--- title: Displacement Current module: Maxwell’s Equations and Electromagnetic Waves moduleNumber: 10 lessonNumber: 1 order: 1001 summary: > Ampère's law asks for the current through a surface bounded by a loop, but a charging capacitor breaks it: slide the surface off the wire and into the gap and the enclosed conduction current drops to zero, while the magnetic field around the loop plainly does not. Maxwell's repair is to count a changing electric flux as itself a source of magnetic circulation. We derive the displacement-current term $\varepsilon_0\,\d\Phi_E/\d t$, show that charge continuity demands it, compute the magnetic field it produces inside a charging capacitor, and see how it closes the Ampère–Maxwell law so that electric and magnetic fields can sustain one another as a wave. topics: [Maxwell’s Equations and Electromagnetic Waves] draft: false sources: - book: Tipler & Mosca ref: "Ch. 30 — Maxwell’s Equations and Electromagnetic Waves; §30-1" --- ## Charging Capacitors and Surface Ambiguity The magnetostatic form of Ampère's law relates circulation of magnetic field around a closed curve $C$ to the conduction current through a surface bounded by that curve: $$ \oint_C\vec B\mathbin{\cdot}\d\vec\ell=\mu_0 I_{\rm cond}. $$ For steady continuous current, every surface sharing the same boundary curve intersects the same current. A charging capacitor exposes the limitation. Consider a circle around a wire leading to one capacitor plate. A spanning surface that cuts the wire contains conduction current $I$. A second spanning surface stretched through the plate gap contains no conduction current. The magnetic circulation around the same boundary curve cannot depend on which mathematical surface was selected. $$ % caption: One circular loop $C$ around the lead bounds two capping surfaces. The flat % cap cuts the wire and encloses the conduction current $I$; the bulged cap passes % through the plate gap and encloses no conduction current, yet the circulation around % $C$ must be single-valued. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,very thick] (0.40,1.60)--(2.35,1.60); \draw[black,very thick] (3.65,1.60)--(5.60,1.60); \draw[acc,very thick] (2.35,0.85)--(2.35,2.35); \draw[acc,very thick] (3.65,0.85)--(3.65,2.35); \draw[->,acc,very thick] (0.70,1.60)--(1.45,1.60); \node[above] at (1.05,1.66) {I}; \draw[acc,thick,fill=acc!10] (1.75,1.60) ellipse (0.16 and 0.62); \draw[black,dashed,thick] (1.75,2.18) .. controls (3.15,2.45) and (3.00,1.95) .. (3.00,1.60) .. controls (3.00,1.25) and (3.15,0.75) .. (1.75,1.02); \node[above] at (1.55,2.30) {wire cap}; \node[below] at (1.75,0.80) {loop C}; \node[above] at (3.55,2.30) {bulged cap}; \node[below] at (3.00,0.62) {plate gap}; \end{tikzpicture} $$ The missing term follows from charge continuity. If charge on the plate increases, $\d Q/\d t=I_{\rm cond}$ for the ideal charging path. Gauss's law relates charge to electric flux through a closed surface, so a changing electric flux between the plates accompanies the arrival of conduction current at the plate. The magnetic source in an ideal gap is the time variation of electric flux. Free charge accumulates on the plate surfaces rather than traversing the dielectric. ## Displacement Current and Continuity Maxwell defined the displacement current in vacuum as $$ I_{\rm d}=\varepsilon_0\frac{\d\Phi_E}{\d t}, \qquad \Phi_E=\int_S\vec E\mathbin{\cdot}\d\vec A. $$ The name has a historical origin. It does not mean that charged particles traverse an empty capacitor gap with current $I_{\rm d}$. Its unit is ampere because $\varepsilon_0\,\d\Phi_E/\d t$ has the same dimensions as charge per time. The generalized law is $$ \oint_C\vec B\mathbin{\cdot}\d\vec\ell =\mu_0I_{\rm cond} +\mu_0\varepsilon_0\frac{\d\Phi_E}{\d t}. $$ Use the same oriented surface for the conduction-current and electric-flux terms. The boundary direction of $C$ and the positive surface normal are linked by the right-hand rule. Reversing the surface normal reverses the signs of both source terms and of the corresponding circulation convention. $$ % caption: The two terms of the Ampère–Maxwell law for a charging capacitor. The % conduction current $I$ arriving along the lead and the displacement current % $\varepsilon_0\,\d\Phi_E/\d t$ filling the gap carry the same value through any % surface on the loop. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,very thick] (0.40,1.60)--(2.05,1.60); \draw[black,very thick] (4.15,1.60)--(5.80,1.60); \draw[acc,very thick] (2.05,0.70)--(2.05,2.55); \draw[acc,very thick] (4.15,0.70)--(4.15,2.55); \draw[->,acc,very thick] (0.70,1.60)--(1.55,1.60); \draw[->,acc,very thick] (4.55,1.60)--(5.40,1.60); \foreach \x in {2.45,2.85,3.25,3.65,4.00} { \draw[->,black,thick] (\x,0.95)--(\x,2.30); } \node[below] at (1.15,1.55) {conduction I}; \node[below] at (5.00,1.55) {conduction I}; \node[above] at (3.05,2.55) {displacement I}; \node[below] at (3.05,0.62) {growing E in gap}; \end{tikzpicture} $$ The two source terms obey a local continuity relation. In differential form, $$ \nabla\mathbin{\times}\vec B =\mu_0\vec J +\mu_0\varepsilon_0\frac{\partial\vec E}{\partial t}. $$ Taking divergence of both sides gives $$ 0=\mu_0\nabla\mathbin{\cdot}\vec J +\mu_0\frac{\partial\rho}{\partial t}, $$ or $\nabla\mathbin{\cdot}\vec J+\partial\rho/\partial t=0$. Thus, the Ampère–Maxwell law is compatible with local charge conservation. The earlier steady-current form would give zero divergence on the left while allowing charge to accumulate on a