In physics, charge conservation is the principle that the total electric charge in an isolated system never changes.^[1] The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved. Charge conservation, considered as a physical conservation law, implies that the change in the amount of electric charge in any volume of space is exactly equal to the amount of charge flowing into the volume minus the amount of charge flowing out of the volume. In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region, given by a continuity equation between charge density ${\displaystyle \rho (\mathbf {x} )}$ and current density ${\displaystyle \mathbf {J} (\mathbf {x} )}$.

This does not mean that individual positive and negative charges cannot be created or destroyed. Electric charge is carried by subatomic particles such as electrons and protons. Charged particles can be created and destroyed in elementary particle reactions. In particle physics, charge conservation means that in reactions that create charged particles, equal numbers of positive and negative particles are always created, keeping the net amount of charge unchanged. Similarly, when particles are destroyed, equal numbers of positive and negative charges are destroyed. This property is supported without exception by all empirical observations so far.^[1]

Although conservation of charge requires that the total quantity of charge in the universe is constant, it leaves open the question of what that quantity is. Most evidence indicates that the net charge in the universe is zero;^[2][3] that is, there are equal quantities of positive and negative charge.

History

Charge conservation was first proposed by British scientist William Watson in 1746 and American statesman and scientist Benjamin Franklin in 1747, although the first convincing proof was given by Michael Faraday in 1843.^[4][5]

it is now discovered and demonstrated, both here and in Europe, that the Electrical Fire is a real Element, or Species of Matter, not created by the Friction, but collected only.

— Benjamin Franklin, Letter to Cadwallader Colden, 5 June 1747^[6]

Formal statement of the law

Mathematically, we can state the law of charge conservation as a continuity equation: $${\displaystyle {\frac {\mathrm {d} Q}{\mathrm {d} t}}={\dot {Q}}_{\rm {IN}}(t)-{\dot {Q}}_{\rm {OUT}}(t).}$$ where ${\displaystyle \mathrm {d} Q/\mathrm {d} t}$ is the electric charge accumulation rate in a specific volume at time t, ${\displaystyle {\dot {Q}}_{\rm {IN}}}$ is the amount of charge flowing into the volume and ${\displaystyle {\dot {Q}}_{\rm {OUT}}}$ is the amount of charge flowing out of the volume; both amounts are regarded as generic functions of time.

The integrated continuity equation between two time values reads: $${\displaystyle Q(t_{2})=Q(t_{1})+\int _{t_{1}}^{t_{2}}\left({\dot {Q}}_{\rm {IN}}(t)-{\dot {Q}}_{\rm {OUT}}(t)\right)\,\mathrm {d} t.}$$

The general solution is obtained by fixing the initial condition time ${\displaystyle t_{0}}$, leading to the integral equation: $${\displaystyle Q(t)=Q(t_{0})+\int _{t_{0}}^{t}\left({\dot {Q}}_{\rm {IN}}(\tau )-{\dot {Q}}_{\rm {OUT}}(\tau )\right)\,\mathrm {d} \tau .}$$

The condition ${\displaystyle Q(t)=Q(t_{0})\;\forall t>t_{0},}$ corresponds to the absence of charge quantity change in the control volume: the system has reached a steady state. From the above condition, the following must hold true: $${\displaystyle \int _{t_{0}}^{t}\left({\dot {Q}}_{\rm {IN}}(\tau )-{\dot {Q}}_{\rm {OUT}}(\tau )\right)\,\mathrm {d} \tau =0\;\;\forall t>t_{0}\;\implies \;{\dot {Q}}_{\rm {IN}}(t)={\dot {Q}}_{\rm {OUT}}(t)\;\;\forall t>t_{0}}$$ therefore, ${\displaystyle {\dot {Q}}_{\rm {IN}}}$ and ${\displaystyle {\dot {Q}}_{\rm {OUT}}}$ are equal (not necessarily constant) over time, then the overall charge inside the control volume does not change. This deduction could be derived directly from the continuity equation, since at steady state ${\displaystyle \partial Q/\partial t=0}$ holds, and implies ${\displaystyle {\dot {Q}}_{\rm {IN}}(t)={\dot {Q}}_{\rm {OUT}}(t)}$.

In electromagnetic field theory, vector calculus can be used to express the law in terms of charge density ρ (in coulombs per cubic meter) and electric current density J (in amperes per square meter). This is called the charge density continuity equation $${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.}$$

The term on the left is the rate of change of the charge density ρ at a point. The term on the right is the divergence of the current density J at the same point. The equation equates these two factors, which says that the only way for the charge density at a point to change is for a current of charge to flow into or out of the point. This statement is equivalent to a conservation of four-current.

Mathematical derivation

The net current into a volume is $${\displaystyle I=-\iint _{S}\mathbf {J} \cdot d\mathbf {S} }$$ where S = ∂V is the boundary of V oriented by outward-pointing normals, and dS is shorthand for NdS, the outward pointing normal of the boundary ∂V. Here J is the current density (charge per unit area per unit time) at the surface of the volume. The vector points in the direction of the current.

From the Divergence theorem this can be written $${\displaystyle I=-<!--\; -\; -->{\iiint <!--\; \iiint \; -->}_V(\mathrm{\nabla <!--\; \nabla \; -->}\cdot <!--\; \cdot \; -->\mathbf{J}}$$Zdroj:https://en.wikipedia.org?pojem=Conservation_of_charge
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