143 lines
5.3 KiB
TeX
143 lines
5.3 KiB
TeX
\subsection{Charge Conservation and Charging Processes}
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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.
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\dfn{Charge conservation and basic charging processes}{Let $q$ denote the net charge of an object or subsystem, measured in coulombs, and let
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\[
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Q_{\mathrm{tot}}=\sum_i q_i
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\]
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denote the total charge of a chosen system.
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Charge conservation states that for an isolated system,
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\[
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Q_{\mathrm{tot},f}=Q_{\mathrm{tot},i}.
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\]
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In ordinary AP charging problems, objects become charged because charge is redistributed:
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\begin{enumerate}[label=\bfseries\tiny\protect\circled{\small\arabic*}]
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\item \emph{Charging by friction}: rubbing two materials can transfer electrons from one object to the other.
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\item \emph{Charging by contact}: when objects touch, charge can transfer between them before they separate.
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\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.
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\end{enumerate}
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In each case, the total charge of the full isolated system remains constant.}
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\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.}
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\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 $q_{1,f},q_{2,f},\dots$.
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\begin{enumerate}[label=\bfseries\tiny\protect\circled{\small\arabic*}]
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\item For any isolated system,
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\[
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\sum_k q_{k,f}=\sum_k q_{k,i}
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\qquad\text{or}\qquad
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\Delta Q_{\mathrm{tot}}=0.
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\]
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\item For charge transfer between two objects $A$ and $B$ within an isolated system,
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\[
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\Delta q_A+\Delta q_B=0,
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\]
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so
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\[
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q_{A,f}-q_{A,i}=-(q_{B,f}-q_{B,i}).
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\]
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\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:
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\[
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q_{A,f}=q_{B,f}=\frac{q_{A,i}+q_{B,i}}{2}.
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\]
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\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
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\[
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\Delta q_{\mathrm{Earth}}=-q_f.
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\]
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\end{enumerate}}
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\qs{Worked AP-style problem}{Three identical small conducting spheres $A$, $B$, and $C$ are far apart initially. Let their initial charges be
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\[
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q_{A,i}=+8.0\,\mathrm{nC},
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\qquad
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q_{B,i}=-2.0\,\mathrm{nC},
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\qquad
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q_{C,i}=0.
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\]
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First, sphere $A$ is touched to sphere $B$ and then separated. Next, sphere $B$ is touched to sphere $C$ and then separated.
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Find:
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\begin{enumerate}[label=(\alph*)]
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\item the charge on each sphere after the first contact,
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\item the final charge on each sphere after the second contact, and
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\item the number of electrons transferred during the second contact.
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\end{enumerate}
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Take the elementary charge magnitude to be $e=1.60\times 10^{-19}\,\mathrm{C}$.}
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\sol Because the spheres are identical, whenever two of them touch and then separate, they share the total charge equally.
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For the first contact, apply charge conservation to spheres $A$ and $B$:
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\[
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q_{A,i}+q_{B,i}=(+8.0\,\mathrm{nC})+(-2.0\,\mathrm{nC})=+6.0\,\mathrm{nC}.
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\]
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Since the spheres are identical, after they separate each has half of this total charge:
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\[
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q_{A}=q_{B}=\frac{+6.0\,\mathrm{nC}}{2}=+3.0\,\mathrm{nC}.
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\]
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So after the first contact,
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\[
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q_{A}=+3.0\,\mathrm{nC},
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\qquad
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q_{B}=+3.0\,\mathrm{nC},
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\qquad
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q_{C}=0.
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\]
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Now sphere $B$ touches sphere $C$. Just before this second contact, their total charge is
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\[
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q_{B}+q_{C}=(+3.0\,\mathrm{nC})+0=+3.0\,\mathrm{nC}.
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\]
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Again they are identical, so after separation they share this total equally:
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\[
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q_{B,f}=q_{C,f}=\frac{+3.0\,\mathrm{nC}}{2}=+1.5\,\mathrm{nC}.
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\]
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Sphere $A$ is not involved in the second contact, so its charge stays
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\[
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q_{A,f}=+3.0\,\mathrm{nC}.
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\]
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Therefore the final charges are
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\[
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q_{A,f}=+3.0\,\mathrm{nC},
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\qquad
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q_{B,f}=+1.5\,\mathrm{nC},
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\qquad
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q_{C,f}=+1.5\,\mathrm{nC}.
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\]
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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
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\[
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\Delta q_B=(+1.5-3.0)\,\mathrm{nC}=-1.5\,\mathrm{nC}.
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\]
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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
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\[
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|\Delta q|=1.5\times 10^{-9}\,\mathrm{C}.
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\]
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Let $N$ denote the number of electrons transferred. Then
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\[
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N=\frac{|\Delta q|}{e}=\frac{1.5\times 10^{-9}}{1.60\times 10^{-19}}=9.375\times 10^9.
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\]
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To two significant figures,
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\[
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N\approx 9.4\times 10^9\text{ electrons}.
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\]
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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.
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