\subsection{Charge Redistribution, Contact, Induction, and Grounding}

This subsection explains how charge moves on conductors, why touching conductors exchange charge until they reach one potential, and how induction plus grounding can leave an object with a net charge without direct contact.

\dfn{Charge redistribution, contact, induction, and grounding}{Let $q_A,q_B,\dots$ denote the net charges on conductors $A,B,\dots$, and let $V_A,V_B,\dots$ denote their electric potentials.

\begin{enumerate}[label=\bfseries\tiny\protect\circled{\small\arabic*}]
\item \emph{Charge redistribution}: mobile charge in a conductor moves through the material until electrostatic equilibrium is reached.
\item \emph{Charging by contact}: if conductors touch or are connected by a wire, charge can flow between them until the connected conductors reach one common potential.
\item \emph{Charging by induction}: a nearby charged object causes charges in another object to separate without direct contact.
\item \emph{Grounding}: connecting an object to Earth allows charge to flow between the object and Earth, which acts as a very large charge reservoir.
\end{enumerate}}

\nt{Charge redistribution in a conductor continues until electrostatic equilibrium is reached. In electrostatics, that means the electric field inside the conductor is zero and all parts of one connected conductor are at the same potential. Therefore contact problems are usually solved with charge conservation plus an equal-potential idea. For identical small conducting spheres, symmetry makes equal potential equivalent to equal final charge, but for conductors of different size or shape equal potential does not generally mean equal charge.}

\ex{Illustrative example}{Two identical small conducting spheres $A$ and $B$ are far apart initially. Let their initial charges be
\[
q_{A,i}=+6.0\,\mathrm{nC},
\qquad
q_{B,i}=0.
\]
If the spheres are touched together and then separated, charge is conserved and the identical spheres must finish with equal charge. The total charge is
\[
q_{\mathrm{tot}}=q_{A,i}+q_{B,i}=+6.0\,\mathrm{nC},
\]
so each sphere ends with
\[
q_{A,f}=q_{B,f}=\frac{q_{\mathrm{tot}}}{2}=+3.0\,\mathrm{nC}.
\]
This is a contact example: the charge does not disappear or appear; it redistributes until the connected conductors reach electrostatic equilibrium.}

\mprop{Practical relations and qualitative rules}{Let $q_{\mathrm{tot}}$ denote the total charge of a chosen isolated system, let $q_{A,i},q_{B,i}$ and $q_{A,f},q_{B,f}$ denote initial and final charges, let $q_{\mathrm{object},f}$ denote the final charge of a grounded object, and let $\Delta q_{\mathrm{Earth}}$ denote the charge change of Earth.

\begin{enumerate}[label=\bfseries\tiny\protect\circled{\small\arabic*}]
\item For any isolated system,
\[
q_{\mathrm{tot},f}=q_{\mathrm{tot},i}.
\]

\item If two identical small conducting spheres touch and then separate,
\[
q_{A,f}=q_{B,f}=\frac{q_{A,i}+q_{B,i}}{2}.
\]

\item Induction without grounding redistributes charge but does not change the net charge of the induced object. For an initially neutral conductor, the near side becomes opposite in sign to the external object, the far side becomes the same sign, and the net charge remains zero.

\item Grounding is charge exchange with Earth. If a positive object is nearby, electrons can flow from Earth onto the grounded conductor. If a negative object is nearby, electrons can flow from the conductor to Earth.

\item In the usual induction-charging sequence, the ground connection is removed first and the external charged object is removed second. Then the conductor is left with a net charge opposite in sign to the inducing object. If the object-Earth system is initially neutral, charge conservation for that system gives
\[
q_{\mathrm{object},f}+\Delta q_{\mathrm{Earth}}=0.
\]
\end{enumerate}}

\qs{Worked AP-style problem}{A neutral metal sphere $S$ is mounted on an insulating stand. A negatively charged rod is brought near the left side of the sphere but does not touch it. While the rod remains in place, the sphere is briefly connected to Earth by a wire. The grounding wire is then removed, and finally the rod is taken away.

Find:
\begin{enumerate}[label=(\alph*)]
\item the signs of the induced charges on the left and right sides of the sphere before the grounding wire is attached,
\item the direction of electron flow while the sphere is grounded,
\item the net charge left on the sphere after the full sequence, and
\item whether charge conservation is violated by the sphere ending with a net charge.
\end{enumerate}}

\sol Before the sphere is grounded, the rod is negative, so it repels mobile electrons in the metal sphere. Those electrons shift toward the right side, farther from the rod.

Therefore, before grounding,
\begin{itemize}
\item the left side of the sphere is induced to be positive, and
\item the right side of the sphere is induced to be negative.
\end{itemize}

Even though the charges have separated, the sphere is still overall neutral at this stage because no charge has entered or left the sphere.

Now the sphere is connected to Earth while the negative rod remains nearby. The excess electrons on the sphere are repelled by the negative rod, and the grounding wire gives those electrons a path to leave. Thus electrons flow
\[
\text{from the sphere to Earth}.
\]

Next, the grounding wire is removed while the rod is still present. Because the sphere is no longer connected to Earth, the electrons that left cannot return. The sphere has lost some electrons, so it now has a net positive charge.

Finally, the rod is taken away. With the rod gone, the remaining positive charge is no longer pulled to one side, so it redistributes over the outer surface of the sphere. The final result is that the sphere is left
\[
\text{positively charged}.
\]

Charge conservation is not violated. The correct isolated system is the sphere together with Earth. During grounding, electrons moved from the sphere to Earth, so the sphere became positive and Earth gained an equal amount of negative charge. If the final charge on the sphere is $q_{S,f}>0$, then
\[
q_{S,f}+\Delta q_{\mathrm{Earth}}=0.
\]

So the total charge of the combined system is unchanged; the process is charge transfer, not charge creation.
