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A resistor is an electrical circuit component whose intended behavior is electrical resistance. An ideal ohmic resistor obeys Ohm's law, $V=IR$, and converts electrical energy into heat at power $P=I^{2}R$ through Joule heating. It provides the dissipative term in a series RLC circuit.

Whether increasing electrical resistance increases Joule heating depends on what the supply holds fixed. Consider separate ideal resistor experiments with $R=5\Omega$ and $R=10\Omega$, each maintained at its specified temperature. At fixed electric current $I_0=1\mathrm{A}$, doubling $R$ changes the required electric potential difference from $5$ to $10\mathrm{V}$ and doubles power from $5$ to $10\mathrm{W}$. At fixed electric potential difference $V_0=5\mathrm{V}$, doubling $R$ halves electric current from $1$ to $0.5\mathrm{A}$ and halves power from $5$ to $2.5\mathrm{W}$. These are stipulated model values, not measured data.

The comparison also predicts feedback when a particular resistor's electrical resistance rises with temperature: a supply maintaining electric current increases its heating power, whereas one maintaining terminal electric potential difference reduces it. Predicting the actual temperature still needs the material response $R(T)$ and its heat transfer to the surroundings; a nominal supply setting does not establish which drive is actually constant at the resistor.

Drive a series RLC circuit with electric potential difference $V(t)=V_0\cos (\omega t)$, where $V_0>0$ and $\omega >0$. After the free transient decays, the electric current has the same angular frequency as the source:For fixed positive $R,L,C,V_0$, the electric current amplitude is largest exactly at $\omega ={\omega }_0=1/\sqrt{LC}$. The two electrical reactance contributions cancel there, leaving only the drop from electrical resistance. In particular,Every solution approaches this periodic response because its difference from the displayed solution is a decaying free response. The peak remains at ${\omega }_0$ even for an overdamped RLC response; free ringing is not required for a maximum of driven electric current. ^[ref].

For example, stipulate $L=10\mathrm{mH}$, $C=1\mu \mathrm{F}$, $R=20\Omega$, and $V_0=1\mathrm{V}$. Then ${\omega }_0=10^4{\mathrm{s}}^{-1}$, $f_0={\omega }_0/(2\pi )\approx 1591.55\mathrm{Hz}$, and the peak electric current amplitude is $50\mathrm{mA}$. Holding the source amplitude fixed is essential when comparing responses at different frequencies.

Three frequencies must be distinguished. The free underdamped RLC response rings at ${\omega }_d=\sqrt{{\omega }_0^2-{\alpha }^2}$. Driven electric current peaks at ${\omega }_0$. Driven electric charge has amplitudewhose peak occurs at $\sqrt{{\omega }_0^2-2{\alpha }^2}$ only if $\alpha <{\omega }_0/\sqrt{2}$. Otherwise it decreases from its zero-frequency limit $C{V}_0$, with no positive-frequency peak. The different peak locations follow because electric current is the time derivative of electric charge, multiplying a sinusoidal amplitude by $\omega$. The limiting responses are $I_{\mathrm{amp}}\sim V_{0}C\omega$ as $\omega \to 0^{+}$ and $I_{\mathrm{amp}}\sim V_0/(L\omega )$ as $\omega \to \infty$.

The Q factor measures selectivity:The average Joule heating power is $R{I}_{\mathrm{amp}}^2/2$. Its half-maximum angular frequencies and their separation are exactlyThese follow by setting $|\omega L-1/(\omega C)|=R$. In the numerical example, $Q=5$ and $\Delta \omega =2000{\mathrm{s}}^{-1}$. Reducing $R$ makes a taller, narrower electric current peak; varying $C$ tunes the selected frequency. Measuring the drop $v_R=Ri$ gives a band-pass filter that suppresses very low and very high frequencies while passing a range near resonance. This is useful for selecting a radio signal, though the range must be broad enough to pass its modulation. ^[ref].

An RL circuit combines electrical resistance $R$ and self-inductance $L$. In the series model, the same electric current $I$ flows through both components. An ideal battery maintains terminal electric potential difference $V$; the drops in the direction of positive electric current obey Ohm's law and the inductor relation:This is Kirchhoff's voltage law with the inductor's terminal drop, rather than its oppositely signed induced electromotive force. For constant $R>0$, $L>0$, and $V$, the first-order linear differential equation has solutionThe time constant $\tau$ measures exponential decay of the difference from the final electric current. ^[ref].

