Charge and Conductors

Electric charge is an intrinsic property of matter. Protons carry $+e$ and electrons carry $-e$, where the elementary-charge magnitude is

An isolated object's net charge has the form $Q=Ne$, with integer $N$. Macroscopic charge appears continuous because one coulomb contains approximately $6.24\times 10^{18}$ elementary charges. Quantization does not imply that every charging process transfers one electron at a time; large collections of electrons can move through a conducting path.

Charge is conserved. Friction, contact, and induction redistribute charge among objects or exchange it with a ground connection; they do not create net charge from nothing. If two initially neutral objects are rubbed together and one acquires $-Q$, the other acquires $+Q$ when the pair is isolated. A grounded object is not an isolated system, since Earth can supply or remove charge through the grounding path.

Conductors contain mobile charge carriers. In ordinary metals, conduction electrons move through a fixed positive-ion lattice. An applied electric field shifts those electrons until electrostatic equilibrium is reached. Insulators also contain charge, but their electrons remain bound to atoms or molecules; an applied field can distort those bound charge distributions without permitting charge to move through the body.

Both materials can hold net charge; the distinction is carrier mobility. A conductor in electrostatic equilibrium has no electric field in its bulk, since a nonzero field would continue to accelerate mobile carriers. Excess conductor charge resides at its surface. An insulator can retain localized deposited charge because its carriers do not redistribute freely.

Charge transfer as a conservation statement

The system boundary determines the charge balance. A glass rod, cloth, and ground form different systems in contact charging and grounding experiments. For the isolated rod-cloth pair,

The sign convention records electron transfer directly. If $n$ electrons move from cloth to rod, $\Delta Q_{\mathrm{rod}}=-ne$ and $\Delta Q_{\mathrm{cloth}}=+ne$. Grounding adds Earth to the system boundary; charge can then cross the wire while total charge of rod, wire, and Earth remains conserved.

In ordinary solids, transferred charge is carried by electrons. A positive cloth charge is an electron deficit relative to neutrality.

Conductors in electrostatic equilibrium

Mobile charge in a conductor continues moving while a bulk electric field remains. Electrostatic equilibrium therefore requires

Excess charge resides on the surface, where its distribution produces zero field inside the conducting material. The surface is equipotential; a tangential field would drive charge along it and contradict equilibrium.

The exterior field need not be uniform. Its normal direction at the conducting surface follows from equilibrium, while its magnitude depends on local surface-charge density and is treated with Gauss's law.

Insulators and polarization

Bound charge in an insulator can shift slightly in an applied electric field. A small induced dipole is commonly represented by

where $\alpha$ is polarizability. The material remains neutral when the positive and negative bound charges merely separate by a small distance.

Polarization charge appears at material boundaries because induced dipoles do not cancel there. It changes the internal field and can attract a nearby charge in a nonuniform field even when the insulator is neutral.

Induction and grounding

A nearby charged object polarizes a conductor. Grounding supplies a path for one sign of mobile charge to leave or enter; removing the ground before removing the inducing charge leaves the conductor with net charge opposite to the inducer.

If the inducing charge is removed before the ground connection is broken, electrons return through the wire and the conductor is left neutral.

Electroscope detection.

An electroscope detects charge through repulsion of like charge on its conducting leaves. Contact or induction redistributes charge from the terminal through the stem to both leaves; their separation increases as their mutual electrostatic repulsion increases.

Leaf divergence is not a direct charge measurement without calibration. Geometry, leaf mass, humidity, and leakage affect the angle. It still indicates the presence of charge and supports induction demonstrations.

Conductor cavities and shielding

An empty cavity entirely enclosed by a conductor has zero electric field in electrostatic equilibrium. A Gaussian surface within the conducting material has zero flux and encloses no net charge; external fields redistribute outer-surface charge without penetrating the cavity.

A charge placed inside the cavity changes the result: induced charge appears on the inner wall, and the exterior surface adjusts to conserve the conductor's total charge.

Curvature and surface charge.

Surface charge density is generally nonuniform. High curvature regions require a larger normal exterior field to maintain one conductor potential; electrostatic equilibrium therefore concentrates charge near a sharp point.

The enhanced field near a sharp tip can initiate air breakdown. This is a local boundary effect; charge does not reside only at tips.

