Charge conservation

In

${\displaystyle \rho (\mathbf {x} )}$ 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:

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:

The general solution is obtained by fixing the initial condition time ${\displaystyle t_{0}}$, leading to the integral equation:

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:

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

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 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

where S = ∂V is the boundary of V oriented by outward-pointing , 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

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

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

(1)

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

So, by the Leibniz integral rule

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

(2)

Equating (1) and (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:

i.e.,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:

${\displaystyle {\frac {d}{dt}}Q_{\Omega }={\frac {d}{dt}}\iiint _{\Omega }\rho \mathrm {d} V=-}$ ${\displaystyle {\scriptstyle \partial \Omega }}$ ${\displaystyle \mathbf {J} \cdot {\rm {d}}\mathbf {S} =-I_{\partial \Omega }.}$

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

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

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

In quantum mechanics the scalar field is equivalent to a

of the charged particle:

${\displaystyle \psi '=e^{iq\chi }\psi }$

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 ${\displaystyle |\psi |^{2}}$.^[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.^[

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

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]

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

1. ^
   ISBN 9781107014022
   .
2. ^ S. Orito; M. Yoshimura (1985). "Can the Universe be Charged?". Physical Review Letters. 54 (22): 2457–2460.
   PMID 10031347
   .
3. ^ E. Masso; F. Rota (2002). "Primordial helium production in a charged universe". Physics Letters B. 545 (3–4): 221–225.
   S2CID 119062159
   .
4. ISBN 978-0-520-03478-5
   .
5. ISBN 978-0813524429
   . benjamin franklin william watson charge conservation.
6. ^ 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.
7. ISBN 978-0-521-88021-3
   .
8. ^
   ISBN 978-1-108-49999-6
   .
9. ^ A.S. Goldhaber; M.M. Nieto (2010). "Photon and Graviton Mass Limits". Reviews of Modern Physics. 82 (1): 939–979.
   S2CID 14395472
   .; see Section II.C Conservation of Electric Charge
10. ^ S.Y. Chu (1996). "Gauge-Invariant Charge Nonconserving Processes and the Solar Neutrino Puzzle". Modern Physics Letters A. 11 (28): 2251–2257.
    doi:10.1142/S0217732396002241
    .
11. ^ 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.
    doi:10.1016/0370-2693(79)90048-0
    .
12. ^ Patrignani, C. et al (Particle Data Group) (2016). "The Review of Particle Physics" (PDF). Chinese Physics C. 40 (100001). Retrieved March 26, 2017.
13. ^
    doi:10.1088/0954-3899/37/7A/075021
    .
14. S2CID 206265225.{{cite journal}}: CS1 maint: multiple names: authors list (link
    )
15. ^ Back, H.O.; et al. (
    doi:10.1016/S0370-2693(01)01440-X.{{cite journal}}: CS1 maint: multiple names: authors list (link
    )
16. ^
    S2CID 124865855. {{cite book}}: |journal= ignored (help
    )
17. ^
    PMID 10035254
    .
18. ^
    doi:10.1016/S0370-2693(99)01091-6
    . This is the most stringent of several limits given in Table 1 of this paper.
19. ^ Norman, E.B.;
    S2CID 41992809
    . Link is to preprint copy.

Further reading

• ISBN 978-0-8122-4121-1
  .