capacitor plate on the right. ### Field in a charging parallel-plate gap Take circular parallel plates of radius $R$, separated by a small distance. Assume uniform electric field between central portions of the plates and negligible fringing. If plate charge changes at rate $I$, then $$ E=\frac{Q}{\varepsilon_0\pi R^2}, \qquad \varepsilon_0\frac{\d}{\d t}\left(E\pi R^2\right) =\frac{\d Q}{\d t}=I. $$ The total displacement current through a surface covering the full plate area equals the conduction current in the lead. An Amperian circle of radius $rR, $$ within the ideal plate and lead geometry. The inside result rises linearly with radius; the outside result falls as inverse radius. Both expressions give the same value at $r=R$. $$ % caption: Top view of the charging circular capacitor. An Amperian circle of radius % $r=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,thick] (2.6,1.7) circle (1.25); \draw[acc,thick,dashed] (2.6,1.7) circle (0.72); \filldraw[black] (2.6,1.7) circle (0.045); \draw[->,acc,very thick] (2.6,2.42) arc (90:18:0.72); \draw[->,acc,very thick] (2.6,0.98) arc (270:198:0.72); \node[below left] at (2.55,1.68) {axis}; \node[below] at (2.6,0.30) {plate rim R}; \node[right] at (3.34,2.02) {path r}; \node at (3.02,1.44) {B}; \end{tikzpicture} $$ The calculation has a restricted geometry. Near a plate edge, electric field is not uniform and the circular-field symmetry is modified by leads and fringing. At high frequency, conductor current distribution, radiation, and wave propagation require a time-dependent field solution beyond the quasi-static plate model. The displacement term remains part of Maxwell's law in those regimes; the simple area-fraction calculation does not. ## Field Distribution and Measurement A probe placed in the gap must distinguish the predicted azimuthal magnetic component from background fields and lead fields. Survey the source-current path and arrange the return lead so that its field is either modelled or reduced by close pairing. Record probe orientation around the capacitor axis. Reversing the charging current reverses the gap magnetic component, while static background remains unchanged. A sinusoidal charging current produces displacement current and gap magnetic field that change sign each half cycle. A phase-sensitive measurement can compare the magnetic signal with the current in the lead. The ideal quasi-static result predicts the same current amplitude in the lead and through the complete gap surface. A frequency-dependent phase shift can indicate instrumentation delay, lead inductance, dielectric loss, or a departure from the assumed field geometry. Report the plate radius, plate separation, current waveform, frequency, source and return lead geometry, probe active area, sensor orientation, and the coordinate relative to the capacitor axis. The field prediction is a signed component. Comparing it with an unsigned magnitude meter erases the circulation direction needed to test the Ampère–Maxwell law. ### Surface independence from charge continuity Let $S_1$ and $S_2$ be two oriented surfaces with the same boundary $C$. Their union, with one orientation reversed, forms a closed surface surrounding a volume $V$. The difference between conduction currents through the two surfaces equals the rate at which charge in that volume changes: $$ I_{{\rm cond},1}-I_{{\rm cond},2} =-\frac{\d Q_V}{\d t}. $$ Gauss's law gives $Q_V=\varepsilon_0\oint_{\partial V}\vec E\mathbin{\cdot}\d\vec A$. With the surface orientations linked consistently, the electric-flux difference satisfies $$ \Phi_{E,1}-\Phi_{E,2}= \frac{Q_V}{\varepsilon_0}. $$ Taking a time derivative yields $$ I_{{\rm cond},1} +\varepsilon_0\frac{\d\Phi_{E,1}}{\d t} =I_{{\rm cond},2} +\varepsilon_0\frac{\d\Phi_{E,2}}{\d t}. $$ The generalized current has the same value through every surface sharing $C$, and this holds without assigning any physical charge transport to the empty capacitor gap. Conduction current and displacement current are two terms required by one local conservation law. $$ % caption: Two oriented surfaces $S_1$ and $S_2$ share the loop $C$. Joined with one % orientation reversed they close around the accumulating plate charge $Q$, so the % difference of their conduction currents equals $-\d Q/\d t$ and the generalized % current is the same through both. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[acc,thick] (1.1,1.7) .. controls (2.5,2.55) and (4.1,2.55) .. (5.5,1.7); \draw[black,thick] (1.1,1.7) .. controls (2.5,0.85) and (4.1,0.85) .. (5.5,1.7); \filldraw[black] (1.1,1.7) circle (0.05); \filldraw[black] (5.5,1.7) circle (0.05); \filldraw[draw=acc,fill=acc!15] (3.3,1.7) circle (0.09); \node[above] at (3.3,2.52) {surface S1}; \node[below] at (3.3,0.88) {surface S2}; \node[right] at (3.42,1.62) {charge Q}; \node[left] at (1.02,1.7) {loop C}; \end{tikzpicture} $$ The sign relation is easiest to check with a physical capacitor. Select the surface normal through the plate gap in the direction of increasing electric field. During charging, $\d\Phi_E/\d t$ is positive. The boundary direction supplied by the right-hand rule matches the magnetic circulation measured around the incoming wire. Changing the boundary direction reverses the circulation, the surface normal, and the flux derivative together. An isolated sign reversal of only one factor signals an inconsistent surface convention. The derivation also identifies the scope of the field quantities. The electric flux must be calculated from the total electric