Charge conservation

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.{{cite book

|title=Electricity in the 17th and 18th centuries: a study of early Modern physics

|last=Heilbron

|author-link=John L. Heilbron

|first=J.L.

|year=1979

|publisher=University of California Press

|isbn=978-0-520-03478-5

|page=330

|url=https://books.google.com/books?id=UlTLRUn1sy8C&pg=PA330}}{{cite book

| last1 = Purrington

| first1 = Robert D.

| title = Physics in the Nineteenth Century

| publisher = Rutgers University Press

| date = 1997

| pages = [https://archive.org/details/physicsinninetee0000purr/page/33 33]

| url = https://archive.org/details/physicsinninetee0000purr

| url-access = registration

| quote = benjamin franklin william watson charge conservation.

| isbn = 978-0813524429

}}

{{Quote|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.|sign=Benjamin Franklin |source=Letter to Cadwallader Colden, 5 June 1747{{cite book

|url = http://www.franklinpapers.org/franklin/framedVolumes.jsp?vol=3&page=141b

|year = 1961

|title = The Papers of Benjamin Franklin

|volume = 3

|page = 142

|publisher = Yale University Press

|access-date = 2010-11-25

|archive-url = https://web.archive.org/web/20110929081257/http://www.franklinpapers.org/franklin/framedVolumes.jsp?vol=3&page=141b

|archive-date = 2011-09-29

|url-status = dead

}}}}

{{see also|Continuity equation}}

Mathematically, we can state the law of charge conservation as a continuity equation:

$$\backslash frac\{\backslash mathrm\{d\}Q\}\{\backslash mathrm\{d\}t\}\; =\; \backslash dot\; Q\_\{\backslash rm\{IN\}\}(t)\; -\; \backslash dot\; Q\_\{\backslash rm\{OUT\}\}(t).$$

where $\backslash mathrm\{d\}Q/\backslash mathrm\{d\}t$ is the electric charge accumulation rate in a specific volume at time {{mvar|t}}, $\backslash dot\; Q\_\{\backslash rm\{IN\}\}$ is the amount of charge flowing into the volume and $\backslash dot\; Q\_\{\backslash rm\{OUT\}\}$ 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:

$$Q(t\_2)\; =\; Q(t\_1)\; +\; \backslash int\_\{t\_1\}^\{t\_2\}\backslash left(\backslash dot\; Q\_\{\backslash rm\{IN\}\}(t)\; -\; \backslash dot\; Q\_\{\backslash rm\{OUT\}\}(t)\backslash right)\backslash ,\backslash mathrm\{d\}t.$$

The general solution is obtained by fixing the initial condition time $t\_0$, leading to the integral equation:

$$Q(t)\; =\; Q(t\_0)\; +\; \backslash int\_\{t\_0\}^\{t\}\backslash left(\backslash dot\; Q\_\{\backslash rm\{IN\}\}(\backslash tau)\; -\; \backslash dot\; Q\_\{\backslash rm\{OUT\}\}(\backslash tau)\backslash right)\backslash ,\backslash mathrm\{d\}\backslash tau.$$

The condition $Q(t)=Q(t\_0)\backslash ;\backslash forall\; t\; >\; t\_0,$ 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:

$$\backslash int\_\{t\_0\}^\{t\}\backslash left(\backslash dot\; Q\_\{\backslash rm\{IN\}\}(\backslash tau)$$

- \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, $\backslash dot\; Q\_\{\backslash rm\{IN\}\}$ and $\backslash dot\; Q\_\{\backslash rm\{OUT\}\}$ 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 $\backslash partial\; Q/\backslash partial\; t=0$ holds, and implies $\backslash dot\; Q\_\{\backslash rm\{IN\}\}(t)\; =\; \backslash dot\; Q\_\{\backslash rm\{OUT\}\}(t)$.

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

$$\backslash frac\{\backslash partial\; \backslash rho\}\; \{\backslash partial\; t\}\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; =\; 0.$$

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.

The net current into a volume is

$$I\; =\; -\; \backslash iint\_S\backslash mathbf\{J\}\backslash cdot\; d\backslash mathbf\{S\}$$

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 written

$$I\; =\; -\; \backslash iiint\_V\; \backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\backslash right)\; dV$$

Charge conservation requires that the net current into a volume must necessarily equal the net change in charge within the volume.

