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In the classical equilibrium circuit, the capacitor stores $C{V}^2/2$. Equipartition assigns this quadratic term mean energy $k_{B}T/2$, soThe mean voltage is zero and its variance is $k_{B}T/C$. Resistance determines the equilibration time and noise spectrum, not this full-band equilibrium variance. A finite-bandwidth or appreciably loading readout can measure less than the ideal result.

Equipartition gives mean energy $k_{B}T/2$ for each independent quadratic term in a classical canonical Hamiltonian. For example, $p^2/(2m)$ and $k{x}^2/2$ each contribute this amount in a harmonic oscillator. It is a thermal average, not an instantaneous equality; constraints, nonquadratic terms or quantum level spacings large compared with $k_{B}T$ require a different calculation.

The fluctuation-dissipation theorem connects equilibrium fluctuations with the dissipative part of linear response. If an applied force couples as $-f(t)x$, use response to $e^{-i\omega t}$ and define the two-sided angular-frequency spectrum $S_{xx}(\omega )=\int e^{i\omega t}⟨\delta x(t)\delta x(0)⟩dt$. The classical relation isFor a viscously damped particle, $m\dot{v}=-\zeta v+\xi (t)$, the corresponding white-noise force satisfies $⟨\xi (t)\xi (t^{\prime })⟩=2\zeta k_{B}T\delta (t-t^{\prime })$. Noise replenishes energy lost by friction; their strengths cannot be varied independently at fixed equilibrium temperature.

Electrically, the one-sided voltage-noise spectrum per hertz of an equilibrium resistor is $4k_{B}TR$. A capacitor in parallel filters it toThe same equilibrium voltage variance follows from equipartition. Increasing R makes the spectrum taller and narrower without changing its integrated area. This thermodynamic connection between dissipative conductance and thermal electrical fluctuations is the content of the classical noise calculation.^[ref]

A proton-precession magnetometer measures the scalar magnitude $F=|\mathbf{B}|$ of the local magnetic field. Its sensor contains a proton-rich liquid, such as a hydrocarbon or suitable water-based fluid. A strong current through a coil briefly polarizes the nuclear magnetic moments, producing a small net magnetization. The polarizing field is directed sufficiently across the local magnetic field to leave a transverse component when the current is switched off.

The protons then undergo Larmor precession about the ambient magnetic field. Their rotating net magnetization induces a decaying alternating voltage in a receiving coil. Measuring the frequency determines the field:For example, $50\mu \mathrm{T}$ gives about $2.13\mathrm{kHz}$. The known proton gyromagnetic ratio provides an absolute frequency-to-field scale; this instrument does not measure the field's direction. Strong spatial gradients cause different parts of the liquid to precess at different rates and shorten the usable signal, which matters very close to a buried steel object.^[ref]

A polymer exhibiting piezoelectricity. A compliant, low-density film can accept larger bending or strain than a brittle ceramic, while low permittivity can yield a large voltage coefficient despite a modest charge coefficient. Extractable energy still depends on mechanical input, capacitance and electrical loading.

The 2002 experiment described in Manipulating the Quantum State of an Electrical Circuit demonstrates Ramsey interferometry in a circuit using superconductivity. An aluminum island between two small Josephson junctions and a larger readout Josephson junction formed the loop; a dilution refrigerator cooled it. Manipulation used the charge/flux bias point least sensitive to first-order fluctuations. Microwave pulse amplitude and duration were calibrated using Rabi oscillations. Two phase-coherent $\pi /2$ pulses then enclosed a variable free interval. A current pulse switched the large Josephson junction to finite voltage with state-dependent probability; a preamplifier and threshold discriminator counted repeated trials.^[ref]

As a readout control, switching was tested against pulse height at thermal equilibrium. The Ramsey interferometry oscillation period agreed with inverse frequency detuning, supporting relative quantum phase accumulation against the external oscillator. The decaying envelope indicated finite quantum coherence, while reduced contrast was attributed provisionally to population relaxation during readout. This supports controlled coherent evolution over a finite interval, without establishing perfect isolation or identifying one exclusive noise mechanism.^[ref]

An RC circuit contains electrical resistance and capacitance. For a series resistor $R$ charging a capacitor $C$ from a constant voltage $V$, Kirchhoff's voltage law and $i=Cd{v}_C/dt$ give $RCd{v}_C/dt+v_C=V$.

With an initially uncharged capacitor, $v_C(t)=V(1-e^{-t/(RC)})$. The time constant $\tau =RC$ is the time to reach $1-e^{-1}$ of the final voltage. With the source removed, the same model predicts exponential discharge. Measuring this response provides a simple way to determine an unknown capacitance or to construct a smoothing low-pass filter.

For positive loop resistance, coupled mechanical and circuit equations relax toward constant sliding speed on sufficiently long rails. The steady current balances the downhill gravitational force, and motional electromotive force balances the resistive voltage. The undamped zero-resistance limit has no relaxation, so its late-time limit is singular.

When $R_s$ is in series with a parallel RLC circuit, the ratio of parallel-network voltage to voltage across $R_s$ is its impedance divided by $R_s$. Its peak is $R/R_s$, and its phase passes from $+\pi /2$ through zero to $-\pi /2$ as angular frequency rises through $1/\sqrt{LC}$. This is not the transfer ratio relative to the total input voltage, which includes the additional series voltage drop and has peak $R/(R+R_s)$. Identifying voltage endpoints is necessary before using the voltage divider formula.

