Charge retention

Preservation of charge describes the physical fact that in every closed system the sum of the existing electrical charge remains constant. When charged particles are created or destroyed, this always happens in the same quantities with opposite signs . The fact that individual charges cannot be generated or destroyed also follows from the validity of Gaussian law together with the theory of relativity (see Gaussian law ).

Corresponding charge conservation laws exist in various calibration theories , such as quantum chromodynamics (conservation of the color charge , associated calibration group SU (3)) and the calibration theory of the electro-weak interaction (calibration group SU (2) x U (1)), which are in the standard model of elementary particle physics Quantum electrodynamics generalized.

Derivation of the differential equation for the conservation of the electric charge

As with the conservation of mass ( continuity equation ), the conservation of charge can also be formulated as a differential equation. It is derived in the following.

Let us assume a contiguous area of ​​space with the volume (volume content) that is enclosed by a surface . ${\displaystyle V}$ ${\displaystyle \mathbf {S} }$

The current that flows out of the space area is

${\displaystyle I=-\iint \limits _{S}\mathbf {j} \cdot d\mathbf {S} }$

where integration takes place over the surface of the spatial area and the product is to be understood as the inner product of the current density vector with the normal vector of the surface.

With Gauss's integral theorem it follows:

${\displaystyle I=-\iiint \limits _{V}\left(\operatorname {div} \mathbf {j} \right)dV}$.

And since the current flow out of the spatial area is equal to the temporal change of the charge in the spatial area, the following applies:

${\displaystyle {\frac {dq}{dt}}=-\iiint \limits _{V}\left(\operatorname {div} \mathbf {j} \right)dV}$.

Charge and charge density are over

${\displaystyle q=\iiint \limits _{V}\rho dV}$.

connected so that one obtains:

${\displaystyle 0=\iiint \limits _{V}\left({\frac {\partial \rho }{\partial t}}+\operatorname {div} \mathbf {j} \right)dV}$.

Since this applies to every connected spatial area, it also applies to the borderline case of an infinitely small spatial area, for a point in space :

${\displaystyle {\frac {\partial \rho }{\partial t}}+\operatorname {div} \mathbf {j} =0}$.

It is of the same mathematical form as the continuity equation following from the conservation of mass (one only has to replace the charge density with mass density, etc.).

The conservation of the electric charge is also implicit in the Maxwell equations :

${\displaystyle {\begin{aligned}\operatorname {div} \mathbf {E} &={\frac {\rho }{\varepsilon _{0}}}\\\operatorname {rot} \mathbf {B} -\mu _{0}\,\varepsilon _{0}\,{\frac {\partial \mathbf {E} }{\partial t}}&=\mu _{0}\,\mathbf {j} .\end{aligned}}}$

Because the divergence of a rotation vanishes, the second equation follows when the divergence is formed

${\displaystyle {\begin{aligned}-\mu _{0}\,\varepsilon _{0}\,\operatorname {div} {\frac {\partial \mathbf {E} }{\partial t}}&=\mu _{0}\,\operatorname {div} \mathbf {j} .\end{aligned}}}$

If the time derivative of the first equation is inserted into this relationship, it follows

${\displaystyle {\frac {\partial \rho }{\partial t}}+\operatorname {div} \mathbf {j} =0}$.

Noether theorem

According to Noether's theorem , every conservation law is associated with a symmetry property of the respective theory, ie its invariance under gauge transformations . In the case of (quantum) electrodynamics this is the invariance under global gauge transformations (gauge group U (1), multiplication with a complex phase factor) of the wave function of charged particles:

${\displaystyle \psi (x)\rightarrow \exp {(i\alpha )}\psi (x)}$.

Use of language

In the apparent contradiction to the maintenance of charge, there is the expression of a charge generation, for example through friction . However, this means a local accumulation of charges of one sign, ie a charge separation and not a “generation”.