field through the spanning surface, including fields from all charges relevant to the chosen geometry. A plate-area formula is valid only when plate charge, field direction, and fringing approximation have been stated. Replacing total flux with one local field sample can give a displacement-current estimate with correct units but incorrect surface dependence. ### Displacement-current density In vacuum, define the displacement-current density $$ \vec J_{\rm d}=\varepsilon_0 \frac{\partial\vec E}{\partial t}. $$ The integral of its normal component over a surface is $I_{\rm d}$. This notation applies when the electric field varies across the surface. In the central region of a broad parallel-plate capacitor, $\partial\vec E/\partial t$ is approximately uniform. Near the edge, fringing produces radial components and a nonuniform displacement-current density. The total flux derivative remains the relevant source term in the Ampère–Maxwell law. The density has the unit $\mathrm{A/m^2}$. It can be compared with conduction current density in a wire, but the comparison does not make the two microscopic mechanisms identical. Conduction current density describes motion of free charge. Vacuum displacement-current density describes a time-varying electric field. In a material dielectric, polarization response must also be included in a material model; the vacuum expression above remains the form used for the present capacitor calculation. $$ % caption: Displacement-current density $\varepsilon_0\,\partial\vec E/\partial t$ % between finite plates. In the central region it is uniform and axial; near the edge % the field bends and spreads, so the source is the surface integral of the normal % component over the whole gap. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[acc,very thick] (0.70,2.55)--(5.45,2.55); \draw[black,very thick] (0.70,0.62)--(5.45,0.62); \foreach \x in {1.55,2.25,2.95,3.65,4.35} { \draw[->,acc,thick] (\x,0.95)--(\x,2.22); } \draw[->,black,thick] (0.92,0.95) .. controls (0.20,1.44) and (0.20,1.96) .. (0.92,2.22); \draw[->,black,thick] (5.23,0.95) .. controls (5.95,1.44) and (5.95,1.96) .. (5.23,2.22); \node[above] at (3.05,2.55) {upper plate}; \node[below] at (3.05,0.62) {lower plate}; \node[below] at (0.56,1.55) {edge}; \node[below] at (5.60,1.55) {edge}; \end{tikzpicture} $$ A circular path in a uniform gap has radial $B(r)$ variation set by the enclosed surface area. A plot of the normalized field $B(r)/B(R)$ rises as $r/R$ inside the plates and falls as $R/r$ outside. The break in slope at $r=R$ reflects the end of the growing enclosed-flux area, not a physical discontinuity in the ideal field. A finite plate edge smooths that break. $$ % caption: Normalized magnetic field $B(r)/B(R)$ around the charging capacitor. Inside % the gap it rises as $r$; outside the plate radius it falls as $1/r$; the two branches % meet at the plate edge $r=R$. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (0.50,0.50)--(5.80,0.50) node[right] {radius r}; \draw[->,black] (0.70,0.30)--(0.70,3.00) node[above] {B ratio}; \draw[acc,very thick] (0.70,0.50)--(3.05,2.45); \draw[acc,very thick] plot[domain=3.05:5.50,samples=120] (\x,{0.50+4.78/(\x-0.60)}); \draw[black,dashed] (3.05,0.50)--(3.05,2.45); \node[below] at (3.05,0.50) {R}; \node[above] at (1.65,1.95) {inside}; \node[above] at (4.55,1.55) {outside}; \end{tikzpicture} $$ ### Phase and quasi-static validity If capacitor charge varies sinusoidally, let $$ Q(t)=Q_0\cos\omega t. $$ Then conduction current in the lead and displacement current through the complete gap surface are $$ I_{\rm cond}(t)=\frac{\d Q}{\d t} =-\omega Q_0\sin\omega t, \qquad I_{\rm d}(t)=I_{\rm cond}(t). $$ The electric field follows charge and is one quarter cycle out of phase with current. The magnetic field in the gap follows the generalized current and therefore has the current phase in the ideal quasi-static model. A measurement that compares gap electric field with lead current should expect this phase relation; a measurement that compares gap magnetic field with lead current should expect matching sign once the coordinate and sensor directions have been fixed. $$ % caption: Phase relations for sinusoidal charging. Plate charge and gap electric field % $E$ share one phase; lead current, displacement current, and the gap magnetic field % $B$ lead them by a quarter cycle. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (0.50,1.62)--(5.82,1.62) node[right] {time}; \draw[->,black] (0.70,0.30)--(0.70,3.00) node[above] {signal}; \draw[acc,very thick] plot[domain=0.82:5.55,samples=180] (\x,{1.62+0.95*cos(112.5*(\x-0.82))}); \draw[black,very thick] plot[domain=0.82:5.55,samples=180] (\x,{1.62+0.95*sin(112.5*(\x-0.82))}); \node[acc] at (1.95,0.70) {charge, E}; \node[black] at (3.30,2.78) {current, B}; \end{tikzpicture} $$ The quasi-static capacitor result assumes electromagnetic propagation time across the apparatus is much shorter than the source period. A dimensionless check is $\omega L/c\ll1$, where $L$ is the largest relevant plate, lead, or observation dimension. It also assumes that source-current distribution and plate charge remain well represented by the lumped geometry. At higher frequency, lead inductance, radiation, dielectric dispersion, and distributed wave effects change the field pattern. Maxwell's law remains valid; the uniform-gap model needs replacement. ### Current-density balance For the uniform circular-plate model, the displacement-current density has magnitude $$ J_{\rm