Charge conservation

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 $\rho (\mathbf {x} )$ 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.^[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.

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- There's a law in physics that has stood the test of time. Laws come and go. Sometimes we discover new things. We have to scrap them, ammend them, adjust them, tweak them, throw them away, but there's one law that has been around for a long time and no one has ever, ever tried to damage this law or discovered any experiment that has shown it to be wrong, and it's called the law of conservation of charge. And this is electric charge, is what we're talking about in this particular example. So what does this mean? Well, imagine you had a box and inside of this box I'm gonna put some charges. So let's say we have a particle here and it's charge is positive two coulombs. And then we have another charge flying around in here, and it has a charge of negative three coulombs. And we have another charge over here that's got, I don't know, positive five coulombs. These are flying around. What the law of conservation of charge says is if this box is closed up, in the sense that no charge can enter or exit. So I'm not going to let any charge come in and I'm not gonna let any charge go out. If that's the case, the total charge inside of this region of space has to be constant when you add it all up. So if you want a mathematical statement, I like math, the mathematical statement is that if you add up, the sigma is the fancy letter for adding up, all the charges in a given region, as long as, here's the asterisk, as long as no charges are incoming or outgoing, then the total amount of charge in that region of space has to be a constant. This math looks complicated, it's actually easy. All I'm saying is that if you add up all this charge... Positive two coulombs plus five coulombs minus three coulombs, you'll get a number and what that number represents is the total amount of charge in there. Which is going to be, five plus two is seven, minus three is four. Positive four coulombs. You ever open up this box, you're always going to find four coulombs in there. Now this sounds possibly obvious. You might be like, duh. If you don't let any of these charges go in or out, of course you're only going to find four coulombs in there because you've just got these three charges. But not necessarily. Physicists know if you collide two particles, these things don't have to maintain their identity. I might end up with eight particles in here at some later point in time. And if I add up all their charges, I'll still get four. That's the key idea here. That's why this is not just a frivolous sort of meaningless trivial statement. This is actually saying something useful, because if these protons, they're not because this is a positive two coulomb and the proton has a very different charge, but for the sake of argument, say this was a proton, runs into some other particle, an electron, really fast. If there's enough energy, you might not even end up with a proton and an electron. You might end up with neurons or top quarks or if this is another proton, you end up with Higgs particles or whatever. And so at some later point in time, here's why this law is important and not trivial, because if this really is closed up and the only stuff going on in there is due to these and whatever descendants particles they create, at some later point in time I may end up with, like, say this one, it doesn't even have to have the same charge. Maybe this one's positive one coulomb. And I end up with a charge over here that has negative seven coulombs. If these were fundamental particles, they would have charges much smaller than this, but to get the idea across, big numbers are better. And let's say this is negative four coulombs. And then you end up with some other particle, some other particle you didn't even have there. None of these particles were there before. And some charge q. Now we end up with these four different particles. These combined, there was some weird reaction and they created these particles. What is the charge of this q? This is a question we can answer now, and it's not even that hard. We know the charge of all the others. We know that if you add up all of these, you've got to add up to the same amount of charge you had previously, because the law of conservation of charge says is if you don't let any charge in or out, the total charge in here has to stay the same. So let's just do it. What do we do? We add them all up. We say that positive one plus negative seven coulombs plus negative four coulombs plus whatever charge this unknown, mystery particle is. We know what that has to equal. What does that have to equal? It has to equal the total charge, because this number does not change. This was the total charge before, positive four coulombs. That means it has to be the total charge afterward in there. That's what the law of conservation of charge says. So that has to equal positive four. Well, negative seven and negative four is negative 11, plus one is negative 10. So I get negative 10 coulombs, plus... Oh, you know what, these q's look like nines, sorry about that. This is law of conservation of charge. I'm gonna add a little tail. This isn't the law of conservation of nines. So this is a little q. This is a little q, not a nine. And so plus q equals four. Now we know that charge has to have a charge of 14 coulombs in order to satisfy this equation. But you don't even really need a box. I mean, nobody really does physics in cardboard box, so let's say we're doing an experiment and there was some particle x, an x particle. And it had a certain amount of charge, it had, say, positive three coulombs. That would be enormous for a particle, but for the sake of argument, say it has positive three coulombs. Well, it decays. Sometimes particles decay, they literally disappear, turn into other particles. Let's say it turns into y particle and z particle. Just give them random names. And you discover that this y particle had a charge of positive two coulombs and this z particle had a charge of negative one coulomb. Well, is this possible? No, this is not possible. If you discover this, something went wrong because this side over here, you started with positive three coulombs. Over here you've gotta end up, according to the law of conservation of charge, with positive three coulombs, but positive two coulombs minus one coulomb, that's only one coulomb. You're missing two coulombs over here. Where'd the other two coulombs go? Well, there had to be some sort of mystery particle over here that you missed. Something happened. Either your detector messed up or it just didn't detect a particle that had another amount of charge. How much charge should it have? This whole side's gotta add up to three. So if you started off with three, over here, these two together, y and z, are only one coulomb. That means that the remainder, the two coulombs, the missing two coulombs, has to be here. So you must've had some particle or some missed charge that has positive two coulombs. Is that another y particle? Maybe, that's why physics is fun. Maybe it is in there, maybe you missed another one. Let me ask you this. So let's say we get rid of all these charges. Here's one that freaks people out sometimes. Take this. Let's say this had no charge. No charge, it was uncharged. You got some particle with zero coulombs. Is it possible to end up with particles that have charge? Yeah, it can happen. In fact, if you have a photon that has no charge, it's possible for this photon to turn into charged particles. How is that possible? Doesn't that break the law of conservation of charge? No, but you've gotta make sure that whatever charge this gets, say positive three coulombs, then this one's going to have to have negative three coulombs so that the total amount of charge over here is zero coulombs just like it was before. So this is weird, but yeah, photon, a beam of light, can turn into an electron, but that means it has to also turn into an anti-electron because it has to have no total charge over here. And an anti-electron has the same charge as an electron, but positive instead of negative. Which is why it's called a positron. Anti-electrons are call positrons because they're the same as electrons, just positive. You don't really need to know that. In fact, you don't need to know a lot about particle physics, that's the whole point here. Just knowing conservation of charge lets you make statements about particle physics because you know the charge has to be conserved and that's a powerful tool in analyzing these reactions in terms of what's possible and what's not possible.