Charging by induction.

Induction produces retained charge without contact. A positive inducer polarizes a grounded conductor; electrons enter from Earth. Disconnecting the ground while the inducer remains fixes the excess electrons on the conductor, and removing the inducer last leaves net negative charge.

The final conductor charge is opposite the inducing charge. Reversing the removal order allows the added electrons to return to Earth, restoring neutrality rather than charging the conductor.

Capacitor surface charge.

Connecting a voltage source transfers electrons until opposite conductor surfaces hold equal charge magnitude. For broad parallel plates, the free surface-charge density is $\sigma =Q/A$ and the central electric field is approximately

Charge resides on conductor surfaces because the electrostatic field inside each plate is zero. The equal and opposite charges do not annihilate across the gap: their separation stores energy in the electric field between the plates.

Faraday-cage field mapping

An electrostatic conductor shields an enclosed empty region because its mobile charge adjusts until the conductor and cavity boundary are equipotential. The interior field condition is

when no charge is placed in the cavity. The exterior field is distorted around the shell; an empty cavity remains field-free.

This result applies to electrostatic fields. Openings, finite conductivity, and time-varying fields require an electromagnetic treatment.

Charge relaxation.

Free charge in a conducting material redistributes on the relaxation timescale

where $\epsilon$ is permittivity and $\sigma$ is conductivity. A bulk charge density decays as $\rho (t)={\rho }_0{e}^{-t/\tau }$ in the simple homogeneous model.

Metals have extremely short relaxation times, so electrostatic equilibrium is often an accurate approximation after ordinary switching transients. Poor conductors and dielectrics retain localized charge far longer; the relaxation model then separates conductivity-driven redistribution from bound-charge polarization.

Charge-conservation measurement.

A Faraday cup transfers an inserted object's charge to an enclosing conductor, where an electrometer measures the resulting potential or charge. Repeated measurements of an isolated transfer obey

The instrument does not create charge. It changes the accessible conductor geometry and detects the redistribution required by electrostatic equilibrium; leakage paths and incomplete insertion are experimental sources of charge-balance error.

Surface-charge persistence in insulators.

Deposited charge on an insulator remains localized when bulk conductivity is small. The local surface density may decay through leakage approximately as

with a leakage time set by material, humidity, and available conduction paths.

The persistence is not absolute. Contamination, moisture, and ionized air provide leakage paths that reduce charge over time. The contrast with a conductor is the timescale and mobility of carriers, not the absence of charge in the material.

Grounded-conductor field termination.

Ground fixes a conductor's potential relative to Earth and permits charge exchange. An ideal grounded conductor obeys

while induced surface charge adjusts so that field lines terminate normally on the surface rather than entering the conducting bulk.

The grounded plane removes the need to specify its total charge in advance. The charge is determined by the boundary condition and can flow to or from Earth until the potential condition and zero interior field are both satisfied.

Normal-field discontinuity.

A surface charge changes the normal electric field across a boundary. A thin Gaussian pillbox gives

In a conductor at electrostatic equilibrium, $E_{\perp ,\mathrm{in}}=0$, so the exterior normal field is $\sigma /{\epsilon }_0$.

The boundary condition concerns the normal component. Tangential field at an ideal conductor surface vanishes in electrostatic equilibrium, since otherwise surface charges would continue to move.

Microscopic charge counts and macroscopic neutrality

The coulomb measures net imbalance, not the total charge carried by all electrons in an object. Current fixes its operational scale: a current of $1\mathrm{A}$ carries $1\mathrm{C}$ through a cross section in $1\mathrm{s}$. The elementary charge is so small that laboratory charges contain enormous integer counts:

Neutrality is a cancellation between large positive and negative contributions. A copper atom has $Z=29$ protons and, when neutral, 29 electrons. Removing one electron changes the atom's charge by $+e$; adding one changes it by $-e$. The nucleus does not have to move for a metal object to become positively charged. An electron deficit relative to the neutral state produces the positive net charge.

The total electronic charge in a neutral object sets a scale for the net-charge imbalance. A $3.10\mathrm{g}$ copper penny contains

and hence $N_e=29N_{\mathrm{at}}=8.53\times 10^{23}$ electrons. Their charge is $-1.37\times 10^5\mathrm{C}$, balanced by the nuclei to leave zero net charge. Transferring $10^{10}$ electrons during ordinary rubbing changes this enormous inventory by a tiny fractional amount while creating an easily measurable external electric effect.

