---
title: Charge and Conductors
module: Electric Fields
moduleNumber: 1
lessonNumber: 1
order: 101
summary: >
  Rub two objects together and one pulls electrons from the other; nothing is
  created, only moved. We define what electric charge is — conserved, additive,
  and quantized in units of $e$ — and why a conductor's mobile carriers rearrange
  until its interior field vanishes and its surface sits at one potential. We follow
  charge through contact, induction, and grounding, treat the field-free cavity that
  turns a conductor into a shield, and mark where finite conductivity and leakage set
  the limits of the electrostatic picture.
topics: [Electric Fields]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 21 — The Electric Field I; §§21-1–21-2 Charge; Conductors and Insulators"
---

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

$$
e=1.602\times10^{-19}\ \mathrm C.
$$

An isolated object's net charge has the form $Q=Ne$, with integer $N$. Macroscopic
charge appears continuous because one coulomb contains
approximately $6.24\times10^{18}$ 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,

$$
\Delta Q_{\rm rod}+\Delta Q_{\rm cloth}=0.
$$

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

$$
% caption: Contact electrification as charge bookkeeping. Transferring $n$
% electrons from the cloth to the rod leaves the rod at $-ne$ and the cloth at
% $+ne$; the dashed boundary encloses the isolated pair, whose total charge
% stays zero.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, dashed] (-1.7,-1.55) rectangle (6.3,1.55);
  \node[black, anchor=north] at (2.3,1.5) {isolated pair: total charge zero};
  \node[draw=black, fill=black!8, minimum width=22mm, minimum height=13mm] (rod) at (0,0) {rod};
  \node[draw=black, fill=black!8, minimum width=22mm, minimum height=13mm] (cloth) at (4.6,0) {cloth};
  \draw[->, acc, very thick] (cloth.west) -- (rod.east) node[midway, above] {n electrons};
  \node[black, anchor=north] at (0,-0.9) {net negative};
  \node[black, anchor=north] at (4.6,-0.9) {net positive};
\end{tikzpicture}
$$

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

$$
\vec E_{\rm bulk}=\vec 0.
$$

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.

$$
% caption: A conductor in electrostatic equilibrium. Mobile charge has
% rearranged onto the surface so the bulk field vanishes ($\vec E=\vec0$); just
% outside, the field meets the surface along the normal, its magnitude set by the
% local surface-charge density.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill[black] (0,-0.9) rectangle (5,0.9);
  \draw[black] (0,-0.9) rectangle (5,0.9);
  \node at (2.5,0) {E = 0 in the bulk};
  \foreach \x in {0.7,1.5,2.3,3.1,3.9,4.7}{
    \node[black] at (\x,1.02) {+};
    \draw[->, acc, thick] (\x,1.2) -- (\x,1.85);
  }
\end{tikzpicture}
$$

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

$$
\vec p=\alpha\vec E,
$$

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

$$
% caption: A molecule polarized by a uniform field $\vec E$ (pointing right). The
% bound electron cloud shifts left and the nucleus right, separating their centres
% into an induced dipole moment $\vec p=\alpha\vec E$ aligned with the field; the
% molecule stays neutral.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \y in {-1.0,-0.5,0,0.5,1.0}{\draw[->, black] (-3.1,\y) -- (-2.0,\y);}
  \node[black, anchor=south] at (-2.55,1.05) {applied E};
  \fill[black!8] (-0.15,0) ellipse (1.1 and 0.65);
  \draw[black] (-0.15,0) ellipse (1.1 and 0.65);
  \node[black, anchor=south] at (-0.15,0.66) {electron cloud};
  \filldraw[draw=black, fill=black!20] (-0.55,0) circle (0.08);
  \filldraw[draw=black, fill=black!8] (0.6,0) circle (0.16);
  \node[black!70] at (0.6,0) {+};
  \node[black, anchor=west] at (0.8,0) {nucleus};
  \draw[->, acc, very thick] (-0.55,-1.15) -- (0.6,-1.15) node[midway, below] {dipole p};
\end{tikzpicture}
$$

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.