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:

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

where

\mathrm {d} Q/\mathrm {d} t

is the electric charge accumulation rate in a specific volume at time

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}}^{t}\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}}^{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,

{\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 $ρ$ (in coulombs per cubic meter) and electric current density $J$ (in amperes per square meter). This is called the charge density continuity equation

{\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

I=-\iint _{S}\mathbf {J} \cdot d\mathbf {S}

where

S = \partial V

is the boundary of

V

oriented by outward-pointing normals, and

d S

is shorthand for

N dS

, the outward pointing normal of the boundary

\partial 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

I=-\iiint _{V}\left(\nabla \cdot \mathbf {J} \right)dV

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

{\frac {dq}{dt}}=-\iiint _{V}\left(\nabla \cdot \mathbf {J} \right)dV

(1)

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

q=\iiint \limits _{V}\rho dV

So, by the Leibniz integral rule

{\frac {dq}{dt}}=\iiint _{V}{\frac {\partial \rho }{\partial t}}dV

(2)

Equating (1) and (2) gives

0=\iiint _{V}\left({\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} \right)dV.

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

{\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =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=\nabla \cdot (\nabla \times \mathbf {B} )=\nabla \cdot \left(\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +\varepsilon _{0}{\frac {\partial }{\partial t}}\nabla \cdot \mathbf {E} \right)=\mu _{0}\left(\nabla \cdot \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:

{\frac {d}{dt}}Q_{\Omega }={\frac {d}{dt}}\iiint _{\Omega }\rho \mathrm {d} V=-

$\oiint$

{\scriptstyle \partial \Omega }

\mathbf {J} \cdot {\rm {d}}\mathbf {S} =-I_{\partial \Omega }.

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.^[7] 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 $\phi$ . However the full symmetry is more complicated, and also involves the vector potential $\mathbf {A}$ . 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 $\chi$ :

\phi '=\phi -{\frac {\partial \chi }{\partial t}}\qquad \qquad \mathbf {A} '=\mathbf {A} +\nabla \chi .

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

\psi '=e^{iq\chi }\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 $|\psi |^{2}$ .^[8]

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]} 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.^[9] Gauge invariance also implies quantization of hypothetical magnetic charges.^[8]

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.^[10]^[11]

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.^[12] 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.^[13] 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 6.6×10²⁸ years (90% Confidence Level),^[14]^[15]

but there are theoretical arguments that such single-photon decays will never occur even if charge is not conserved.^[16] Charge disappearance tests are sensitive to decays without energetic photons, other unusual charge violating processes such as an electron spontaneously changing into a positron,^[17] and to electric charge moving into other dimensions. The best experimental bounds on charge disappearance are:

	e → anything	mean lifetime is greater than 6.4×10²⁴ years (68% CL)^[18]
	n → p + ν + ν	charge non-conserving decays are less than 8 × 10⁻²⁷ (68% CL) of all neutron decays^[19]