d}=\varepsilon_0\frac{\partial E}{\partial t} =\frac{I}{\pi R^2} $$ in the central plate area. It is parallel to the changing electric field. Integrating it over the full circular gap recovers $I$. The relation provides a local density description and a global current check: $$ \int_{A_{\rm plate}}\vec J_{\rm d}\mathbin{\cdot}\d\vec A =J_{\rm d}\pi R^2=I. $$ At a smaller radius $r$, the enclosed generalized current is $J_{\rm d}\pi r^2$, producing the earlier linear magnetic-field profile. The current-density picture makes the area fraction explicit. It also shows why treating the full current $I$ as enclosed by every small circular path in the gap would give an incorrect inverse-radius field near the axis. A conducting wire has a different current-density distribution. The current through its cross-section is $\int\vec J\mathbin{\cdot}\d\vec A$, where $\vec J$ usually follows the conductor and can be nonuniform near contacts, at high frequency, or in a material with varying conductivity. At the interface between a wire and capacitor plate, free charge accumulates on the plate surface. The continuity equation connects the incoming conduction current to the growing surface charge. The displacement-current density in the gap connects that changing charge to magnetic circulation across a surface that does not cut the wire. The density form should not be used to claim that every local electric-field change in an apparatus produces a measurable magnetic field independent of source geometry. The Ampère–Maxwell law integrates the normal component over a surface bounded by the chosen loop. Spatial contributions can reinforce or cancel. Symmetry is what reduces the circular-capacitor problem to one scalar $B(r)$; a rectangular plate, off-axis lead, or nearby return current generally requires a vector field calculation. > **Worked example (field in a charging capacitor gap).** Circular plates of radius > $R=30.0\ \mathrm{mm}$ carry a charging current $I=2.50\ \mathrm A$. Find the magnetic > field on a circular path of radius $r=20.0\ \mathrm{mm}$ inside the gap. The path > encloses the area fraction > > $$ > \frac{r^2}{R^2} > =\frac{(0.0200)^2}{(0.0300)^2} > =0.444, > $$ > > so the enclosed displacement current is $I_{{\rm d},r}=0.444\,(2.50\ \mathrm A)=1.11\ \mathrm A$. > The Ampère–Maxwell law gives > > $$ > B(0.0200\ \mathrm m) > =\frac{\mu_0(1.11\ \mathrm A)} > {2\pi(0.0200\ \mathrm m)} > =1.11\times10^{-5}\ \mathrm T. > $$ > > The direct formula $B=\mu_0Ir/(2\pi R^2)$ returns the same value, confirming that the > enclosed current and the path radius refer to one circle. Substituting the full > $2.50\ \mathrm A$ into the exterior expression $\mu_0I/(2\pi r)$ would give > $2.5\times10^{-5}\ \mathrm T$, too large by exactly the reciprocal of the area fraction. The ideal field direction follows the right-hand rule around the direction of increasing electric flux. In a three-dimensional setup, draw the selected surface normal first, determine whether $\d\Phi_E/\d t$ is positive or negative, then curl the right-hand fingers around that normal to identify the magnetic circulation. A current reversal reverses charge growth, electric-flux change, displacement current, and magnetic field together. Uncertainty in the calculated field is set by current, path radius, and plate radius. For $r **Worked example (peak current of a 100 pF capacitor).** A $100\ \mathrm{pF}$ capacitor > driven at $f=1.00\ \mathrm{MHz}$ with a $10.0\ \mathrm V$ peak voltage draws peak lead > current > > $$ > I_0=\omega C V_0=(2\pi)(1.00\times10^6)(100\times10^{-12})(10.0) > =6.28\ \mathrm{mA}. > $$ > > The same capacitor at $1.00\ \mathrm{kHz}$ draws only $6.28\ \mathrm{\mu A}$: the charging > rate scales with frequency while voltage amplitude and plate geometry stay fixed. A > low-frequency current measurement cannot be carried over to a radio-frequency capacitor > without revisiting lead, dielectric, and instrument-bandwidth assumptions. ### Surface orientation and sign The sign of $I_{\rm d}$ follows the selected surface normal. Choose normal $\hat n$ through the plate gap. If electric field in that direction grows, $\d\Phi_E/\d t>0$ and $I_{\rm d}$ is positive. The positive direction around boundary $C$ follows the right-hand rule: curl fingers around $C$ with the thumb along $\hat n$. A magnetic probe tangent to the circle reports a signed component only after its positive axis has been related to that circulation. $$ % caption: Orientation convention. Choosing the gap normal $\hat n$ fixes the sign of % the electric flux and, by the right-hand rule, the positive circulation sense around % the boundary loop $C$; reversing $\hat n$ reverses both. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,thick,dashed] (2.9,1.5) ellipse (1.2 and 0.42); \draw[->,acc,very thick] (2.9,1.5)--(2.9,2.85); \draw[->,acc,very thick] (4.1,1.5) arc (0:150:1.2 and 0.42); \node[above] at (2.9,2.85) {normal n}; \node[below] at (2.9,1.00) {loop C}; \node[right] at (3.72,2.02) {circulation}; \end{tikzpicture} $$ Several sign errors have distinct experimental signatures. Reversing source leads should reverse the gap magnetic signal and the measured lead current. Reversing only the probe cable or changing the data-acquisition channel polarity reverses the reported signal while leaving physical field direction unchanged. A phase shift of one half cycle can therefore arise from circuit polarity or sensor calibration; compare it with an independently measured current direction before attributing it to a