$$
% caption: Charging by induction. (1) A positive inducer draws conduction
% electrons to the near face, leaving the far face positive. (2) Grounding the far
% side lets more electrons flow in from Earth. (3) Breaking the ground before
% removing the inducer leaves the conductor with a net negative charge $-ne$.
% Electrons are drawn as small filled dots and positive excess as plus signs.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % panel 1
  \fill[black] (0,-0.6) rectangle (1.7,0.6);
  \draw[black] (0,-0.6) rectangle (1.7,0.6);
  \node[black, anchor=south] at (0.85,0.64) {1};
  \filldraw[draw=black, fill=black!8] (-0.6,0) circle (0.16);
  \node[black!70] at (-0.6,0) {+};
  \foreach \y in {-0.28,0,0.28}{\fill[acc] (0.28,\y) circle (0.055);}
  \foreach \y in {-0.28,0,0.28}{\node[black] at (1.42,\y) {+};}
  % panel 2
  \begin{scope}[xshift=3.7cm]
    \fill[black] (0,-0.6) rectangle (1.7,0.6);
    \draw[black] (0,-0.6) rectangle (1.7,0.6);
    \node[black, anchor=south] at (0.85,0.64) {2};
    \filldraw[draw=black, fill=black!8] (-0.6,0) circle (0.16);
    \node[black!70] at (-0.6,0) {+};
    \foreach \y in {-0.28,0,0.28}{\fill[acc] (0.28,\y) circle (0.055);}
    \draw[black] (1.7,-0.42) -- (2.5,-0.42);
    \draw[black] (2.3,-0.55) -- (2.7,-0.55);
    \draw[black] (2.38,-0.66) -- (2.62,-0.66);
    \node[black, anchor=north] at (2.5,-0.68) {earth};
    \draw[->, acc, thick] (2.55,0.3) -- (1.82,0.3) node[midway, above] {electrons};
  \end{scope}
  % panel 3
  \begin{scope}[xshift=7.7cm]
    \fill[black] (0,-0.6) rectangle (1.7,0.6);
    \draw[black] (0,-0.6) rectangle (1.7,0.6);
    \node[black, anchor=south] at (0.85,0.64) {3};
    \foreach \x/\y in {0.4/0.25,0.85/-0.12,1.3/0.25,0.6/-0.28,1.05/0.12}{\fill[acc] (\x,\y) circle (0.055);}
    \node[acc, anchor=north] at (0.85,-0.64) {net negative};
  \end{scope}
\end{tikzpicture}
$$

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

$$
E=\frac{\sigma}{\varepsilon_0}.
$$

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

$$
\vec E_{\rm cavity}=\vec 0
$$

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

$$
\tau=\frac{\varepsilon}{\sigma},
$$

where $\varepsilon$ is permittivity and $\sigma$ is conductivity. A bulk charge
density decays as $\rho(t)=\rho_0e^{-t/\tau}$ 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

$$
Q_{\rm initial}=Q_{\rm cup}+Q_{\rm remaining}.
$$

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

$$
\sigma_s(t)=\sigma_{s0}e^{-t/\tau_{\rm leak}},
$$

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

$$
V_{\rm conductor}=0,
$$

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

$$
E_{\perp,\rm out}-E_{\perp,\rm in}=\frac{\sigma}{\varepsilon_0}.
$$

In a conductor at electrostatic equilibrium, $E_{\perp,\rm in}=0$, so the exterior
normal field is $\sigma/\varepsilon_0$.

$$
% caption: A Gaussian pillbox straddling a conductor surface. Its lower face lies
% in the field-free bulk ($\vec E=\vec0$) and its upper face samples the exterior
% normal field; only the enclosed surface charge contributes flux, giving the jump
% $E_\perp=\sigma/\varepsilon_0$.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill[black] (0,-1.2) rectangle (5,0);
  \draw[black] (0,0) -- (5,0);
  \node[black, anchor=west] at (0.15,-0.85) {conductor};
  \node at (3.7,-0.6) {E = 0};
  \foreach \x in {1.9,2.3,2.7,3.1}{\node[black] at (\x,0.16) {+};}
  \draw[acc, dashed, thick] (1.7,-0.55) rectangle (3.3,0.78);
  \node[acc, anchor=east] at (1.6,-0.35) {pillbox};
  \foreach \x in {2.1,2.5,2.9}{\draw[->, acc, thick] (\x,0.16) -- (\x,1.3);}
  \node[acc, anchor=west] at (3.42,1.05) {E outside};
\end{tikzpicture}
$$

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.

