= Charge conservation
{title2=$\partial_t\rho+\nabla\cdot\mathbf J=0$}
{wiki}
= Conservation of electric charge
{synonym}
= Electric charge conservation
{synonym}
<Charge conservation> says that the total signed <electric charge> of an isolated collection stays constant. More generally, a collection closed to <electric charge> exchange has constant total <electric charge>, even if <energy> crosses its boundary. This is the charge bookkeeping relevant to a <closed system> or an <isolated system>; isolation from <energy> exchange is not required. Positive and negative <electric charge> count with opposite signs. Creating or removing an <electron> and a <Positron> together is compatible with <charge conservation>, because their combined <electric charge> is zero. https://www.feynmanlectures.caltech.edu/I_04.html#Ch4-S4[Feynman, Leighton and Sands, The Feynman Lectures on Physics, I.4.4: net charge conservation]{ref}.
Imagine a fixed box initially containing <electric charge> $Q(0)=10\,\mu\mathrm C$. Stipulate incoming <conventional current> $I_{\mathrm{in}}=3\,\mu\mathrm A$ and outgoing <conventional current> $I_{\mathrm{out}}=1\,\mu\mathrm A$ throughout $0\le t\le2\,\mathrm s$, with no other boundary transport. Since one <ampere> is one <coulomb> per second, the stored <electric charge> grows at $2\,\mu\mathrm{C/s}$:
$$
\frac{dQ}{dt}=I_{\mathrm{in}}-I_{\mathrm{out}}=2\,\mu\mathrm{C/s},\qquad Q(t)=10\,\mu\mathrm C+(2\,\mu\mathrm{C/s})t,\qquad Q(2\,\mathrm s)=14\,\mu\mathrm C.
$$
If instead the outgoing <electric current> also equals $3\,\mu\mathrm A$, the stored <electric charge> remains $10\,\mu\mathrm C$ while <electric charge> passes through. These are stipulated transport examples; constant terminal <electric currents> alone do not guarantee a time-independent interior <charge density>.
\Image[/charge-conservation-control-volume.png]
{title=Signed <electric current> through a fixed <control volume>}
{description=The dashed box is an accounting boundary. Outward <unit normals> point left at the inlet and right at the outlet: the respective contributions to outward <electric current> are $-I_{\mathrm{in}}$ and $+I_{\mathrm{out}}$. Arrow lengths are schematic.}
{source=charge-conservation-control-volume.py}
\Image[/charge-storage-versus-steady-current.png]
{title=Stored <electric charge> with unequal and balanced terminal <electric currents>}
{description=Synthetic curves from the stipulated rates over $0\le t\le2\,\mathrm s$, with the same initial <electric charge>; these are not measurements. Balanced transport keeps total <electric charge> fixed, and is compatible with steady transport.}
{source=charge-storage-versus-steady-current.py}
Locally, <charge conservation> is the <charge continuity equation>, an instance of the general <continuity equation>:
$$
\partial_t\rho+\nabla\cdot\mathbf J=0.
$$
Here $\rho(\mathbf r,t)$ is signed volume <charge density> and $\mathbf J(\mathbf r,t)$ is <current density>. A positive <divergence> of <current density> means net outward transport from a small neighborhood, so its <charge density> decreases. For a fixed <control volume> $V$ with <closed surface> $\partial V$ and outward <unit normal> $\mathbf n$, the equivalent integrated balance is
$$
Q_V(t)=\int_V\rho\,dV,\qquad \frac{dQ_V}{dt}=-\oint_{\partial V}\mathbf J\cdot\mathbf n\,dA=-I_{\mathrm{out,net}}.
$$
The <surface integral> is signed: incoming positive <electric charge> contributes negatively to outward <electric current>, and outgoing positive <electric charge> contributes positively. Integrate the <charge continuity equation> over fixed $V$ and apply the <divergence theorem> to obtain this balance. Zero net boundary transport gives constant $Q_V$; the global and local statements describe the same <conservation law> at different scales. https://www.feynmanlectures.caltech.edu/II_13.html#Ch13-S2[Feynman, Leighton and Sands, II.13.2: local and integrated charge conservation]{ref}.
In a steady transport state, <charge density> and <current density> are time-independent, so the <charge continuity equation> reduces to $\nabla\cdot\mathbf J=0$. At an <electrical circuit> junction, choose each terminal <electric current> $I_k$ positive outward and assume the terminals account for all boundary transport. Then
$$
\sum_k I_k=-\frac{dQ_{\mathrm{junction}}}{dt}.
$$
<Kirchhoff's current law>, $\sum_k I_k=0$, follows when accumulation of <electric charge> at the junction is negligible. During storage, retain the same balance with its nonzero <derivative>; <charge conservation> continues to hold. This makes <charge conservation> useful for checking <electrical circuit> models and for tracking the charging of a <capacitor>. https://openstax.org/books/university-physics-volume-2/pages/10-3-kirchhoffs-rules[Ling, Sanny and Moebs, University Physics II, §10.3: Kirchhoff's junction rule]{ref}.
The beautiful consistency with <Maxwell equations> is that the <electric field> and <magnetic field> already enforce this transport balance. For sufficiently smooth fields in vacuum <SI>, take the <divergence> of the <Ampère-Maxwell equation>, use that the <divergence of a curl is zero>, and substitute <Gauss's law>, $\nabla\cdot\mathbf E=\rho/\epsilon_0$. Constant <vacuum permittivity> $\epsilon_0$ gives
$$
0=\nabla\cdot\mathbf J+\epsilon_0\partial_t(\nabla\cdot\mathbf E)=\nabla\cdot\mathbf J+\partial_t\rho.
$$
The <displacement current> term supplies the part needed when <electric charge> accumulates. Historically, Maxwell added that term to extend the steady-current equation; see the existing <charge conservation from Maxwell equations> for this connection. https://www.feynmanlectures.caltech.edu/II_18.html#Ch18-S1[Feynman, Leighton and Sands, II.18.1: Maxwell's added term and charge conservation]{ref}.
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