Contact sharing between identical conductors

Equal sharing relies on equal geometry and an identical electrical environment. Spheres of different radii reach the same potential during contact, but their final charges generally differ in magnitude. Nearby charged bodies also remove the exchange symmetry by polarizing the connected pair.

The word contact names the physical connection, rather than a requirement that the two bodies become one solid object. A wire joining separated spheres supplies the same carrier path. An insulating support does not. Charge can redistribute only while a conducting route remains available before the bodies are separated.

Induction with two initially neutral spheres.

Induction separates charge without transferring charge from the external rod to the spheres. Begin with two neutral identical metal spheres touching. A positive rod near sphere $A$ attracts conduction electrons from sphere $B$ toward $A$. At this stage the pair remains neutral:

Separating the spheres while the rod remains nearby traps the imbalance on each isolated sphere. If $n$ electrons have shifted from $B$ to $A$, the final charges are

Removing the rod after separation allows each sphere's own surface charge to spread more uniformly, while preserving its net charge. Removing the rod before the spheres are separated produces a different final state: the connected conductors restore a neutral distribution because the carrier path remains available. The order of operations therefore follows directly from the available conducting paths and charge conservation.

Grounding as an enlarged system boundary.

Earth is treated as a very large conductor in electrostatic demonstrations. A wire from a small sphere to ground permits electrons to cross the boundary of the small sphere's system. A positive rod near the sphere first creates an electron-rich near face and an electron-poor far face. Connecting the far side to Earth supplies electrons; after the wire is disconnected, the small sphere carries a negative net charge.

Charge conservation applies to the larger sphere-wire-Earth system. Earth receives the corresponding positive charge when electrons leave it. Treating the ground as a reservoir is accurate when the transferred charge is tiny compared with Earth's charge capacity and the connecting path has time to equilibrate. A floating metal table, an insulated person, or an isolated laboratory apparatus cannot be substituted for ground without including its finite charge and geometry in the calculation.

Electroscope readings.

An electroscope contains a conducting terminal, a conducting stem, and two thin conducting leaves isolated from the enclosure. Charge placed on the terminal spreads through the connected metal. Both leaves acquire charge of the same sign and repel, so their separation provides a visible indication of a nonzero charge distribution. The instrument responds to charge already on it and to nearby external charge that polarizes it; those two situations require different interpretations.

In a symmetric leaf model, let each leaf carry charge $q_{\ell }$, let their separation be $s$, and let each leaf have mass $m$. The mutual electrostatic force has magnitude

If each leaf makes angle $\theta$ with the vertical stem, static force balance gives

so that

The separation $s$ itself changes with the angle, so the leaf angle is not linearly proportional to total charge. Leaf length, hinge stiffness, ambient humidity, and the terminal geometry also enter a calibration. A leaf electroscope is therefore a sensitive qualitative detector unless it has been calibrated against known charges.

Contact charging and induction make the leaves diverge for different reasons. Touching a charged rod to the terminal permits net charge transfer. Bringing a rod near without touching redistributes the electroscope's existing charge: the terminal region becomes enriched in the charge attracted by the rod, while the leaves receive the corresponding displaced charge. A neutral electroscope can show leaf divergence during this approach and return to its original state once the rod is removed.

Sign testing starts with an electroscope charged to a known sign. A rod with the same sign repels the charge already on the terminal toward the leaves and increases their separation. A rod with the opposite sign draws that terminal charge upward and reduces the leaf separation. The comparison must be made without contact; contact changes the electroscope's net charge and loses the diagnostic reference state.

Slow leaf collapse after a charging experiment records charge leakage rather than a separate electrostatic force. Moist air, surface contamination, and imperfect insulators provide conducting paths from the terminal to the enclosure or Earth. The leakage rate varies enough with the apparatus that repeated readings need a fixed waiting time and comparable environmental conditions. A hand near the terminal also changes the surrounding charge distribution by induction, which is why electroscope procedures keep hands and other conductors at controlled distances.

Experimental charge checks.