displacement-current model. Do not combine a surface chosen through the wire with electric flux evaluated only in the plate gap. That mixes source terms from different spanning surfaces. Either use a wire-cutting surface, for which the conduction current is direct and flux is included over that full surface, or use a gap surface, for which conduction current is zero and the electric-flux derivative is the generalized current. Both calculations give the same circulation when all terms refer to one consistent oriented surface. ### Model checks The capacitor calculation contains several independent limits: - **Central-region geometry:** plate separation is small compared with plate radius, and the chosen path is far enough from the edge for the stated uniform-field approximation. - **Current closure:** supply and return lead paths are mapped or placed so their magnetic background is smaller than the uncertainty target. - **Quasi-static scale:** apparatus size is small compared with the wavelength at the drive frequency, and the measurement is not dominated by radiated fields. - **Sensor response:** probe bandwidth, phase delay, active area, and orientation are calibrated over the expected signal range. - **Material model:** vacuum or dielectric assumptions are stated before relating charge, voltage, electric field, and flux. A radial test holds the current waveform fixed while varying probe radius. In the ideal circular geometry, a normalized gap scan follows $B\propto r$ inside the plates. Near the edge, departure from that trend estimates the range over which fringing matters. Outside the plates, lead geometry can dominate the simple inverse-radius reference. A complete model records the plate radius and the full current circuit. ### Auxiliary fields in a material gap The vacuum form of the Ampère–Maxwell law uses total magnetic field $\vec B$, free-charge current density $\vec J$, and vacuum electric field $\vec E$. Matter adds polarization and magnetization, so experiments often use the auxiliary fields $\vec D$ and $\vec H$: $$ \nabla\mathbin{\times}\vec H =\vec J_{\rm free}+\frac{\partial\vec D}{\partial t}, \qquad \vec D=\varepsilon_0\vec E+\vec P, \qquad \vec B=\mu_0(\vec H+\vec M). $$ The source term on the right contains free conduction current and the time rate of electric displacement. Polarization $\vec P$ represents charge displacement inside atoms and molecules; magnetization $\vec M$ represents magnetic dipole response. A linear isotropic material over a stated operating range can be described by $\vec D=\varepsilon\vec E$ and $\vec B=\mu\vec H$. Those relations are material models, with frequency, temperature, bias, and geometry limits. Ferroelectric, lossy, anisotropic, or magnetic materials require measured constitutive data over the stated range of conditions. The free-source form over a gap surface $S$ reads $$ \oint_C\vec H\mathbin{\cdot}\d\vec\ell =\int_S\vec J_{\rm free}\mathbin{\cdot}\d\vec A +\frac{\d}{\d t}\int_S\vec D\mathbin{\cdot}\d\vec A. $$ A dry insulating gap has negligible free current through its interior, so the second term carries the terminal-current balance. A leaky dielectric has both terms. In that case, the source current splits into a conductive leakage contribution and a displacement contribution; a single capacitance value cannot describe the entire current record. A stated integration surface separates these terms cleanly. The surface must cross the same material region whose $\vec D$ model is being used. Terminal measurements commonly use a complex voltage and current at one frequency. With the passive sign convention, a practical small-signal capacitor model is $$ \widetilde I=(G+\mathrm i\omega C)\widetilde V. $$ The conductance $G$ represents loss or leakage within the stated frequency window, and $C$ represents the stored-energy component. A lossless ideal capacitor has $G=0$, with current one quarter cycle ahead of voltage. A nonzero $G$ shifts the terminal-current phase toward the voltage. The phase result separates a material-loss contribution from the RMS-current magnitude: reactive charge-storage contribution when the model applies. Connect the field calculation to the measurement through common phasor definitions. If the voltage monitor records $\widetilde V$, a current shunt records $\widetilde I$, and the probe records $\widetilde B_{\phi}$, retain the channel polarity and reference delay for all three. The gap model predicts $\widetilde B_{\phi}$ from the relevant generalized current. A current magnitude from one instrument combined with a phase from another instrument has no interpretable sign until their timing references have been reconciled. $$ % caption: Small-signal current phasor of a lossy capacitor. The in-phase component % $GV$ (material loss) lies along the voltage reference; the quadrature component % $\omega CV$ (charge storage) is perpendicular; their sum is the terminal current. