Charge conservation explained

\rho( x ) and current density J ( 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;^{[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]}

Formal statement of the law

See also: Continuity equation.

Mathematically, we can state the law of charge conservation as a continuity equation:$$\backslash frac\; =\; \backslash dot\; Q\_(t)\; -\; \backslash dot\; Q\_(t).$$where

d Q/ d t is the electric charge accumulation rate in a specific volume at time, • Q _{\rm{IN} } is the amount of charge flowing into the volume and • 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:$$Q(t\_2)\; =\; Q(t\_1)\; +\; \backslash int\_^\backslash left(\backslash dot\; Q\_(t)\; -\; \backslash dot\; Q\_(t)\backslash right)\backslash ,\backslash mathrmt.$$

The general solution is obtained by fixing the initial condition time

t _{0} , leading to the integral equation:$$Q(t)\; =\; Q(t\_0)\; +\; \backslash int\_^\backslash left(\backslash dot\; Q\_(\backslash tau)\; -\; \backslash dot\; Q\_(\backslash tau)\backslash right)\backslash ,\backslash mathrm\backslash tau.$$

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:$$\backslash int\_^\backslash left(\backslash dot\; Q\_(\backslash tau)\; -\; \backslash dot\; Q\_(\backslash tau)\backslash right)\backslash ,\backslash mathrm\backslash tau\; =\; 0\backslash ;\backslash ;\backslash forall\; t>t\_0\backslash ;\backslash implies\backslash ;\backslash dot\; Q\_(t)\; =\; \backslash dot\; Q\_(t)\backslash ;\backslash ;\backslash forall\; t>t\_0$$therefore, • Q _{\rm{IN} } and • 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 • Q _{\rm{IN} }(t) = \dot Q_(t).

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

The term on the left is the rate of change of the charge density at a point. The term on the right is the divergence of the current density 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$$I\; =\; -\; \backslash iint\_S\backslash mathbf\backslash cdot\; d\backslash mathbf$$where is the boundary of oriented by outward-pointing normals, and is shorthand for, the outward pointing normal of the boundary . Here 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$$I\; =\; -\; \backslash iiint\_V\; \backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\backslash right)\; dV$$

Charge conservation requires that the net current into a volume must necessarily equal the net change in charge within the volume.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 ruleEquating and gives$$0\; =\; \backslash iiint\_V\; \backslash left(\backslash frac\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; \backslash right)\; dV.$$Since this is true for every volume, we have in general$$\backslash frac\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; 0.$$

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:$$0\; =\; \backslash nabla\backslash cdot\; (\backslash nabla\backslash times\; \backslash mathbf)\; =\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash mu\_0\; \backslash left(\backslash mathbf\; +\; \backslash varepsilon\_0\; \backslash frac\; \backslash right)\; \backslash right)\; =\; \backslash mu\_0\backslash left(\backslash nabla\backslash cdot\; \backslash mathbf\; +\; \backslash varepsilon\_0\backslash frac\backslash nabla\backslash cdot\; \backslash mathbf\backslash right)\; =\; \backslash mu\_0\backslash left(\backslash nabla\backslash cdot\; \backslash mathbf\; +\backslash frac\backslash right)$$i.e.,$$\backslash frac\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; 0.$$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

\phi . However the full symmetry is more complicated, and also involves the vector potential 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 \chi :\phi' = \phi - (\partial \chi)/(\partial t) A ' = A + \nabla \chi.

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

\psi' = e ^{i} \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

|\psi| ^{2} .^[7]

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. 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.^[8] 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.^{[9] [10]}

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.^[11] 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.^[12] The best experimental test comes from searches for the energetic photon from an electron decaying into a neutrino and a single photon:

e → ν + γ

mean lifetime is greater than (90% Confidence Level),^[13] ^[14]

but there are theoretical arguments that such single-photon decays will never occur even if charge is not conserved.^[15] Charge disappearance tests are sensitive to decays without energetic photons, other unusual charge violating processes such as an electron spontaneously changing into a positron,^[16] and to electric charge moving into other dimensions.The best experimental bounds on charge disappearance are:

	e → anything	mean lifetime is greater than (68% CL)^[17]
	n → p + ν +	charge non-conserving decays are less than 8 × 10^−27 (68% CL) of all neutron decays^[18]

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

Further reading

• Book: Lemay , J.A. Leo . The Life of Benjamin Franklin, Volume 3: Soldier, Scientist, and Politician . 2008 . Leo Lemay . . 978-0-8122-4121-1 . Chapter 2: Electricity . https://books.google.com/books?id=NL5bcRP5aRAC&pg=PA58.