For a stipulated example, take electrical resistance $R=2\Omega$, self-inductance $L=0.5\mathrm{H}$, terminal electric potential difference $V=6\mathrm{V}$, and initial electric current $I(0)=0$. ThenWith $t$ measured numerically in seconds, this is $I(t)=3(1-e^{-4t})$ in amperes. At one time constant,and the terminal electric potential difference drops areTheir sum is exactly $6\mathrm{V}$. At $t=0^{+}$, the inductor takes the entire terminal electric potential difference while the electric current is still zero. At steady electric current, the inductor's terminal drop is zero and the entire drop is across the electrical resistance.

The final magnetic energy stored by the inductor is $U_{\infty }=L{I}_{\infty }^2/2=2.25\mathrm{J}$. At one time constant, the electric current has reached about $63.21\%$ of its final value, but its magnetic energy has reached only about $39.96\%$:This difference follows from the square of electric current in the inductor's magnetic energy.

For release, take $t=0$ at the start of a separate experiment with initial electric current $3\mathrm{A}$. Remove the battery from the model while maintaining a closed path through the same electrical resistance and inductor. ThenNow $v_L=-v_R$: the induced electromotive force supports the continuing positive electric current as its magnitude decreases. This closed-loop model describes exponential decay, not opening the sole path and imposing an instantaneous jump to zero electric current. The inductor's magnetic energy and the power of Joule heating areTheir time constant is $\tau /2=0.125\mathrm{s}$. The total energy dissipated by Joule heating is exactly the initial magnetic energy:The second diagram in inductor shows this energy transfer. A finite closed path lets us predict both how long the electric current persists and how much energy becomes heat. ^[ref].

For a uniform plane-wave beam of cross section $A$, mean intensity is $P/A=c{B}_{\mathrm{rms}}^2/{\mu }_0$. Hence $B_{\mathrm{rms}}=\sqrt{{\mu }_{0}P/(cA)}$. A sinusoidal peak amplitude is $\sqrt{2}$ times the rms amplitude. A Gaussian beam requires a stated waist or diameter convention before inferring its on-axis field from total power.

Arthur Leonard Schawlow's 1981 lecture describes an external-cell arrangement: a beam splitter makes opposed beams, an optical chopper interrupts the pump, and a lock-in amplifier reads the probe detector with the chopper reference. Pump-on reduces absorption and raises the detector signal; scanning the laser finds the narrow resonance. Its first application used iodine; pressure and laser jitter limited the resolution.^[ref]

For an original detector model, let $P_0$ be the probe power reaching a photodetector without sample attenuation and let ${\tau }_{\mathrm{off}},{\tau }_{\mathrm{on}}$ be the sample's probe optical depths. After subtracting a stable detector background, the transmitted powers areA positive synchronous change means less probe photon absorption, provided incident probe power is stable and scattered pump light is excluded. For small optical-depth changes the fractional signal is approximately ${\tau }_{\mathrm{off}}-{\tau }_{\mathrm{on}}$. A lock-in amplifier's voltage also depends on gain, reference phase and chopping waveform; it is not automatically this dimensionless ratio.

Useful controls are blocking the pump, detuning far from the feature, reversing or adjusting beam alignment, and repeating the scan at several pump light intensities. Frequency calibration, sample pressure, transit through the beams and laser frequency fluctuations must be assessed separately. Nearly opposed beams leave a residual Doppler shift: for wave vectors ${\mathbf{k}}_1,{\mathbf{k}}_2$, the simultaneous-resonance condition gives $2\delta =({\mathbf{k}}_1+{\mathbf{k}}_2)\cdot \mathbf{v}$. Exact opposition removes this transverse sensitivity in the first-order model.

For a charge $q$ of mass $m$ bound by spring constant $k=m{\omega }_0^2$, the undamped equation $m\ddot{x}+kx=q{E}_0{e}^{-i\omega t}$ has complex amplitude $x_0=q{E}_0/[m({\omega }_0^2-{\omega }^2)]$. The induced electric dipole moment is $p_0=q{x}_0$. Averaged electric dipole radiation power, divided by incident light intensity, givesThe Rayleigh scattering limit $\omega \ll {\omega }_0$ gives $\sigma \sim {\sigma }_T(\omega /{\omega }_0{)}^4$, proportional to ${\lambda }^{-4}$. For ${\omega }_0\ll \omega \ll m{c}^2/\hslash$, the charge behaves as free and the Thomson cross-section is recovered. At resonance the undamped steady-state formula is invalid; a finite damping mechanism or radiation reaction must be included. The formula describes independent weakly driven scatterers, not arbitrary coherent scattering from an extended medium.