{{NumBlk||$$\backslash frac\{dq\}\; \{dt\}\; =\; -\; \backslash iiint\_V\backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\backslash right)\; dV$$|{{EquationRef|1}}}}

The total charge q in volume V is the integral (sum) of the charge density in V

$$q\; =\; \backslash iiint\backslash limits\_V\; \backslash rho\; dV$$

So, by the Leibniz integral rule

{{NumBlk||$$\backslash frac\{dq\}\{dt\}\; =\; \backslash iiint\_V\; \backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; dV$$|{{EquationRef|2}}}}

Equating ({{EquationNote|1}}) and ({{EquationNote|2}}) gives

$$0\; =\; \backslash iiint\_V\; \backslash left(\; \backslash frac\{\backslash partial\; \backslash rho\}\; \{\backslash partial\; t\}\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; \backslash right)\; dV.$$

Since this is true for every volume, we have in general

$$\backslash frac\{\backslash partial\; \backslash rho\}\; \{\backslash partial\; t\}\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; =\; 0.$$

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:$$0\; =\; \backslash nabla\backslash cdot\; (\backslash nabla\backslash times\; \backslash mathbf\{B\})\; =\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash mu\_0\; \backslash left(\backslash mathbf\{J\}\; +\; \backslash varepsilon\_0\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; \backslash right)\; \backslash right)\; =\; \backslash mu\_0\backslash left(\backslash nabla\backslash cdot\; \backslash mathbf\{J\}\; +\; \backslash varepsilon\_0\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash nabla\backslash cdot\; \backslash mathbf\{E\}\backslash right)\; =\; \backslash mu\_0\backslash left(\backslash nabla\backslash cdot\; \backslash mathbf\{J\}\; +\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\backslash right)$$i.e.,$$\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; =\; 0.$$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:

: {{oiint|preintegral=$\backslash frac\{d\}\{dt\}Q\_\backslash Omega\; =\; \backslash frac\{d\}\{dt\}\; \backslash iiint\_\{\backslash Omega\}\; \backslash rho\; \backslash mathrm\{d\}V\; =\; -$|intsubscpt=$\{\backslash scriptstyle\; \backslash partial\; \backslash Omega\; \}$|integrand=$\backslash mathbf\{J\}\; \backslash cdot\; \{\backslash rm\; d\}\backslash mathbf\{S\}\; =\; -\; I\_\{\backslash partial\; \backslash Omega\}.$}}

In particular, in an isolated system the total charge is conserved.

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.{{cite book

| last = Bettini

| first = Alessandro

| title = Introduction to Elementary Particle Physics

| publisher = Cambridge University Press

| year = 2008

| location = UK

| pages = 164–165

| url = https://books.google.com/books?id=HNcQ_EiuTxcC&pg=PA164

| isbn = 978-0-521-88021-3}} This is related to the fact that the electric and magnetic fields are not changed by different choices of the value representing the zero point of electrostatic potential $\backslash phi$. However the full symmetry is more complicated, and also involves the vector potential $\backslash mathbf\{A\}$. The full statement of gauge invariance is that the physics of an electromagnetic field are unchanged when the scalar and vector potential are shifted by the gradient of an arbitrary scalar field $\backslash chi$:

:$\backslash phi\text{'}\; =\; \backslash phi\; -\; \backslash frac\; \{\backslash partial\; \backslash chi\}\{\backslash partial\; t\}\; \backslash qquad\; \backslash qquad\; \backslash mathbf\{A\}\text{'}\; =\; \backslash mathbf\{A\}\; +\; \backslash nabla\; \backslash chi.$

In quantum mechanics the scalar field is equivalent to a phase shift in the wavefunction of the charged particle:

:$\backslash psi\text{'}\; =\; e^\{i\; q\; \backslash chi\}\backslash psi$

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 $|\backslash psi|^2$.{{Cite book |last1=Sakurai |first1=J. J. |url=http://dx.doi.org/10.1017/9781108499996 |title=Modern Quantum Mechanics |last2=Napolitano |first2=Jim |date=2017-09-21 |publisher=Cambridge University Press |doi=10.1017/9781108499996 |isbn=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.