A closed antidot edge orbit can tunnel-couple extended channels on opposite edges. When its electrochemical energy aligns with the reservoirs, resonant backscattering reduces conductance from an otherwise transmitting integer plateau. Different connectivity can instead add a transmission path and produce peaks. Gate voltage changes both the orbit and tunnel couplings; charging and spin alter the simplest fixed-area Aharonov-Bohm resonance sequence.

For the conductance-weighted random walk on a graph, set $d(x)={\sum }_v{c}_{xv}$. The voltage $h(u)=P_u(T_x<T_y)$ has boundary values one and zero. The total electric current is ${\sum }_v{c}_{xv}(1-h(v))=1/R(x,y)$. The first-step analysis therefore gives the displayed positive-return-time formula. Wiring the exterior of finite sets and taking limits gives $P_x(T_x^{+}=\infty )=1/(d(x)R(x,\infty ))$.

An ideal voltage source drives a series resistor feeding a parallel inductor and capacitor. In the $e^{i\omega t}$ convention, the branch voltage ratio is the title expression. It is a band-pass response with unit peak at $1/\sqrt{LC}$, phase from $+\pi /2$ through zero to $-\pi /2$, and half-power bandwidth $1/(RC)$. At resonance the input current vanishes while equal and opposite inductor/capacitor currents remain finite. Reactive branch-current magnitudes must be combined as phasors, not added as positive scalars.

A series resistor $R$ and capacitor $C$ feed a parallel pair with the same values. Measuring the parallel pair's voltage relative to the return gives this unloaded transfer function for phasors proportional to $e^{i\omega t}$. The voltage divider applied to $Z_s=R+1/(i\omega C)$ and $Z_p=R/(1+i\omega RC)$ proves the formula. The amplitude peaks at $\omega =1/(RC)$ with ratio $1/3$ and zero relative phase. At low and high positive frequencies, the transfer function tends respectively to $i\omega RC$ and $-i/(\omega RC)$. Reversing output polarity changes its sign, while loading changes the response itself.

An ideal short circuit fixes zero electric potential difference across a terminal pair, corresponding to zero load electrical impedance. At the end of a lossless transmission line, its voltage-wave reflection coefficient is $-1$ and the terminal voltage is zero. The available current is still limited by source and circuit impedances; real short circuits can dissipate substantial Joule heating.

The short-circuit current is the terminal electric current of a device when its terminal voltage is zero. For a solar cell, it is usually denoted $I_{\mathrm{sc}}$ and is obtained from the illuminated current–voltage curve where the two terminals are connected through a sufficiently low-resistance measuring path.

A nonzero short-circuit current establishes photogenerated charge collection, but the delivered terminal power $P=VI$ is zero at an ideal short circuit. Useful electrical output instead requires a finite operating voltage, often near the maximum power point. The short-circuit current depends on illumination, active area and collection efficiency; short-circuit current density divides it by the specified area, allowing devices of different sizes to be compared.

For locally linear electron confinement potential energy $U(y)$, two Zeeman splitting branches cross the same Fermi energy at positions differing by the energy splitting divided by $|U^{\prime }|$. The expression neglects self-consistent electrostatic screening and exchange interaction enhancement. If the quoted slope is an electrostatic voltage gradient instead of an electron-energy gradient, its magnitude must be multiplied by the elementary charge. A screened compressible quantum Hall edge strip is not a single sharp crossing with an imposed bare slope.

For an equal-duty square wave alternating between $\pm A$ with period $T$, the steady periodic capacitor voltage has extrema $\pm v_p$. A positive half-period maps $-v_p$ to $A-(A+v_p)e^{-T/(2RC)}=v_p$, proving the title formula. The negative half-period is its sign-reversed copy. Short periods give a small nearly triangular output; long periods give input-like plateaux with exponential transitions.

On a uniform lossless transmission line, incident and reflected sinusoidal waves give voltage maxima $|V^{+}|(1+|\Gamma |)$ and minima $|V^{+}|(1-|\Gamma |)$. Their ratio is the voltage standing wave ratio, $(1+|\Gamma |)/(1-|\Gamma |)$, where $\Gamma$ is the voltage-wave reflection coefficient. A matched termination gives one, while perfect reflection gives an infinite ratio. These expressions assume a common real characteristic impedance and no propagation attenuation between the compared extrema.

Coupling in which a magnetic material's preferred strain transfers mechanical stress to a piezoelectric constituent, producing an electrical response. It depends on interfaces, elastic load sharing and electrical boundary conditions; a composite's measured voltage is not by itself an intrinsic single-phase magnetoelectric coefficient.

Let a constant input be $x=(k+f)\Delta$ with $0\le f\le 1$, between consecutive quantization levels $k\Delta$ and $(k+1)\Delta$. Add independent centered uniform dither $U\sim \mathrm{Uniform}[-\Delta /2,\Delta /2]$ before nearest-level rounding, and assume that neither code clips.

The upper code is selected exactly when $U\ge (1/2-f)\Delta$. This subinterval occupies fraction $f$ of the uniform distribution's support, so $Q=k\Delta +\Delta B$ with $B$ a Bernoulli random variable of parameter $f$. Thus $\mathbb{E}[Q]=x$ and $\mathrm{Var}(Q)=f(1-f){\Delta }^2$. Averaging $N$ independent conversions givesFor example, an input one quarter of a step above the lower code produces the upper code one quarter of the time. Its averaged code can estimate a sub-step voltage, with a finite remaining variance. Centering, independence and absence of clipping are assumptions of this calculation; a shifted dither distribution can bias the result.

A lossless uniform transmission line has a forward travelling-wave voltage and current related by $V_0=Z_0{I}_0$ with $Z_0=\sqrt{L/C}$. Arbitrary independent DC voltage and current fields cannot simply acquire the same forward wave factor while retaining Maxwell consistency.