For a sphere of radius $a$, the scattering efficiency is its effective scattering area divided by the projected geometric cross-section $\pi a^2$. It is a dimensionless quantity. For Rayleigh scattering by a small sphere with relative refractive index $m$ and incident wavenumber $k$,At fixed wavelength and fixed refractive index, the Rayleigh scattering cross-section scales as $a^6$, whereas the scattering efficiency scales as $a^4$. Doubling radius multiplies the former by $64$ and the latter by $16$. An efficiency here is an effective scattering area relative to a reference area, rather than a fraction of all light in an arbitrary beam.

For a worked toy example, take $a=10\mathrm{nm}$, $m=1.5$, and $\lambda =550\mathrm{nm}$. Then $ka=2\pi (10/550)\simeq 0.1142$ and $(m^2-1)/(m^2+2)=5/17$. Substitution gives Rayleigh scattering cross-section $\sigma \simeq 1.234\times 10^{-20}{\mathrm{m}}^2$ and scattering efficiency $Q_{\mathrm{sca}}\simeq 3.929\times 10^{-5}$. At incident light intensity $1\mathrm{W}{\mathrm{m}}^{-2}$, the scattered power is approximately $1.234\times 10^{-20}\mathrm{W}$. These are leading small-particle estimates, not a measured air-molecule cross-section.

The Seebeck coefficient $S(T)$ quantifies the thermoelectric voltage response per small temperature difference at zero electric current. Use $S=-dV/dT$ along a homogeneous material, with voltage understood through electrochemical potential. Its units are volts per kelvin, commonly $\mu V/K$. The word thermopower is also used, despite this being a voltage response rather than power; some authors use that word for $|S|$, so check their convention. ^[ref].

For the A-minus-B terminal order in the Seebeck effect experiment,Thus a small temperature difference around $T_0$ gives $V_{AB}\approx [S_A(T_0)-S_B(T_0)]\Delta T$. The finite-interval ratio $V_{AB}/\Delta T$ is an average of the coefficient difference, not necessarily its value at either endpoint. Ordinary two-material measurements determine a relative Seebeck coefficient: to recover $S_A$ from the measured difference, add a separately known $S_B$. Neither the choice of a zero for voltage nor the use of a familiar reference material makes $S_B$ identically zero.

A concrete research example is Martin and colleagues' 2021 certification experiment. They measured small temperature differences and voltages on six silicon–germanium specimens at 13 base temperatures between $295$ and $900\mathrm{K}$. At each setting they acquired 40 voltage-and-temperature-difference pairs and used linear regression with a free intercept, so a constant voltage offset did not become part of the slope. Platinum voltage leads required a reference-material correction; their terminal order gave $S_{\mathrm{measured}}=S_{\mathrm{Pt}}-S_{\mathrm{specimen}}$, hence $S_{\mathrm{specimen}}=S_{\mathrm{Pt}}-S_{\mathrm{measured}}$. This is an actual measurement protocol, unlike the toy plot above. ^[ref].

Microscopically, a temperature gradient changes charge carrier transport until an opposing electrical response balances it at zero net electric current. In the usual simple semiconductor picture, electrons and electron holes give opposite Seebeck coefficient signs, and their contributions are weighted by their electrical conductivities. The Seebeck coefficient is a transport property: density of states alone does not determine it, since carrier motion and scattering also matter. Its usefulness extends beyond temperature sensing to diagnosing which carriers transport electric charge. ^[ref].

Consider a stipulated series circuit containing an ideal resistor of electrical resistance $R=100\Omega$ and an ideal capacitor of capacitance $C=1\mu \mathrm{F}$. The same electric current passes through both. Their terminal electric potential differences add by Kirchhoff's voltage law, soA quick derivation uses $i=Cd{v}_C/dt$: taking the derivative of the complex exponential multiplies the capacitor voltage phasor by $\mathrm{i}\omega$, hence ${\hat{V}}_C=\hat{I}/(\mathrm{i}\omega C)$. Dimensionally, $1/(\omega C)$ has units $s/F=V/A=\Omega$, matching $R$.

At frequency $f=1000\mathrm{Hz}$, the angular frequency is $\omega =2\pi f$. The signed capacitive electrical reactance, modulus and complex argument of the total electrical impedance areApply $V_{\mathrm{rms}}=1.00\mathrm{V}$ across the complete series circuit. Its root mean square electric current isWith voltage as the zero-phase reference, the current phasor has complex argument $+57.858092^{\circ }$: electric current leads. The phasor drops have moduli $V_{R,\mathrm{rms}}=0.532018\mathrm{V}$ and $V_{C,\mathrm{rms}}=0.846733\mathrm{V}$; these add as complex numbers, not as positive magnitudes. Their squared magnitudes sum to $(1.00\mathrm{V}{)}^2$.