Notes

^ ^a ^b Purcell, Edward M.; Morin, David J. (2013). Electricity and magnetism (3rd ed.). Cambridge University Press. p. 4. ISBN 9781107014022.
^ S. Orito; M. Yoshimura (1985). "Can the Universe be Charged?". Physical Review Letters. 54 (22): 2457–2460. Bibcode:1985PhRvL..54.2457O. doi:10.1103/PhysRevLett.54.2457. PMID 10031347.
^ E. Masso; F. Rota (2002). "Primordial helium production in a charged universe". Physics Letters B. 545 (3–4): 221–225. arXiv:astro-ph/0201248. Bibcode:2002PhLB..545..221M. doi:10.1016/S0370-2693(02)02636-9. S2CID 119062159.
^ Heilbron, J.L. (1979). Electricity in the 17th and 18th centuries: a study of early Modern physics. University of California Press. p. 330. ISBN 978-0-520-03478-5.
^ Purrington, Robert D. (1997). Physics in the Nineteenth Century. Rutgers University Press. pp. 33. ISBN 978-0813524429. benjamin franklin william watson charge conservation.
^ The Papers of Benjamin Franklin. Vol. 3. Yale University Press. 1961. p. 142. Archived from the original on 2011-09-29. Retrieved 2010-11-25.
^ Bettini, Alessandro (2008). Introduction to Elementary Particle Physics. UK: Cambridge University Press. pp. 164–165. ISBN 978-0-521-88021-3.
^ ^a ^b Sakurai, J. J.; Napolitano, Jim (2017-09-21). Modern Quantum Mechanics. Cambridge University Press. ISBN 978-1-108-49999-6.
^ A.S. Goldhaber; M.M. Nieto (2010). "Photon and Graviton Mass Limits". Reviews of Modern Physics. 82 (1): 939–979. arXiv:0809.1003. Bibcode:2010RvMP...82..939G. doi:10.1103/RevModPhys.82.939. S2CID 14395472.; see Section II.C Conservation of Electric Charge
^ S.Y. Chu (1996). "Gauge-Invariant Charge Nonconserving Processes and the Solar Neutrino Puzzle". Modern Physics Letters A. 11 (28): 2251–2257. Bibcode:1996MPLA...11.2251C. doi:10.1142/S0217732396002241.
^ S.L. Dubovsky; V.A. Rubakov; P.G. Tinyakov (2000). "Is the electric charge conserved in brane world?". Journal of High Energy Physics. August (8): 315–318. arXiv:hep-ph/0007179. Bibcode:1979PhLB...84..315I. doi:10.1016/0370-2693(79)90048-0.
^ Patrignani, C. et al (Particle Data Group) (2016). "The Review of Particle Physics" (PDF). Chinese Physics C. 40 (100001). Retrieved March 26, 2017.
^ Particle Data Group (May 2010). "Tests of Conservation Laws" (PDF). Journal of Physics G. 37 (7A): 89–98. Bibcode:2010JPhG...37g5021N. doi:10.1088/0954-3899/37/7A/075021.
^ Agostini, M.; et al. (Borexino Coll.) (2015). "Test of Electric Charge Conservation with Borexino". Physical Review Letters. 115 (23): 231802. arXiv:1509.01223. Bibcode:2015PhRvL.115w1802A. doi:10.1103/PhysRevLett.115.231802. PMID 26684111. S2CID 206265225.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Back, H.O.; et al. (Borexino Coll.) (2002). "Search for electron decay mode e → γ + ν with prototype of Borexino detector". Physics Letters B. 525 (1–2): 29–40. Bibcode:2002PhLB..525...29B. doi:10.1016/S0370-2693(01)01440-X.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ L.B. Okun (1989). "Comments on Testing Charge Conservation and the Pauli Exclusion Principle". Comments on Testing Charge Conservation and Pauli Exclusion Principle (PDF). World Scientific Lecture Notes in Physics. Vol. 19. pp. 99–116. doi:10.1142/9789812799104_0006. ISBN 978-981-02-0453-2. S2CID 124865855. {{cite book}}: |journal= ignored (help)
^ R.N. Mohapatra (1987). "Possible Nonconservation of Electric Charge". Physical Review Letters. 59 (14): 1510–1512. Bibcode:1987PhRvL..59.1510M. doi:10.1103/PhysRevLett.59.1510. PMID 10035254.
^ P. Belli; et al. (1999). "Charge non-conservation restrictions from the nuclear levels excitation of ¹²⁹Xe induced by the electron's decay on the atomic shell". Physics Letters B. 465 (1–4): 315–322. Bibcode:1999PhLB..465..315B. doi:10.1016/S0370-2693(99)01091-6. This is the most stringent of several limits given in Table 1 of this paper.
^ Norman, E.B.; Bahcall, J.N.; Goldhaber, M. (1996). "Improved limit on charge conservation derived from ⁷¹Ga solar neutrino experiments". Physical Review. D53 (7): 4086–4088. Bibcode:1996PhRvD..53.4086N. doi:10.1103/PhysRevD.53.4086. PMID 10020402. S2CID 41992809. Link is to preprint copy.

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