> **Worked example (grounding trace).** An initially neutral conductor is charged by
> induction from a positive external source. The conductor's charge accounting is
>
> $$
> Q_{\rm final}=Q_{\rm initial}+Q_{\rm ground}=0+(-ne)=-ne,
> $$
>
> where $n$ electrons enter through the grounding path while the inducer is held near.
> The inducing charge never touches the conductor, so it transfers no charge directly;
> the net $-ne$ comes entirely from Earth through the ground wire.

## 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:

$$
n=\frac{|Q|}{e},\qquad
e=1.602177\times10^{-19}\ \mathrm C.
$$

> **Worked example (electrons in a laboratory charge).** How many excess electrons
> make up a charge of magnitude $50\ \mathrm{nC}$? Since $|Q|=ne$,
>
> $$
> n=\frac{|Q|}{e}=\frac{50\times10^{-9}\ \mathrm C}{1.602177\times10^{-19}\ \mathrm C}
> =3.12\times10^{11}\ \text{electrons}.
> $$
>
> The count is a huge integer, so the discreteness of charge is invisible at this
> scale: the uncertainty in any macroscopic charge measurement dwarfs a single
> electron, and charge behaves as a continuum in ordinary circuit and electrostatic
> work.

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

$$
N_{\rm at}=\frac{3.10\ \mathrm g}{63.5\ \mathrm{g\,mol^{-1}}}
N_A=2.94\times10^{22}\ \text{atoms},
$$

and hence $N_e=29N_{\rm at}=8.53\times10^{23}$ electrons. Their charge is
$-1.37\times10^5\ \mathrm C$, 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

> **Worked example (contact sharing between identical spheres).** A sphere carrying
> charge $Q$ touches an identical neutral sphere, held far from other charges. While
> joined, the pair acts as one conductor: electrons redistribute until the field inside
> both spheres vanishes, and the exchange symmetry (swapping the two spheres leaves the
> setup unchanged) forces equal final charges. Charge conservation then fixes them,
>
> $$
> Q_A+Q_B=Q,\qquad Q_A=Q_B\ \Rightarrow\ Q_A=Q_B=\tfrac{Q}{2}.
> $$
>
> For an initial $Q=12\ \mathrm{nC}$, each sphere ends at $6\ \mathrm{nC}$.

$$
% caption: Contact sharing between identical conducting spheres. Joined, they act
% as one conductor; symmetry and charge conservation split the charge equally, so
% each sphere carries $Q/2$ after separation.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill[black!8] (0,0) circle (0.6);
  \draw[black, thick] (0,0) circle (0.6);
  \node at (0,0) {$Q$};
  \fill[black] (1.9,0) circle (0.6);
  \draw[black] (1.9,0) circle (0.6);
  \node at (1.9,0) {$0$};
  \node[black, anchor=north] at (0.95,-0.78) {before contact};
  \draw[->, thick] (3.0,0) -- (4.2,0) node[midway, above] {touch};
  \fill[black!8] (5.1,0) circle (0.6);
  \draw[black, thick] (5.1,0) circle (0.6);
  \node at (5.1,0) {$\frac{Q}{2}$};
  \fill[black!8] (7.0,0) circle (0.6);
  \draw[black, thick] (7.0,0) circle (0.6);
  \node at (7.0,0) {$\frac{Q}{2}$};
  \node[black, anchor=north] at (6.05,-0.78) {after separation};
\end{tikzpicture}
$$

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:

$$
Q_A+Q_B=0.
$$

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

$$
Q_A=-ne,\qquad Q_B=+ne.
$$

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.

$$
Q_{\rm sphere,initial}=0,\qquad
\Delta Q_{\rm sphere}=-ne,\qquad
Q_{\rm sphere,final}=-ne.
$$

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

$$
F_e=\frac{kq_\ell^2}{s^2}.
$$

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

$$
T\sin\theta=F_e,\qquad T\cos\theta=mg,
$$

so that

$$
\tan\theta=\frac{kq_\ell^2}{mgs^2}.
$$

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.

$$
% caption: Force balance on a charged gold-leaf electroscope. Like charge on the
% two leaves gives an outward electrostatic force $F_e$; leaf weight $mg$ and the
% hinge set the divergence angle $\theta$, so the reading depends on geometry as
% well as charge.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \filldraw[draw=black, fill=black!8] (0,2.0) circle (0.34);
  \node[black, anchor=west] at (0.46,2.0) {terminal};
  \draw[black, thick] (0,1.66) -- (0,0.5);
  \draw[black, thick] (0,0.5) -- (-0.85,-1.0);
  \draw[black, thick] (0,0.5) -- (0.85,-1.0);
  \node[black] at (-1.02,-0.6) {+};
  \node[black] at (1.02,-0.6) {+};
  \draw[->, acc, thick] (-0.72,-0.72) -- (-1.62,-1.12) node[left] {$F_e$};
  \draw[->, acc, thick] (0.72,-0.72) -- (1.62,-1.12) node[right] {$F_e$};
  \draw[->, black] (-0.85,-1.0) -- (-0.85,-1.78) node[below] {$mg$};
\end{tikzpicture}
$$