Electrostatic demonstrations separate three claims that are often conflated: a body has acquired net charge, its charge has merely redistributed, or charge has crossed a ground connection. Each claim has a different conservation statement. A contact experiment starts with a known total charge and compares the separated conductors. An induction experiment starts with a neutral isolated pair and checks that their final charges sum to zero. A grounding experiment expands the system boundary to include Earth, so the small conductor's nonzero final charge is balanced by an equal and opposite change in Earth.

For two collectors with measured charges $Q_1$ and $Q_2$, the balance residual is

An ideal isolated transfer gives $R=0$. In a real measurement, the result is judged against the uncertainty of both readings and the charge expected to leak through the supports or humid air. A residual with a stable sign usually points to a systematic path or an incomplete transfer rather than a failure of charge conservation.

The identical-sphere experiment has a sharp prediction. If a charged sphere of known charge $Q$ touches an initially neutral identical sphere in a symmetric environment, each separated sphere should read $Q/2$. A large difference between the readings can arise from unequal sphere radii, a nearby charged rod, unequal insulating supports, or separating the spheres before the charge has settled. The comparison tests both the conducting path and the symmetry assumption used in the derivation.

The two-sphere induction experiment has a different signature. After a positive rod is brought near a touching neutral pair and the pair is separated, the near sphere should carry negative charge and the far sphere positive charge. The magnitudes agree only when the pair began neutral and no charge escaped. Measuring a nonzero sum $Q_A+Q_B$ identifies leakage, accidental grounding, or a preexisting charge on one of the spheres. The signs alone do not establish the mechanism; the order of rod approach, separation, and removal establish that evidence.

Grounding tests require a conductor with a verified route to Earth. A thin oxide layer, paint, dry skin, or an insulating floor can leave the nominal ground path open in a circuit diagram but ineffective in the apparatus. A continuity measurement and an electroscope discharge test establish whether the path conducts. Once the path is closed, a positive external rod near the conductor should draw electrons from Earth into the conductor. Opening the ground before withdrawing the rod preserves the negative charge. Reversing the sequence returns the apparatus toward neutrality.

The ideal diagrams assume quasistatic motion, negligible air ionization, and a conducting path with time to equilibrate. Rapid motion, sharp high-voltage electrodes, or a spark introduce charge transfer through air and require a larger system boundary. Account for every path by which charge can cross the selected system boundary.

The charge remains after the source is removed because the conductor is then isolated. If grounding were removed last, electrons would flow back until the conductor became neutral; the trace therefore tests the physical order rather than a memorized sign. Electrostatic equilibration in a conductor.

A metal contains conduction electrons with number density $n$. An electric field inside the bulk exerts force $-e\vec{E}$ on each carrier. In the elementary Drude description, collisions transfer this momentum to the lattice and produce a current density

Here $\tau$ is an effective mean time between momentum-randomizing collisions. Static equilibrium requires $\vec{J}=0$. A finite-conductivity metal therefore rearranges charge until the bulk field vanishes. Any residual field would maintain carrier drift, so static equilibrium requires a zero bulk field.

The rearrangement time follows from charge conservation and Ohm's law. Combining $\partial \rho /\partial t+\nabla \cdot \vec{J}=0$ with $\nabla \cdot \vec{E}=\rho /\epsilon$ gives

Thus a bulk excess-charge density decays as

In a good metal, ${\tau }_r$ is extremely short. Surface charge is the remaining distribution compatible with zero interior field and the imposed external boundary conditions. The charge-transfer and electrostatic-equilibrium figures therefore describe two stages: carrier motion during transient adjustment, then a static surface distribution.

Polarization without conduction.

An insulator responds to a field through displacement of bound positive and negative charge. The dipole moment of one polarized molecule is $\vec{p}$; the macroscopic polarization is dipole moment per volume,

In a linear isotropic dielectric, $\vec{P}={\epsilon }_0{\chi }_e\vec{E}$. The induced bound charge appears through

These relations distinguish polarization from transferred free charge. A neutral dielectric slab placed in a uniform field has equal and opposite bound surface charges, so its total charge remains zero. Polarization separates the centres of positive and negative charge; it does not carry electrons across the sample from one face to the other.

