other chemical species can divert electric current from copper deposition; transient electrode capacitance can also consume part of a short pulse without producing copper. A fragile deposit can detach even when its formation used the expected electric charge. Conversely, retained electrolyte or salts can increase an apparent mass gain. Comparing anode loss with cathode gain helps detect such problems but does not identify every cause. A slope below $M_{\mathrm{Cu}}/(2F)$ alone cannot distinguish competing chemical reactions from lost deposit. Current efficiency measures the fraction of transported electric charge assigned to a specified electrode reaction, rather than the fraction of electrical energy stored chemically.^[ref]

The beautiful connection is that an electrical measurement counts chemical events: integrating electric current and reading a balance provide two ways of following the same electron stoichiometry. This makes electrolysis useful for controlling coating quantity and for testing a charge-to-amount of substance relation. It does not determine coating uniformity, adhesion or the required voltage, which depend on transport and electrode chemical kinetics as well.

Electromotive force is the driving work per unit electric charge around an electrical circuit, measured in volts, despite its name containing "force". For a stationary loop its electromagnetic contribution is $\mathcal{E}={\oint }_C\mathbf{E}\cdot d\mathbf{l}$; for a moving electrical conductor, include the motional electromotive force. A nonzero closed-loop line integral of the electric field cannot be represented by a single-valued electric potential alone. An electromotive force specifies the drive; the resulting electric current also depends on the electrical resistance and self-inductance. ^[ref].

The electron-phonon interaction couples electronic states to the atomic displacements represented by phonons. A lattice vibration changes the potential seen by an electron; its quantized displacement permits phonon absorption and emission. This interaction affects electrical resistance, electronic thermal conductivity, lattice dynamics and pairing in conventional superconductors.