The mean absorbed power uses the root mean square values:The ideal capacitor has zero cycle-averaged absorbed power; all the mean absorbed power becomes Joule heating in the resistor.^[ref] To see why the phase factor appears, average $p(t)=v(t)i(t)$ over a full cycle with $v(t)=\sqrt{2}{V}_{\mathrm{rms}}\cos \omega t$ and $i(t)=\sqrt{2}{I}_{\mathrm{rms}}\cos (\omega t-\varphi )$. The cosine addition formula reduces the average to $V_{\mathrm{rms}}{I}_{\mathrm{rms}}\cos \varphi$; the term at $2\omega$ averages to zero. Independently, $p_C=d(C{v}_C^2/2)/dt$, and the capacitor energy returns to its starting value after each cycle.

At fixed applied $V_{\mathrm{rms}}$, the limit $\omega \to 0^{+}$ gives $|Z|\to \infty$, $\varphi \to -\pi /2$ and $I_{\mathrm{rms}}\to 0$: the ideal capacitor blocks steady electric current. The limit $\omega \to \infty$ gives $Z\to R$, $\varphi \to 0$ and $I_{\mathrm{rms}}\to V_{\mathrm{rms}}/R$. These are limits of the ideal lumped electrical circuit; unwanted lead inductance and component losses can invalidate that model at high frequency. At $\omega RC=1$, the equal legs of the impedance right triangle give $|Z|=\sqrt{2}R$ and $\varphi =-\pi /4$ without numerical evaluation. Measuring both the magnitude and the sign of the phase difference distinguishes this capacitor response from an equally large positive electrical reactance.

In a series RLC circuit, the same electric current $i=q^{\prime }$ flows through the resistor with electrical resistance $R$, the inductor, and the capacitor. Choose $q$ as the signed electric charge on the capacitor plate into which positive $i$ flows. With terminal drops measured in that direction, Kirchhoff's voltage law givesHere $V(t)$ is the applied electric potential difference; assume constant $L>0$, $C>0$, and $R>0$. With the applied source set to zero and the loop kept closed, the free response satisfiesThe angular frequency ${\omega }_0$ is the value for lossless free oscillation; $\alpha$ is the decay parameter. The three possibilities are underdamped RLC response for $\alpha <{\omega }_0$, critically damped RLC response for $\alpha ={\omega }_0$, and overdamped RLC response for $\alpha >{\omega }_0$. ^[ref].

For a fair comparison, start each circuit with the same electric charge $q(0)=q_0$ and zero electric current $q^{\prime }(0)=0$. Set $\tau ={\omega }_{0}t$ and $a=\alpha /{\omega }_0$; then $x=q/q_0$ obeys $d^{2}x/d{\tau }^2+2adx/d\tau +x=0$. The plot compares $a=0.5,1,2$. These dimensionless curves apply to any $L,C$ after rescaling time and electric charge.

The stored energy is the sum of magnetic energy and capacitor energy:Differentiating and using the circuit equation gives the complete power balance,In the free response, $E^{\prime }=-R{i}^2\le 0$: Joule heating accounts exactly for the lost energy. For the plotted initial conditions, $E(0)=q_0^2/(2C)$ and $E/E(0)=x^2+(dx/d\tau {)}^2$. Since every free solution tends to zero for $R>0$,In the lossless limit $R=0$, energy stays constant and shuttles between the capacitor and inductor. Thus an oscillation of electric charge and a decrease of total energy are compatible statements.

The analogy with a damped harmonic oscillator is exact within these ideal models: electric charge corresponds to displacement, electric current to velocity, inductance to mass, $1/C$ to spring constant, and $R$ to the coefficient of a resisting force proportional to velocity. The two stored energy terms play the roles of kinetic energy and elastic potential energy. ^[ref].

Differentiating the driven equation also gives $L{i}^{\prime \prime }+R{i}^{\prime }+i/C=V^{\prime }$. When using this equation to solve for electric current, retain the original constraint $L{i}^{\prime }(0)+Ri(0)+q(0)/C=V(0)$: the initial electric charge fixes the initial rate of change of electric current.