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (1.00,0.80)--(5.40,0.80) node[right] {voltage axis}; \draw[->,black] (1.00,0.80)--(1.00,3.00) node[above] {quadrature}; \draw[->,black,very thick] (1.00,0.80)--(3.20,0.80); \draw[->,acc,very thick] (3.20,0.80)--(3.20,2.50); \draw[->,acc,very thick] (1.00,0.80)--(3.20,2.50); \node[below] at (2.10,0.78) {loss part}; \node[right] at (3.28,1.66) {storage part}; \node[above left] at (3.22,2.50) {total I}; \end{tikzpicture} $$ Material data need a frequency range and a drive amplitude. A capacitance bridge at $1\ \mathrm{kHz}$ can yield a different complex response from a radio-frequency drive because polarization mechanisms, electrode loss, and lead inductance have their own time scales. Record the bias voltage, temperature, waveform, frequency, electrode area, gap thickness, and field amplitude. That record determines whether a measured terminal current can be used to infer a displacement-current density in the gap. ### Energy flow into the field Electric and magnetic fields carry energy through the Poynting vector $$ \vec S=\frac{1}{\mu_0}\vec E\mathbin{\times}\vec B $$ in vacuum. The charging-capacitor geometry provides a direct power check. Between large parallel plates, $\vec E$ points across the gap and the magnetic field circles the charging axis. Their cross product has a radial component. During charging, the energy flux through a cylindrical surface near the plate rim points into the space between the plates, with the direction set by the chosen source polarity. The electromagnetic energy stored in the capacitor enters through the surrounding field region while charge accumulates on the conductors. $$ % caption: Energy flow into a charging capacitor. $\vec E$ spans the gap, $\vec B$ % circles the axis, and the Poynting vector $\vec S=\vec E\times\vec B/\mu_0$ points % radially inward across the cylindrical rim into the region where field energy % accumulates. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[acc,very thick] (1.20,1.05)--(5.10,1.05); \draw[acc,very thick] (1.20,2.35)--(5.10,2.35); \draw[black,dashed] (1.55,0.72) rectangle (4.75,2.68); \foreach \x in {2.15,2.85,3.55,4.05} { \draw[->,black,thick] (\x,1.35)--(\x,2.05); } \node at (1.75,2.10) {E}; \draw[->,acc,very thick] (5.05,1.55)--(4.40,1.55); \draw[->,acc,very thick] (1.25,1.55)--(1.90,1.55); \node[above] at (4.75,1.60) {S}; \node[above] at (1.60,1.60) {S}; \draw[->,black,thick] (4.35,2.58) arc (10:150:0.42 and 0.15); \node[above] at (3.95,2.66) {B}; \end{tikzpicture} $$ At the plate edge in the ideal circular model, $$ E=\frac{V}{d}, \qquad B(R)=\frac{\mu_0 I}{2\pi R}. $$ The side area has magnitude $2\pi Rd$. Multiplying the radial Poynting-vector magnitude by that area gives $$ \left(\frac{EB(R)}{\mu_0}\right)(2\pi Rd) =\left(\frac{V}{d}\right) \left(\frac{\mu_0 I}{2\pi R}\right) \frac{2\pi Rd}{\mu_0} =VI. $$ The field-energy rate agrees with the source power delivered at the capacitor terminals. An ideal capacitor has $Q=CV$, so $$ \frac{\d}{\d t}\left(\frac{Q^2}{2C}\right) =\frac{\d}{\d t}\left(\frac12CV^2\right) =VI. $$ The derivation uses leading-order fields near the rim and a quasi-static source. It does not replace a full fringing-field calculation. It links three independently measured quantities: terminal voltage, terminal current, and rate of change of stored field energy. A lossless capacitor returns the energy to the circuit during discharge; a material-loss term or series resistance transfers part of the supplied energy to heat. Power balance identifies several experimental errors. Integrating a shunt-current record with an uncalibrated voltage trace can create an apparent energy mismatch from channel gain or delay. A series resistor has power $I^2R$ that must be removed from the source-energy budget before comparison with capacitor energy. A dielectric loss appears as an in-phase current component and an accumulated heating term. During a periodic drive, average source power can be zero for an ideal capacitor even though energy moves in and out of the gap twice per cycle. The instantaneous field-energy record remains nonzero and phase dependent. ## Quasi-Static Limits and Waves The parallel-plate result assumes that source changes are communicated across the apparatus on a time scale short compared with the source period. Let $a$ denote the largest relevant dimension of the capacitor, leads, and return path, and let $v$ denote electromagnetic propagation speed in the surrounding medium. The comparison $$ \frac{\omega a}{v}\ll1, \qquad \lambda=\frac{2\pi v}{\omega} $$ expresses the quasi-static requirement. Small $\omega a/v$ permits one current value and one voltage value to describe the apparatus at a given time to the accuracy required by the model. The corresponding wavelength must be large relative to the complete current loop and its return path. At higher frequency, voltage and current vary along conductors, the return path changes the nearby magnetic field, and reflections from connectors or cable ends can alter the terminal waveform. The local Ampère–Maxwell law remains valid. The simple area-fraction expression for $B(r)$ no longer represents the complete field because the assumed cylindrical symmetry and instantaneous circuit relation have failed. Transmission-line variables, distributed capacitance and inductance, and radiated fields become part of the model. A practical frequency study keeps the geometry and source amplitude fixed while stepping frequency. At each point, record terminal voltage, terminal current, probe amplitude, probe phase, and the distance of the probe from the capacitor axis. Normalize the magnetic reading by the measured generalized current. In the quasi-static region, the normalized radial shape should remain stable after probe calibration. A frequency-dependent deformation of that shape, or a phase that varies with position after cable-delay correction, marks a distributed-field effect. The range limit depends on the requested accuracy. A geometry with $\omega a/v=0.05$ may be satisfactory for a qualitative field-direction diagram