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

$$
R=Q_1+Q_2-Q_{\rm initial}.
$$

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

$$
\vec J=\sigma\vec E,
\qquad
\sigma=\frac{ne^2\tau}{m_e}.
$$

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
$\nabla\cdot\vec E=\rho/\varepsilon$ gives

$$
\frac{\partial\rho}{\partial t}
=-\frac{\sigma}{\varepsilon}\rho.
$$

Thus a bulk excess-charge density decays as

$$
\rho(t)=\rho(0)\exp\left(-\frac{t}{\tau_r}\right),
\qquad
\tau_r=\frac{\varepsilon}{\sigma}.
$$

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,

$$
\vec P=\frac{\d\vec p_{\rm net}}{\d V}.
$$

In a linear isotropic dielectric, $\vec P=\varepsilon_0\chi_e\vec E$.
The induced bound charge appears through

$$
\rho_b=-\nabla\cdot\vec P,
\qquad
\sigma_b=\vec P\cdot\hat n.
$$

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:

$$
\int_{\text{surface}}\sigma\,\d A=0.
$$

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

$$
Q=C_m\Delta V.
$$

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

$$
\left(\frac{\delta Q}{Q}\right)^2
=
\left(\frac{\delta C_m}{C_m}\right)^2
+
\left(\frac{\delta V}{\Delta V}\right)^2.
$$

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

$$
E_\perp=\frac{\sigma}{\varepsilon}.
$$

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

$$
E(r)(4\pi r^2)=\frac{Q}{\varepsilon_0},
\qquad
\vec E(r)=\frac{Q}{4\pi\varepsilon_0r^2}\hat r.
$$

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:

$$
\vec E(r)=0\qquad(r<R).
$$

The surface field has magnitude $Q/(4\pi\varepsilon_0R^2)$. Comparison with the
boundary relation $E_\perp=\sigma/\varepsilon_0$ gives the uniform surface density

$$
\sigma=\frac{Q}{4\pi R^2}.
$$

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 $\sigma/\varepsilon_0$, 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

$$
p_e=\frac{\sigma}{2}\left(\frac{\sigma}{\varepsilon_0}\right)
=\frac{\sigma^2}{2\varepsilon_0}
=\frac12\varepsilon_0E_\perp^2.
$$

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 $C=\varepsilon_0A/d$. Holding charge fixed gives

$$
U(Q,d)=\frac{Q^2d}{2\varepsilon_0A},
\qquad
F_d=-\left(\frac{\partial U}{\partial d}\right)_Q
=-\frac{Q^2}{2\varepsilon_0A}.
$$

The negative sign indicates attraction: the force reduces $d$. Dividing by area
recovers $p_e=\sigma^2/(2\varepsilon_0)$. 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:

$$
\sum_{i=1}^{N}Q_i=Q_{\rm total}.
$$

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 $R_2$,

$$
V_1=\frac{Q_1}{4\pi\varepsilon_0R_1},
\qquad
V_2=\frac{Q_2}{4\pi\varepsilon_0R_2}.
$$

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

$$
\frac{Q_1}{Q_2}=\frac{R_1}{R_2},
\qquad
Q_1=\frac{R_1}{R_1+R_2}Q_{\rm total}.
$$

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,

$$
Q_i=\sum_j C_{ij}V_j.
$$

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:

$$
q+Q_{\rm inner}=0,
\qquad
Q_{\rm inner}=-q.
$$

Charge conservation fixes the outer-surface charge:

$$
Q_{\rm outer}=Q-Q_{\rm inner}=Q+q.
$$

Spherical symmetry makes both induced surface densities uniform,

$$
\sigma_{\rm inner}=-\frac{q}{4\pi a^2},
\qquad
\sigma_{\rm outer}=\frac{Q+q}{4\pi b^2}.
$$