The Shockley-Queisser limit is the radiative conversion-efficiency bound for an ideal single-junction solar cell, with one collected electron-electron hole pair per absorbed above-gap photon, carrier thermalization, and radiative recombination required by the reverse optical process. Photons below the band gap are not absorbed, while excess photon energy above it is thermalized. Maximizing the electrical power over the operating electric potential difference and then the band gap gives a ceiling of roughly thirty percent for ordinary unconcentrated sunlight; its precise value depends on the incident spectrum, cell temperature and emission geometry. The original calculation by William Shockley and Hans-Joachim Queisser obtained approximately $30\%$ at $1.1\mathrm{eV}$ for a $6000\mathrm{K}$ blackbody source and a cell at $300\mathrm{K}$.^[ref]

For a periodic signal, the mean squared magnitude over a period. Its units are the square of the signal's units: a voltage signal has mean-square power in ${\mathrm{V}}^2$. Physical electrical power in a resistive load additionally divides this quantity by electrical resistance. Subtracting the squared mean gives the variance of the signal over a uniformly sampled phase; the two quantities coincide for a zero-mean signal.

Silver chloride is a chemical compound with formula $\mathrm{AgCl}$. Its darkening under illumination supplied the chemical readout in the historical ultraviolet discovery described above. That response depends on the incident optical spectrum and the sample's condition; it does not provide an uncalibrated direct measurement of incident power. ^[ref].

Single-slit diffraction is the spreading of light transmitted by one slit. Consider an ideal rectangular slit of width $a$, illuminated normally by a uniform plane wave with a single wavelength $\lambda$ in the surrounding medium. This monochromatic wave has uniform wave amplitude and equal incident wave phases across the slit, providing the required optical coherence. In the scalar Fraunhofer diffraction model, the normalized angular wave intensity isHere $\theta$ is measured from the forward normal in the plane across the slit width. The sinc function uses the unnormalized convention $\mathrm{sinc}u=\sin u/u$. The limit makes the central value one. This ideal envelope neglects additional direction-dependent propagation and detector factors; near the forward direction it describes the normalized screen irradiance. ^[ref].

The noncentral zeros satisfyThe restriction follows from the range of the sine of a real angle. The first minima have $\sin \theta =\pm \lambda /a$ when $\lambda /a\le 1$. If $a<\lambda$, this model has no noncentral zero in the forward hemisphere; equality puts the first zeros at grazing directions, outside the small-angle approximation and off any finite perpendicular screen. ^[ref].

For $0<\lambda /a<1$, define the central width as the separation between the two first minima. Its angular width and its width on a perpendicular screen areThe angular approximation uses radians and requires $\lambda /a\ll 1$; the screen approximation uses $y\approx L\theta$. The tangent relation is exact straight-line geometry, even though the underlying Fraunhofer diffraction field is approximate. This width is distinct from the full width at half maximum.

For $\lambda =500\mathrm{nm}$, $a=0.1\mathrm{mm}$ and $L=1\mathrm{m}$, the positive first minimum isThe dimensionless far-field parameter is $a^2/(\lambda L)=0.02$. Doubling the slit width to $a=0.2\mathrm{mm}$ gives $W\approx 5.00002\mathrm{mm}$ and parameter $0.08$; both curves below use the Fraunhofer diffraction approximation. Choosing $L=2\mathrm{m}$ halves those parameters and doubles both screen widths.

Each curve is normalized to its own $I_0$; the arrows measure between first minima. The calculation uses $\theta =\arctan (y/L)$ before evaluating the sinc function, rather than replacing $\sin \theta$ by $y/L$. The lower diagram shows the exact screen geometry with the vertical scale enlarged. Normalization removes the change in peak brightness when the slit width changes.

The wave interference explanation is especially simple: divide the slit into equal narrow strips and add their complex amplitudes, rather than their wave intensities. With wavenumber $k=2\pi /\lambda$, the continuous slit givesIntegrating the complex exponential gives the endpoint difference; its continuous value at zero is one. Taking the squared complex modulus gives the stated wave intensity. This is the Fourier transform of an interval indicator expressed in angular coordinates. Feynman's continuous-source argument obtains the same pattern by letting the number of sources tend to infinity. ^[ref].

At the positive first minimum, the wave phases span exactly $2\pi$ across the slit. Every strip in one half has a partner $a/2$ away, with equal wave amplitude and phase difference $\pi$, so the pairs cancel. At $\theta =0$, all the wave phases align instead. ^[ref].