and inadequate for a phase-sensitive measurement. Compare repeated measurements at several frequencies with a stated tolerance. A cable delay is often removed by a reference measurement, whereas a changing field shape requires a new physical model. The experimental report should state the dimension used for $a$, the assumed propagation speed, the frequency interval, and the criterion used to accept or reject the quasi-static approximation. ## Error Analysis and Verification A radial gap-field scan tests the model spatially. In the uniform central region, calculate $B/r$; its value should be $\mu_0I/(2\pi R^2)$. A scan outside the plate radius tests $Br$, which should approach $\mu_0I/(2\pi)$ only when the full current loop and field geometry support the ideal approximation. A transition near the rim is expected from fringing and finite plate thickness. The analysis should retain the raw radius, signed probe component, current monitor, and phase reference before forming either normalized quantity. Background subtraction needs a reversal plan. Measure the probe signal with the source current in the positive and negative directions, using identical frequency and drive amplitude. Half the difference isolates components that reverse with source current; half the sum retains static background and offsets. Repeat after moving the return conductor or replacing it with a close paired path. A large change under lead relocation points to a lead-field background rather than a gap-flux contribution. Probe calibration requires amplitude, phase, and spatial-response information. A finite Hall sensor or pickup loop averages magnetic field over its active area. The average can differ from the field at its stated center when the probe spans a strong radial gradient. Calibrate the active-area response in a known field, align the sensitive axis with the predicted azimuthal direction, and propagate positioning uncertainty into the field comparison. A displacement of one millimetre can matter near a small capacitor axis because the predicted field changes with radius. An uncertainty budget should separate current-shunt calibration, voltage-channel calibration, plate radius, probe radius, probe orientation, active-area averaging, return-lead geometry, dielectric response, and model truncation. Radius uncertainty is correlated across every point from one scan. Random probe noise can be reduced by repeat records; a common radius-scale error cannot. Display residuals against radius, frequency, and source polarity. Their shapes identify physical omissions: a constant offset suggests background field, a radius-dependent departure suggests fringing or probe averaging, and a frequency-dependent phase departure suggests timing or distributed propagation. The final data package contains the circuit diagram, plate geometry, return-lead layout, current and voltage waveforms, probe calibration, raw signed readings, radial coordinates, background-reversal records, material parameters, and the stated quasi-static criterion. Those records support a surface-independent Ampère–Maxwell test and distinguish the capacitor-gap source term from artifacts of the surrounding circuit. ### Dimensionless radial checks The ideal circular-plate result can be tested without building its dimensional constants into every graph. Define an inside normalized field and an outside normalized field by $$ b_{\rm in}(r)=\frac{2\pi R^2B_\phi(r)}{\mu_0 I r}, \qquad rR. $$ Each quantity has predicted value one in its respective ideal region. The normalization separates the radial law from the measured current scale. A constant offset in both normalized traces can arise from a current-monitor calibration or probe gain. A rise near the plate rim can arise from fringing, finite plate thickness, or an incorrectly assigned electrical radius. An alternating pattern that follows the source polarity often signals pickup or a probe reference error. The radius $R$ in these expressions is an electrical boundary, not automatically the outer diameter of a metal disk. Charge density rolls off near a rounded edge; guard rings, dielectric overhang, nearby shields, and the connection point alter the field region. Determine the geometry from a drawing and a dimensional survey, then treat the effective radius as a fitted parameter only when the fitting procedure is reported. A free effective-radius fit can conceal a missing return-lead field by absorbing it into the geometry. Finite-element or boundary-element modeling becomes appropriate when the apparatus has a guard ring, noncircular plates, a narrow return conductor, or a frequency near the quasi-static limit. The numerical domain must include every nearby conductor that carries appreciable current or holds a prescribed potential. A model containing the plates but omitting the return path can reproduce a plausible gap electric field while giving the wrong magnetic background at the probe. Boundary placement also matters: an artificial outer boundary placed too close can redirect electric flux and distort the calculated displacement term. Mesh refinement should follow field gradients. Fine cells belong near plate edges, feed points, narrow gaps, and probe locations; large cells can cover remote regions where the fields vary slowly. Repeat the calculation with a denser local mesh and compare the reported quantity, such as the line integral of $\vec B$ around the probe circle or the area-integrated $\partial\vec D/\partial t$. Agreement of the displayed field