\subsection{Charge Conservation and Charging Processes}

This subsection treats electric charge as a conserved quantity and introduces the AP-level charging processes of friction, contact, and induction as charge-bookkeeping ideas.

\dfn{Charge conservation and basic charging processes}{Let $q$ denote the net charge of an object or subsystem, measured in coulombs, and let
\[
Q_{\mathrm{tot}}=\sum_i q_i
\]
denote the total charge of a chosen system.

Charge conservation states that for an isolated system,
\[
Q_{\mathrm{tot},f}=Q_{\mathrm{tot},i}.
\]

In ordinary AP charging problems, objects become charged because charge is redistributed:
\begin{enumerate}[label=\bfseries\tiny\protect\circled{\small\arabic*}]
\item \emph{Charging by friction}: rubbing two materials can transfer electrons from one object to the other.
\item \emph{Charging by contact}: when objects touch, charge can transfer between them before they separate.
\item \emph{Charging by induction}: a nearby charged object causes charge separation in another object; with grounding, charge can enter or leave so the object may be left with a net charge after the process.
\end{enumerate}

In each case, the total charge of the full isolated system remains constant.}

\nt{The sign of charge is bookkeeping. A positive net charge means an electron deficit, while a negative net charge means an electron excess. In ordinary friction, contact, and induction processes, charge is transferred from one place to another; it is not created from nothing. If one part of an isolated system gains $+\Delta q$, the rest of the system must change by $-\Delta q$. When grounding is involved, include the Earth in the system because it can supply or receive the transferred charge.}

\mprop{Practical bookkeeping relations for isolated systems and simple sharing}{Consider a chosen system of objects with initial charges $q_{1,i},q_{2,i},\dots$ and final charges $q_{1,f},q_{2,f},\dots$.

\begin{enumerate}[label=\bfseries\tiny\protect\circled{\small\arabic*}]
\item For any isolated system,
\[
\sum_k q_{k,f}=\sum_k q_{k,i}
\qquad\text{or}\qquad
\Delta Q_{\mathrm{tot}}=0.
\]

\item For charge transfer between two objects $A$ and $B$ within an isolated system,
\[
\Delta q_A+\Delta q_B=0,
\]
so
\[
q_{A,f}-q_{A,i}=-(q_{B,f}-q_{B,i}).
\]

\item If two identical small conducting spheres with initial charges $q_{A,i}$ and $q_{B,i}$ are touched together and then separated, symmetry gives equal final charges:
\[
q_{A,f}=q_{B,f}=\frac{q_{A,i}+q_{B,i}}{2}.
\]

\item If a neutral object is charged by induction while connected to ground, then charge conservation must be applied to the combined object-Earth system. If the object starts neutral and ends with charge $q_f$, then the Earth changes by
\[
\Delta q_{\mathrm{Earth}}=-q_f.
\]
\end{enumerate}}

\qs{Worked AP-style problem}{Three identical small conducting spheres $A$, $B$, and $C$ are far apart initially. Let their initial charges be
\[
q_{A,i}=+8.0\,\mathrm{nC},
\qquad
q_{B,i}=-2.0\,\mathrm{nC},
\qquad
q_{C,i}=0.
\]

First, sphere $A$ is touched to sphere $B$ and then separated. Next, sphere $B$ is touched to sphere $C$ and then separated.

Find:
\begin{enumerate}[label=(\alph*)]
\item the charge on each sphere after the first contact,
\item the final charge on each sphere after the second contact, and
\item the number of electrons transferred during the second contact.
\end{enumerate}

Take the elementary charge magnitude to be $e=1.60\times 10^{-19}\,\mathrm{C}$.}

\sol Because the spheres are identical, whenever two of them touch and then separate, they share the total charge equally.

For the first contact, apply charge conservation to spheres $A$ and $B$:
\[
q_{A,i}+q_{B,i}=(+8.0\,\mathrm{nC})+(-2.0\,\mathrm{nC})=+6.0\,\mathrm{nC}.
\]

Since the spheres are identical, after they separate each has half of this total charge:
\[
q_{A}=q_{B}=\frac{+6.0\,\mathrm{nC}}{2}=+3.0\,\mathrm{nC}.
\]

So after the first contact,
\[
q_{A}=+3.0\,\mathrm{nC},
\qquad
q_{B}=+3.0\,\mathrm{nC},
\qquad
q_{C}=0.
\]

Now sphere $B$ touches sphere $C$. Just before this second contact, their total charge is
\[
q_{B}+q_{C}=(+3.0\,\mathrm{nC})+0=+3.0\,\mathrm{nC}.
\]

Again they are identical, so after separation they share this total equally:
\[
q_{B,f}=q_{C,f}=\frac{+3.0\,\mathrm{nC}}{2}=+1.5\,\mathrm{nC}.
\]

Sphere $A$ is not involved in the second contact, so its charge stays
\[
q_{A,f}=+3.0\,\mathrm{nC}.
\]

Therefore the final charges are
\[
q_{A,f}=+3.0\,\mathrm{nC},
\qquad
q_{B,f}=+1.5\,\mathrm{nC},
\qquad
q_{C,f}=+1.5\,\mathrm{nC}.
\]

To find the number of electrons transferred during the second contact, compute the magnitude of the charge change on either sphere. Sphere $B$ changes from $+3.0\,\mathrm{nC}$ to $+1.5\,\mathrm{nC}$, so
\[
\Delta q_B=(+1.5-3.0)\,\mathrm{nC}=-1.5\,\mathrm{nC}.
\]

The negative change means sphere $B$ gained electrons. Equivalently, sphere $C$ changed from $0$ to $+1.5\,\mathrm{nC}$, so sphere $C$ lost electrons. The magnitude of transferred charge is
\[
|\Delta q|=1.5\times 10^{-9}\,\mathrm{C}.
\]

Let $N$ denote the number of electrons transferred. Then
\[
N=\frac{|\Delta q|}{e}=\frac{1.5\times 10^{-9}}{1.60\times 10^{-19}}=9.375\times 10^9.
\]

To two significant figures,
\[
N\approx 9.4\times 10^9\text{ electrons}.
\]

These electrons moved from sphere $C$ to sphere $B$. This direction makes sense because sphere $B$ became less positive while sphere $C$ became more positive.