The drawing uses $N=8$ equally weighted midpoint samples $x_j/a=-1/2+(j+1/2)/8$, with $j=0,\dots ,7$. At the first minimum their complex amplitudes are a common rotation of the eight roots of unity, so their head-to-tail polygon closes exactly. This finite sum illustrates cancellation; the continuous integral, rather than eight point sources, defines the slit pattern at general angles. The circle shows the continuous range of wave phases; coloured arrows show the eight samples.

A narrower aperture requires a larger angle to accumulate the cancelling phase difference, so narrowing the slit widens the beam. This is a useful limit on geometrical optics: confining light more tightly at an opening increases its subsequent angular spread. At fixed incident irradiance and slit height, reducing $a$ also reduces transmitted power, exposing a practical balance between diffraction-limited resolution and light throughput. In a diffraction grating, this single-slit diffraction envelope modulates the wave interference between different slits.

PHYWE's documented experiment uses a one-metre slide wire, a connection box, an unknown resistor, a known comparison resistor, a supply, and a digital multimeter in steady-current mA mode as the null detector. Its two yellow detector leads join the fixed-resistor junction and the slider. Move the slider until the indicated electric current vanishes; read the two wire lengths on the ruler and infer the unknown electrical resistance from their ratio. The guide advises changing the comparison resistor if no null is reachable or the slider lies near an end.^[ref]

The inference requires uniform electrical resistivity and wire cross-sectional area, so the two wire electrical resistances have the ratio of their lengths, by $R={\rho }_{\mathrm{el}}L/A$. Retaining PHYWE's labeling, its equation (6) is $R_x=R_C{\ell }_1/{\ell }_2$, where $R_C$ is the comparison electrical resistance. This is a different arm placement from our schematic. The guide's example table reports larger discrepancies near wire ends.^[ref]

Useful additional controls distinguish a true null measurement from an insensitive or disconnected detector. Check the detector zero with the supply off, then move the adjustable arm slightly to either side of the candidate balance and verify opposite detector signs. Reconnect the unknown with short, secure leads and repeat the balance. Any lead or contact electrical resistance included in that arm contributes to the inferred value, even though a detector lead carries no electric current at exact balance. For very small unknown electrical resistances, four-terminal sensing addresses this separate connection problem; a detector null alone does not remove it.

Check for Joule heating by repeating at lower supply electric potential difference and with shorter powered intervals, holding ambient temperature fixed. At balance the unknown still dissipates powerHalving $V_s$ quarters this power while halving ideal detector sensitivity. A balance that moves with excitation or elapsed time suggests changing electrical resistance, including self-heating; it is not explained by passive detector loading alone. These are proposed controls, not claimed outcomes of PHYWE's experiment. Standard-resistor accuracy, wire uniformity, contact reproducibility and finite detector resolution still limit the electrical measurement. Large electrical resistances also make unwanted leakage paths consequential.

A fixed electrically small loop of area A with peak sinusoidal current $I_0\cos \omega t$ has magnetic moment $m=AI$. The magnetic dipole radiation law gives mean power ${\mu }_0{A}^2{I}_0^2{\omega }^4/(12\pi c^3)$. Defining it as $I_0^2{R}_{\mathrm{rad}}/2$ yields $R_{\mathrm{rad}}={\mu }_0{A}^2{\omega }^4/(6\pi c^3)$. This equivalent resistance accounts for radiation, not heating in the conductor. Rotating the loop changes its vector moment and cannot in general be modelled by the same scalar current alone.

An ideal contact mechanical constraint supplying a normal reaction and no tangential friction. For a stationary constraining surface, the permitted velocity is a tangent vector, so the normal force has zero power and does no work. For a moving surface, normality alone does not imply zero power in the laboratory frame.

A solar cell converts incident light into electrical power through the photovoltaic effect. A semiconductor p-n junction separates and collects photogenerated electrons and electron holes. An ideal illuminated junction has output current densityHere $J_{\mathrm{sc}}$ is the short-circuit current density, $J_0$ is the diode saturation current, and $V_{\mathrm{oc}}$ is the open-circuit electric potential difference. The conversion efficiency is the largest delivered $VJ$ divided by the incident power per unit area; the short-circuit and open-circuit operating points themselves deliver zero power.

The solar-cell fill factor compares the maximum delivered electrical power with the rectangle formed by the open-circuit electric potential difference and short-circuit electric current. It measures the shape of the illuminated current-voltage characteristic. The conversion efficiency is $\eta =V_{\mathrm{oc}}{I}_{\mathrm{sc}}\mathrm{FF}/P_{\mathrm{in}}$.