{{cite journal

|author1=A.S. Goldhaber |author2=M.M. Nieto |journal=Reviews of Modern Physics

|volume=82

|issue=1

|year=2010

|pages= 939–979

|title=Photon and Graviton Mass Limits

|arxiv=0809.1003

|doi=10.1103/RevModPhys.82.939|bibcode = 2010RvMP...82..939G |s2cid=14395472 }}; see Section II.C Conservation of Electric Charge Gauge invariance also implies quantization of hypothetical magnetic charges.

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.

{{cite journal

|author=S.Y. Chu

|journal=Modern Physics Letters A

|volume=11

|issue=28

|year=1996

|pages= 2251–2257

|title=Gauge-Invariant Charge Nonconserving Processes and the Solar Neutrino Puzzle

|doi=10.1142/S0217732396002241

|bibcode = 1996MPLA...11.2251C }}

{{cite journal

|author1=S.L. Dubovsky |author2=V.A. Rubakov |author3=P.G. Tinyakov |journal=Journal of High Energy Physics

|volume=August

|issue=8

|year=2000

|pages= 315–318

|title=Is the electric charge conserved in brane world?

|arxiv=hep-ph/0007179

|doi=10.1016/0370-2693(79)90048-0

|bibcode = 1979PhLB...84..315I }}

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.{{cite journal

| last1 = Patrignani

| first1 = C. et al (Particle Data Group)

| title = The Review of Particle Physics

| journal = Chinese Physics C

| volume = 40

| issue = 100001

| date = 2016

| url = http://pdg.lbl.gov/2016/tables/rpp2016-tab-baryons-N.pdf

| access-date = March 26, 2017}} Ordinary matter contains equal numbers of positive and negative particles, protons and electrons, in enormous quantities. If the elementary charge on the electron and proton were even slightly different, all matter would have a large electric charge and would be mutually repulsive.

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.

{{cite journal

|author=Particle Data Group

|author-link=Particle Data Group

|journal=Journal of Physics G

|volume=37

|issue=7A

|pages= 89–98

|title=Tests of Conservation Laws

|url=http://pdg.lbl.gov/2010/tables/rpp2010-conservation-laws.pdf

|date=May 2010

|doi=10.1088/0954-3899/37/7A/075021

|bibcode = 2010JPhG...37g5021N }}

The best experimental test comes from searches for the energetic photon from an electron decaying into a neutrino and a single photon:

but there are theoretical arguments that such single-photon decays will never occur even if charge is not conserved.

{{cite book

|author=L.B. Okun

|author-link=Lev Okun

|journal=Comments on Nuclear and Particle Physics

|volume=19

|issue=3

|year=1989

|pages=99–116

|title=Comments on Testing Charge Conservation and Pauli Exclusion Principle

|chapter=Comments on Testing Charge Conservation and the Pauli Exclusion Principle

|url=http://inspirehep.net/record/268594/files/v19-n3-p99.pdf

|doi=10.1142/9789812799104_0006 |series=World Scientific Lecture Notes in Physics

|isbn=978-981-02-0453-2

|s2cid=124865855

}}

Charge disappearance tests are sensitive to decays without energetic photons, other unusual charge violating processes such as an electron spontaneously changing into a positron,

{{cite journal

|author=R.N. Mohapatra

|author-link= Rabindra Mohapatra

|journal=Physical Review Letters

|volume=59

|issue=14

|year=1987

|pages=1510–1512

|title=Possible Nonconservation of Electric Charge

|doi=10.1103/PhysRevLett.59.1510

|bibcode=1987PhRvL..59.1510M

|pmid=10035254

}}

and to electric charge moving into other dimensions.

The best experimental bounds on charge disappearance are:

• 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

{{Reflist|2}}

• {{cite book

|title=The Life of Benjamin Franklin, Volume 3: Soldier, Scientist, and Politician

|last=Lemay

|first=J.A. Leo

|year=2008

|author-link=Leo Lemay

|publisher=University of Pennsylvania Press

|isbn=978-0-8122-4121-1

|chapter=Chapter 2: Electricity

|chapter-url=https://books.google.com/books?id=NL5bcRP5aRAC&pg=PA58}}

Category:Electromagnetism

Category:Conservation laws