image alone is insufficient. The integral quantities appearing in the Ampère–Maxwell law provide the relevant convergence checks. Model-data comparison should use the same observable on both sides. A Hall probe averages one component over a finite active region; sample the calculated field with that same spatial weighting before comparing it to a probe record. A pickup loop measures time rate of magnetic flux, so its response requires integration over loop area and a frequency response correction. A pointwise simulated $B_\phi$ value and an unprocessed loop voltage are different quantities even when their units have been converted after the fact. ### Consistency fits and correlated uncertainty One radial data set can support a joint fit for probe gain, background field, and effective geometry. Let the signed measured component at radius $r_j$ be $B_j$. A central-region model can be written $$ B_j=g\,\frac{\mu_0I r_j}{2\pi R_{\rm eff}^2}+B_{\rm bg}+\epsilon_j, $$ where $g$ represents the calibrated probe-scale correction, $B_{\rm bg}$ represents source-independent background after the reversal analysis, and $\epsilon_j$ is the remaining measurement noise. Independent information about $g$ and $B_{\rm bg}$ should enter the fit as calibration constraints. Leaving all three quantities free with a short radius range can produce a good curve with unphysical parameter values. The current monitor introduces a shared uncertainty. Every normalized data point uses the same current reading, so a scale error correlates the entire radial scan. Plate radius has an even stronger effect inside the gap because the predicted field contains $R^{-2}$. Treating every point as independent would report an artificially small uncertainty in the inferred geometry. A covariance matrix or a separate common-scale term preserves these correlations. Random noise averages down through repeated records; a common shunt calibration factor remains after averaging. The conclusion should name the observable and its validity range. For example, a statement can report agreement of the central-gap quantity $B_\phi/r$ with $\mu_0I/(2\pi R^2)$ within a combined uncertainty over a stated interval of radius and frequency. A separate statement can report the measured onset of edge departure. Such wording retains the conditions under which the capacitor model was tested, the polarity convention used for $B_\phi$, and the evidence that the generalized current was surface independent. ### Time-domain checks A pulsed charge experiment provides a direct comparison between terminal current and electric-flux change. Record the current-shunt waveform $I_{\rm sh}(t)$, the capacitor voltage $V(t)$, and the magnetic-probe waveform on one synchronized time base. Integrating the shunt current gives a charge estimate, $$ Q_{\rm sh}(t)=Q_{\rm sh}(t_0) +\int_{t_0}^{t}I_{\rm sh}(t')\,\d t'. $$ A linear capacitor within a stated frequency range permits comparison of that estimate with $C V(t)$. The difference exposes leakage, dielectric absorption, baseline drift, or an incorrect current-shunt zero. In the vacuum parallel-plate model, the same charge record determines the gap flux through $\Phi_E=Q/\varepsilon_0$. The displacement-current integral then follows the measured charge rate instead of a numerical derivative extracted from a noisy voltage trace. A derivative amplifies high-frequency measurement noise. Low-pass filtering a voltage record can suppress noise while also changing rise time and phase, which alters the inferred $\d\Phi_E/\d t$. The current-integral route has a different sensitivity: shunt offset accumulates into a charge drift. Measure a zero-current interval before and after the pulse, fit the offset over that interval, and retain the correction with the charge analysis. The current and voltage routes should both appear in the final record because their error structures differ. Compare the magnetic trace with the modeled circulation over the full charge pulse. In the central gap, divide the signed probe reading by radius and compare it with the simultaneously measured current after applying the probe transfer function. A time lag that remains after instrument-delay correction can arise from dielectric response, cable propagation, or a return-path field. A lag that reverses when only the probe cable polarity is exchanged belongs to the measurement chain. Pulse duration must exceed the response time of the current shunt, voltage divider, magnetic probe, and digitizer. A probe with a bandwidth lower than the pulse spectrum can round the magnetic waveform and create a spurious disagreement with $\d\Phi_E/\d t$. State the sampling rate, analog bandwidth, trigger source, record length, and filter transfer function. A valid comparison uses the same frequency content in the theoretical waveform and in the calibrated measurement channels. Charge conservation gives an integral end-of-pulse check. When the source current returns to zero and leakage is negligible, the final integrated shunt charge equals the charge inferred from the final voltage. During the rising edge, the gap displacement current equals the lead current for the full plate surface. During the falling edge, both reverse. A magnetic signal that fails to reverse with the charge rate indicates background pickup, an orientation error, or a source path outside the assumed geometry. These time-domain checks make the surface-independent source balance testable without assigning a flow of free charge through the empty gap.