$$
% caption: A conducting shell with a point charge $+q$ at the centre of its
% cavity. Zero field in the metal forces induced charge $-q$ onto the inner wall
% (filled dots); charge conservation puts $Q+q$ on the outer wall (plus signs).
% Outside, the field is that of a point charge $Q+q$ at the centre.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill[black, even odd rule] (0,0) circle (1.9) (0,0) circle (1.05);
  \draw[black] (0,0) circle (1.9);
  \draw[black] (0,0) circle (1.05);
  \filldraw[draw=black, fill=black!8] (0,0) circle (0.14);
  \node[black!70] at (0,0) {+};
  \foreach \a in {30,90,150,210,270,330}{\fill[acc] (\a:1.05) circle (0.05);}
  \foreach \a in {0,60,120,180,240,300}{\node[acc] at (\a:1.9) {+};}
  \node[black] at (0,1.45) {metal};
  \node[black, anchor=north] at (0,-2.05) {E = 0 in the metal};
\end{tikzpicture}
$$

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

$$
\vec E(r)=\frac{Q+q}{4\pi\varepsilon_0r^2}\hat r,
\qquad
V(r)=\frac{Q+q}{4\pi\varepsilon_0r},
\quad r>b.
$$

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

$$
V_{\rm metal}=\frac{Q+q}{4\pi\varepsilon_0b}.
$$

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, $E_r=q/(4\pi\varepsilon_0r^2)$ 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 $Q_{\rm outer}=Q+q$:
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 $0<r<a$,

$$
V(r)=\frac{q}{4\pi\varepsilon_0r}
-\frac{q}{4\pi\varepsilon_0a}
+\frac{Q+q}{4\pi\varepsilon_0b}.
$$

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

$$
-\frac{\d V}{\d r}
=\frac{q}{4\pi\varepsilon_0r^2}=E_r.
$$

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

$$
V(\text{metal})=\text{constant},
\qquad
V(\infty)=0
$$

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

$$
V(Q)=\frac{Q}{4\pi\varepsilon_0R}.
$$

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

$$
U=\int_0^Q\frac{Q'}{4\pi\varepsilon_0R}\,\d Q'
=\frac{Q^2}{8\pi\varepsilon_0R}
=\frac12QV.
$$

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

$$
U=\int_R^\infty
\frac12\varepsilon_0E^2(4\pi r^2)\,\d r,
\qquad
E=\frac{Q}{4\pi\varepsilon_0r^2}.
$$

Substitution produces $Q^2/(8\pi\varepsilon_0R)$. 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

$$
\frac{\partial\rho}{\partial t}
=-\frac{\sigma}{\varepsilon}\rho,
\qquad
\tau_r=\frac{\varepsilon}{\sigma}.
$$

The relaxation time measures how quickly a conductor removes a bulk electric
field after a static charge imbalance is introduced. For copper,
$\sigma\approx5.8\times10^7\ \mathrm{S\,m^{-1}}$ and
$\varepsilon\approx\varepsilon_0$, giving

$$
\tau_r\approx
\frac{8.85\times10^{-12}}{5.8\times10^7}
=1.5\times10^{-19}\ \mathrm s.
$$

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

$$
\delta=\sqrt{\frac{2}{\mu\sigma\omega}}.
$$

A field entering a thick good conductor decays approximately as
$\exp(-x/\delta)$. At $60\ \mathrm{Hz}$ in copper,
$\delta\approx8.5\ \mathrm{mm}$; 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

$$
\hat n\cdot
\left(\vec D_{\rm diel}-\vec D_{\rm cond}\right)
=\sigma_f.
$$

In electrostatic equilibrium $\vec E_{\rm cond}=0$, hence
$\vec D_{\rm cond}=0$ for an ordinary conductor. The normal displacement in
the dielectric is therefore

$$
D_{\perp,\rm diel}=\sigma_f.
$$

In a homogeneous linear dielectric, $\vec D=\varepsilon\vec E$, so

$$
E_{\perp,\rm diel}=\frac{\sigma_f}{\varepsilon}.
$$

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

$$
\sigma_b=\vec P\cdot\hat n,
\qquad
\vec P=\varepsilon_0\chi_e\vec E.
$$

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 $\varepsilon$
has free plate charge densities $+\sigma_f$ and $-\sigma_f$. The field between
the plates is

$$
E=\frac{\sigma_f}{\varepsilon},
\qquad
V=Ed=\frac{\sigma_fd}{\varepsilon}.
$$

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
$\sigma_f=\varepsilon V/d$. 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.