{{Short description|Fundamental physical law – electric charge is continuously conserved in space and time}}{{about|the conservation of electric charge|a general theoretical concept|charge (physics)}}In physics, charge conservation is the principle that the total electric charge in an isolated system never changes.BOOK , 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 rho(mathbf{x}) and current density 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.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;JOURNAL JOURNAL

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.BOOK }}

Formal statement of the law

{{see also|Continuity equation}}Mathematically, we can state the law of charge conservation as a continuity equation: where mathrm{d}Q/mathrm{d}t is the electric charge accumulation rate in a specific volume at time {{mvar|t}}, dot Q_{rm{IN}} is the amount of charge flowing into the volume and 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: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 t_0, leading to the integral equation:Q(t) = Q(t_0) + int_{t_0}^left(dot Q_{rm{IN}}(tau) - dot Q_{rm{OUT}}(tau)right),mathrm{d}tau.The condition 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:int_{t_0}^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, dot Q_{rm{IN}} and 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 partial Q/partial t=0 holds, and implies 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 {{mvar|ρ}} (in coulombs per cubic meter) and electric current density {{math|J}} (in amperes per square meter). This is called the charge density continuity equation The term on the left is the rate of change of the charge density {{mvar|ρ}} at a point. The term on the right is the divergence of the current density {{math|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 isI = - iint_Smathbf{J}cdot dmathbf{S}where {{math|1=S = ∂V}} is the boundary of {{mvar|V}} oriented by outward-pointing normals, and {{math|dS}} is shorthand for {{math|NdS}}, the outward pointing normal of the boundary {{math|∂V}}. Here {{math|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 writtenI = - iiint_V left(nabla cdot mathbf{J}right) dVCharge conservation requires that the net current into a volume must necessarily equal the net change in charge within the volume.{{NumBlk||frac{dq} {dt} = - iiint_Vleft(nabla cdot mathbf{J}right) dV |{{EquationRef|1}}}}The total charge q in volume V is the integral (sum) of the charge density in Vq = iiintlimits_V rho dVSo, by the Leibniz integral rule{{NumBlk||frac{dq}{dt} = iiint_V frac{partial rho}{partial t} dV |{{EquationRef|2}}}}Equating ({{EquationNote|1}}) and ({{EquationNote|2}}) gives Since this is true for every volume, we have in general

Derivation from Maxwell's Laws

The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the modified Ampere's law has zero divergence by the div–curl identity. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives:0 = nablacdot (nablatimes mathbf{B}) = nabla cdot left(mu_0 left(mathbf{J} + varepsilon_0 frac{partial mathbf{E}} {partial t} right) right) = mu_0left(nablacdot mathbf{J} + varepsilon_0frac{partial}{partial t}nablacdot mathbf{E}right) = mu_0left(nablacdot mathbf{J} +frac{partial rho}{partial t}right)i.e.,frac{partial rho}{partial t} + nabla cdot mathbf{J} = 0.By the Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary: In particular, in an isolated system the total charge is conserved.

Connection to gauge invariance

Charge conservation can also be understood as a consequence of symmetry through Noether's theorem, a central result in theoretical physics that asserts that each conservation law is associated with a symmetry of the underlying physics. The symmetry that is associated with charge conservation is the global gauge invariance of the electromagnetic field.BOOK

In quantum mechanics the scalar field is equivalent to a phase shift in the wavefunction of the charged particle: so gauge invariance is equivalent to the well known fact that changes in the overall phase of a wavefunction are unobservable, and only changes in the magnitude of the wavefunction result in changes to the probability function |psi|^2.BOOK, Sakurai, J. J.,weblink Modern Quantum Mechanics, Napolitano, Jim, 2017-09-21, Cambridge University Press, 978-1-108-49999-6, Gauge invariance is a very important, well established property of the electromagnetic field and has many testable consequences. The theoretical justification for charge conservation is greatly strengthened by being linked to this symmetry.{{Citation needed|date=December 2023}} For example, gauge invariance also requires that the photon be massless, so the good experimental evidence that the photon has zero mass is also strong evidence that charge is conserved.JOURNAL Even if gauge symmetry is exact, however, there might be apparent electric charge non-conservation if charge could leak from our normal 3-dimensional space into hidden extra dimensions.JOURNAL JOURNAL

Experimental evidence

Simple arguments rule out some types of charge nonconservation. For example, the magnitude of the elementary charge on positive and negative particles must be extremely close to equal, differing by no more than a factor of 10−21 for the case of protons and electrons.JOURNAL The best experimental tests of electric charge conservation are searches for particle decays that would be allowed if electric charge is not always conserved. No such decays have ever been seen.JOURNAL The best experimental test comes from searches for the energetic photon from an electron decaying into a neutrino and a single photon:{| border="0" cellpadding="2"

2em}}	style="width: 8em"	2em}}	mean lifetime is greater than {{val	e=28	Confidence limit>Confidence Level),JOURNAL , Agostini, M.; et al. (Borexino Coll.) , 2015 , Test of Electric Charge Conservation with Borexino , Physical Review Letters , 115, 23, 231802 , 10.1103/PhysRevLett.115.231802 , 2015PhRvL.115w1802A , 1509.01223 , 26684111, 206265225 , JOURNAL , Back, H.O.; et al. (Borexino Coll.) , Physics Letters B , 525 , 1–2 , 2002 , 29–40 , Search for electron decay mode e → γ + ν with prototype of Borexino detector , 10.1016/S0370-2693(01)01440-X , 2002PhLB..525...29B, free ,

but there are theoretical arguments that such single-photon decays will never occur even if charge is not conserved.BOOK Charge disappearance tests are sensitive to decays without energetic photons, other unusual charge violating processes such as an electron spontaneously changing into a positron,JOURNAL and to electric charge moving into other dimensions.The best experimental bounds on charge disappearance are:{| border="0" cellpadding="2"

2em}}	style="width: 8em"	anything >	6.4	u=years}} (68% CL)JOURNAL , P. Belli , DAMA/NaI , Physics Letters B , 465 , 1–4 , 1999 , 315–322 , Charge non-conservation restrictions from the nuclear levels excitation of 129Xe induced by the electron's decay on the atomic shell , 10.1016/S0370-2693(99)01091-6 , 1999PhLB..465..315B, etal, This is the most stringent of several limits given in Table 1 of this paper.

	ν}}	charge non-conserving decays are less than 8 × 10−27 (68% CL) of all neutron decaysJOURNAL , Norman , E.B. , Bahcall , J.N. , John N. Bahcall , Goldhaber , M. , Maurice Goldhaber , Physical Review , D53 , 7 , 1996 , 4086–4088 , Improved limit on charge conservation derived from 71Ga solar neutrino experiments ,weblink , 10.1103/PhysRevD.53.4086 , 10020402 , 1996PhRvD..53.4086N , 41992809 , Link is to preprint copy ,

See also

• Capacitance
• Charge invariance
• Conservation Laws and Symmetry
• Introduction to gauge theory – includes further discussion of gauge invariance and charge conservation
• Kirchhoff's circuit laws – application of charge conservation to electric circuits
• Maxwell's equations
• Relative charge density
• Franklin's electrostatic machine

Notes

{{Reflist|2}}

Further reading

• BOOK