Permanent molecular dipoles and induced electronic dipoles have different microscopic origins. Both contribute to $\vec{P}$, and both reduce the field inside a dielectric relative to the field created by the free source charges. The reduction depends on material response, temperature, and field strength; the linear relation is an approximation, not a universal identity. Induction as a boundary-value problem.

Induction changes a conductor's surface-charge distribution without transferring charge across an insulating gap. A positive external source attracts conduction electrons toward the near surface and leaves positive excess charge on the far surface. For an isolated conductor, the induced charges sum to zero:

Grounding changes that condition. The conductor and Earth become one conducting system, and electrons can move through the ground wire while the external source is present. Removing the ground connection before removing the source can leave a nonzero net charge on the conductor. Reversing that order restores the initial neutral state. The induction sequence follows this order of operations, which is part of the physical specification.

A closed conducting shell provides electrostatic shielding when it reaches equilibrium. A Gaussian surface contained entirely in the conducting material has $\vec{E}=0$ everywhere, hence its enclosed charge is zero. A charge placed in an empty cavity induces charge on the inner wall. The outer-surface charge then adjusts to satisfy the shell's specified total charge. Shielding applies to static or slowly changing fields after charge rearrangement. It does not imply that an arbitrary time-varying electromagnetic field is excluded from every enclosure.

Faraday-cup measurement and uncertainty.

A Faraday cup transfers an unknown charge to an enclosed conductor. The cup and electrometer form a capacitance $C_m$, so the measured potential change gives

The measurement is indirect: $C_m$ must be calibrated, leakage must be negligible over the acquisition time, and the cup must enclose the transferred charge. If uncertainties in capacitance and voltage are independent, first-order propagation gives

A Faraday cup is a conservation test as much as a charge detector. Repeated transfers with a known electron count test linearity, reversing the transfer direction checks the sign convention, and a slow drift with an empty cup indicates leakage or offset current. Surface field and curvature.

The field just outside a conductor follows from a pillbox Gaussian surface that straddles the material boundary. The flux through the side wall vanishes as its height tends to zero. The interior face contributes nothing in equilibrium. The outer face gives

Only the normal component can remain at the surface. Any tangential component would drive surface carriers and violate electrostatic equilibrium. The normal-field-discontinuity figure represents this boundary condition: field magnitude is set locally by surface-charge density, while field direction is normal to the conducting surface.

Surface curvature affects $\sigma$ through the global boundary-value problem. A sharply curved protrusion generally carries a larger surface charge density than a broad nearly planar region at the same conductor potential. The local field becomes correspondingly larger. This concentration explains corona discharge near sharp electrodes, but curvature alone is not an independent formula for field strength; the surrounding conductors, total charge, and applied potentials determine the complete solution.

A conductor with a cavity and no charge inside has zero cavity field in electrostatic equilibrium. With a cavity charge $q$, the inner surface carries total induced charge $-q$. The result follows from a Gaussian surface lying in the metal around the cavity. If the conductor's total charge is $Q$, its outer surface carries $Q+q$. These integrated charges constrain the solution but do not specify the local density $\sigma (\vec{r})$ on an irregular surface. Spherical conductor as a boundary-value calculation.

A conducting spherical shell of radius $R$ carrying total charge $Q$ makes the surface-charge and shielding statements quantitative. Spherical symmetry requires a radial exterior field whose magnitude depends only on $r$. A Gaussian sphere of radius $r>R$ encloses $Q$, so

A Gaussian surface with $r<R$ lies in the empty cavity. It encloses no charge, but zero enclosed charge alone does not prove that the field vanishes; symmetry and the absence of interior sources do. For a conducting shell in electrostatic equilibrium with no cavity charge, the field is zero throughout the cavity and throughout the metal:

The surface field has magnitude $Q/(4\pi {\epsilon }_0{R}^2)$. Comparison with the boundary relation $E_{\perp }=\sigma /{\epsilon }_0$ gives the uniform surface density

This is a special consequence of spherical symmetry. An irregular conductor at the same total charge has a nonuniform surface density and cannot be represented by $Q/(4\pi R^2)$.

Place a point charge $q$ at the centre of the cavity. The metal remains field-free, so a Gaussian surface in the metal must enclose zero net charge. The inner surface therefore carries $-q$. If the shell's initial net charge is $Q$, charge conservation gives outer-surface charge $Q+q$. The induced density is uniform only for a central charge. Off-centre cavity charges induce a strongly nonuniform inner distribution, although the integrated inner charge remains $-q$.