Notes and References

1. Book: Purcell , Edward M. . Morin, David J. . 2013 . Electricity and magnetism . 3rd . Cambridge University Press . 9781107014022 . 4.
2. S. Orito . M. Yoshimura . Physical Review Letters . 54 . 22 . 1985 . 2457–2460 . Can the Universe be Charged? . 10.1103/PhysRevLett.54.2457 . 10031347 . 1985PhRvL..54.2457O .
3. E. Masso . F. Rota . Physics Letters B . 545 . 3–4 . 2002 . 221–225 . Primordial helium production in a charged universe . astro-ph/0201248 . 10.1016/S0370-2693(02)02636-9. 2002PhLB..545..221M . 119062159 .
4. Book: Heilbron , J.L. . Electricity in the 17th and 18th centuries: a study of early Modern physics . John L. Heilbron . 1979 . University of California Press . 978-0-520-03478-5 . 330 .
5. Book: Purrington . Robert D. . Physics in the Nineteenth Century . Rutgers University Press . 1997 . 33 . registration . benjamin franklin william watson charge conservation. . 978-0813524429 .
6. Book: Bettini , Alessandro . Introduction to Elementary Particle Physics . Cambridge University Press . 2008 . UK . 164–165 . 978-0-521-88021-3.
7. Book: Sakurai . J. J. . Modern Quantum Mechanics . Napolitano . Jim . 2017-09-21 . Cambridge University Press . 10.1017/9781108499996 . 978-1-108-49999-6.
8. A.S. Goldhaber . M.M. Nieto . Reviews of Modern Physics . 82 . 1 . 2010 . 939–979 . Photon and Graviton Mass Limits . 0809.1003 . 10.1103/RevModPhys.82.939. 2010RvMP...82..939G . 14395472 .

   see Section II.C Conservation of Electric Charge

9. S.Y. Chu . Modern Physics Letters A . 11 . 28 . 1996 . 2251–2257 . Gauge-Invariant Charge Nonconserving Processes and the Solar Neutrino Puzzle . 10.1142/S0217732396002241 . 1996MPLA...11.2251C .
10. S.L. Dubovsky . V.A. Rubakov . P.G. Tinyakov . Journal of High Energy Physics . August . 8 . 2000 . 315–318 . Is the electric charge conserved in brane world? . hep-ph/0007179 . 10.1016/0370-2693(79)90048-0. 1979PhLB...84..315I .
11. Patrignani . C. et al (Particle Data Group) . The Review of Particle Physics . Chinese Physics C . 40 . 100001 . 2016 . March 26, 2017.
12. Particle Data Group . Particle Data Group . Journal of Physics G . 37 . 7A . 89–98 . Tests of Conservation Laws . May 2010 . 10.1088/0954-3899/37/7A/075021. 2010JPhG...37g5021N .
13. Agostini, M.; et al. (Borexino Coll.) . 2015 . Test of Electric Charge Conservation with Borexino . . 115 . 23 . 231802 . 10.1103/PhysRevLett.115.231802 . 2015PhRvL.115w1802A . 1509.01223 . 26684111. 206265225 .
14. 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 .
15. Book: L.B. Okun . Lev Okun . Comments on Nuclear and Particle Physics . 19 . 3 . 1989 . 99–116 . Comments on Testing Charge Conservation and Pauli Exclusion Principle . Comments on Testing Charge Conservation and the Pauli Exclusion Principle . 10.1142/9789812799104_0006 . World Scientific Lecture Notes in Physics . 978-981-02-0453-2 . 124865855 .
16. R.N. Mohapatra . Rabindra Mohapatra . Physical Review Letters . 59 . 14 . 1987 . 1510–1512 . Possible Nonconservation of Electric Charge . 10.1103/PhysRevLett.59.1510 . 1987PhRvL..59.1510M . 10035254 .
17. 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.
18. 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 . 10.1103/PhysRevD.53.4086 . 10020402 . 1996PhRvD..53.4086N . 41992809 . Link is to preprint copy .