The same calculation distinguishes shielding from simple force cancellation. The vanishing cavity field follows from a conductor's mobile carriers and the electrostatic boundary condition; it does not follow merely from adding a few oppositely directed Coulomb-force vectors. A time-dependent source can penetrate before charge redistribution is complete, and apertures or finite conductivity alter the ideal result. Electrostatic pressure and mechanical force.

Surface charge does more than set the boundary field. It also produces a mechanical stress on the conductor. Consider a small flat patch with surface density $\sigma$. The total field immediately outside is $\sigma /{\epsilon }_0$, but the field acting on the charge within the patch excludes the patch's own contribution. An infinitesimally thin locally flat sheet gives half the total exterior field. The outward force per unit area is therefore

This electrostatic pressure tends to pull a charged conductor outward. The same result follows from energy. At fixed charge, increasing the separation of two oppositely charged plates lowers their capacitance and raises field energy; an external agency must supply work against their attraction. At fixed voltage, a battery exchanges energy with the field, so force calculations require the full system: plates, source, and field.

A parallel-plate capacitor of area $A$, separation $d$, and negligible fringing has $C={\epsilon }_{0}A/d$. Holding charge fixed gives

The negative sign indicates attraction: the force reduces $d$. Dividing by area recovers $p_e={\sigma }^2/(2{\epsilon }_0)$. This calculation connects the surface-charge figure to a measurable force instead of treating $\sigma$ as a purely geometric label.

Charge conservation in a network of conductors

For several isolated conductors, charge conservation applies to the entire collection:

Electrostatic equilibrium imposes a further condition on each connected conducting component: it has one potential. A wire connecting two initially charged metal spheres allows carriers to move until both spheres have the same potential. For well-separated spheres with radii $R_1$ and $R_2$,

With $V_1=V_2$ and $Q_1+Q_2=Q_{\mathrm{total}}$,

The larger sphere receives more charge because equal potential requires charge proportional to radius in this far-separation approximation. Equal charge is not the equilibrium condition. Equal potential is.

The approximation fails when the spheres are close enough for each sphere's field to alter the other's potential appreciably. In that case the charges and potentials are related by capacitance coefficients,

The off-diagonal terms encode mutual influence. This is the electrostatic predecessor of the capacitance-network calculations developed later.

Limits of the ideal-conductor model

The statement $\vec{E}=0$ inside a conductor assumes electrostatic equilibrium. Finite conductivity, finite observation time, and time-varying sources alter the response. At low frequencies a conductor screens fields over a short relaxation time. At higher frequencies, induction and magnetic fields cannot be ignored; the relevant penetration scale becomes the skin depth rather than the static relaxation time.

A conductor also ceases to be an ideal electrostatic boundary when the exterior field ionizes nearby gas or exceeds a material's dielectric strength. Corona discharge near a sharp point removes charge through the surrounding medium. Breakdown begins most readily at a sharp point, where high curvature raises the local $E_{\perp }$ until the air, not the metal, becomes the limiting material.

Conducting shell with a central cavity charge

A conducting shell has inner radius $a$, outer radius $b$, and initial net charge $Q$. A point charge $q$ is placed at the centre of the empty cavity. Electrostatic equilibrium requires zero field in the conducting material, so a Gaussian surface with $a<r<b$ encloses zero net charge:

Charge conservation fixes the outer-surface charge:

Spherical symmetry makes both induced surface densities uniform,

Outside the shell, the field and potential are those of net charge $Q+q$ at the centre:

Within the metal, $\vec{E}=0$ and the potential is constant,

The cavity potential includes the point-charge term and the constants generated by both charged surfaces. Its radial derivative still gives the point-charge field, $E_r=q/(4\pi {\epsilon }_0{r}^2)$ for $0<r<a$. The constant offset affects potential values but not the force on a test charge in the cavity.

For $Q=0$, a central charge $q$ induces $-q$ on the inner wall and $+q$ on the outer wall. The shell remains neutral overall, while its exterior field is the same as that of $q$ alone. Grounding changes the condition $Q_{\mathrm{outer}}=Q+q$: Earth can provide charge, and the outer charge becomes whatever value enforces the grounded potential boundary. Potential throughout the cavity.

The central-charge shell calculation has spherical symmetry. It therefore gives a closed form for potential at every radius. For $0<r<a$,

The first term is the point-charge potential. The second term is the constant potential of the uniformly charged inner surface, and the third term comes from the outer surface. Differentiation gives

The surface terms disappear from the field because they are constants inside a spherical shell. They remain essential when comparing the potential of the cavity to the potential of the metal or to infinity.

An off-centre cavity charge breaks spherical symmetry. The inner induced charge still integrates to $-q$, but its density is largest on the nearby wall. The outer surface carries total charge $Q+q$. Neither surface density is uniform, and the elementary Gaussian-surface argument no longer determines the local field in the cavity. The boundary condition is instead

for an isolated shell with the usual reference. Solving the off-centre problem requires a boundary-value method such as image charges or a multipole expansion. The central result should not be extrapolated by replacing $r$ with an arbitrary distance to the cavity wall.

Energy of charging a conductor.

An isolated conducting sphere of radius $R$ has potential

Bringing charge from infinity in increments $\mathbf{d}Q$ requires work $V(Q)\mathbf{d}Q$. Integration gives the self-energy

The same result has a field interpretation. Exterior field energy is

Substitution produces $Q^2/(8\pi {\epsilon }_{0}R)$. Agreement between the charging work and field-energy integrals checks both the inverse-square field and the potential reference.

A sphere near another conductor cannot use this isolated-sphere energy without correction. The neighbour changes surface charge distribution and introduces mutual electrostatic energy. Capacitance becomes a relation among several potentials and charges rather than $Q/V$ for one isolated object. Charge relaxation and electromagnetic screening.

Charge relaxation and skin depth describe different physical limits. In the electrostatic regime, a bulk charge density obeys

The relaxation time measures how quickly a conductor removes a bulk electric field after a static charge imbalance is introduced. For copper, $\sigma \approx 5.8\times 10^7\mathrm{S}{\mathrm{m}}^{-1}$ and $\epsilon \approx {\epsilon }_0$, giving

This value explains why static equilibrium is effectively immediate on ordinary laboratory time scales. It does not describe penetration of an alternating electromagnetic wave.

For sinusoidal fields of angular frequency $\omega$, conduction and induction produce a skin depth

A field entering a thick good conductor decays approximately as $\exp (-x/\delta )$. At $60\mathrm{Hz}$ in copper, $\delta \approx 8.5\mathrm{mm}$; at radio frequency it can be much smaller. The static statement $\vec{E}=0$ in a conductor is therefore a limiting boundary condition. A finite-frequency field occupies a surface layer and drives currents within it.

The two scales answer different questions. Use ${\tau }_r$ when a charge configuration is allowed to settle. Use $\delta$ when a time-varying field is specified. Mixing them treats a dynamic electromagnetic problem as though it were electrostatics. Conductor–dielectric boundary conditions.

At a boundary between a conductor and a linear dielectric, the electric field and displacement field separate free charge from polarization charge. A pillbox crossing the interface gives

In electrostatic equilibrium ${\vec{E}}_{\mathrm{cond}}=0$, hence ${\vec{D}}_{\mathrm{cond}}=0$ for an ordinary conductor. The normal displacement in the dielectric is therefore

In a homogeneous linear dielectric, $\vec{D}=\epsilon \vec{E}$, so

The free charge ${\sigma }_f$ resides on the conductor. The dielectric also develops bound surface charge, determined by polarization:

These charges have distinct origins. Free charge has crossed a conducting path or was deposited on a conductor. Bound charge represents shifted positive and negative charge centres in the dielectric. Treating their sum as a single surface density obscures how changing the dielectric changes field while a fixed conductor charge remains fixed.

A parallel-plate capacitor filled with dielectric permittivity $\epsilon$ has free plate charge densities $+{\sigma }_f$ and $-{\sigma }_f$. The field between the plates is

At fixed free charge, inserting a dielectric lowers $E$ and $V$ by the relative permittivity. At fixed voltage, the source supplies additional free charge until ${\sigma }_f=\epsilon V/d$. These two conditions produce different energy and force changes; a capacitor problem is incomplete until it states whether the plates are isolated or connected to a voltage source.