Charge density

Physical size
Surname Space charge density
Formula symbol ${\ displaystyle \ rho}$
Size and
unit system
unit dimension
SI A · s · m -3 I · T · L −3
Gauss ( cgs ) Fr · cm −3 M 1/2  · L −3/2  · T −1
esE ( cgs ) Fr · cm −3 M 1/2  · L −3/2  · T −1
emE ( cgs ) abC · cm -3 L -5/2 · M 1/2
Physical size
Surname Surface charge density
Formula symbol ${\ displaystyle \ sigma}$
Size and
unit system
unit dimension
SI A · s · m -2 I · T · L −2
Gauss ( cgs ) Fr · cm −2 M 1/2  · L −1/2  · T −1
esE ( cgs ) Fr · cm −2 M 1/2  · L −1/2  · T −1
emE ( cgs ) abC · cm -2 L -3/2 · M 1/2
Physical size
Surname Line charge density
Formula symbol ${\ displaystyle \ lambda}$
Size and
unit system
unit dimension
SI A · s · m -1 I · T · L -1
Gauss ( cgs ) Fr · cm -1 M 1/2  · L 1/2  · T −1
esE ( cgs ) Fr · cm -1 M 1/2  · L 1/2  · T −1
emE ( cgs ) abC · cm -1 L -1/2 · M 1/2

The electrical charge density is a physical quantity from electrodynamics that describes a charge distribution. Since there are both positive and negative charges, both positive and negative values ​​are also possible for the charge density.

Since charges can also be distributed on surfaces or along a thin wire, the charge density can be described by the following quantities:

• the charge per volume ( space charge density ), common symbol  ρ (rho)
• the charge per area ( surface charge density ), common symbol  σ (sigma)
• the charge per length ( line charge density ), common symbol  λ (lambda).

Limitation of the surface charge density

The achievable surface charge density is limited by corona discharge into the surrounding air if the maximum field strength of about 10 5  V / m is exceeded:

${\ displaystyle \ sigma _ {\ mathrm {max}} = 2 \ cdot E _ {\ mathrm {max}} \ cdot \ varepsilon _ {0} \ cdot \ varepsilon _ {r} \ approx 1 {,} 8 \ cdot 10 ^ {- 6} \ mathrm {As / m ^ {2}}.}$

So every negatively charged square centimeter carries the excess charge 1.8 · 10 −10  As, which corresponds to 1.1 · 10 9  freely moving electrons . About a million times more electrons are bound to the atomic cores of the metal surface (see also Influenz # of electrons involved ).

The surface charge density on the right half of the metal ball is negative because the electrons escape there due to the repulsion by the negative charge shown on the left; on the left hemisphere the surface charge density is positive because electrons are now missing there.

Similar sizes

A quantity corresponding to the surface charge density σ is the electrical flux density (also called electrical excitation, dielectric displacement or displacement density ), a vector that is perpendicular to the surface in question . In contrast, σ is a scalar (and under certain circumstances it is equal to the absolute value ). ${\ displaystyle {\ vec {D}}}$ ${\ displaystyle | {\ vec {D}} |}$

Not to be confused with the charge density, the charge carrier density , i.e. the number of protons, electrons etc. per unit of space, area or length, as well as the electron density calculated in density functional theory .

definition

The definition of the space charge density is similar to that of the mass density :

${\ displaystyle \ rho ({\ vec {r}}) = {\ frac {\ mathrm {d} Q} {\ mathrm {d} V}} \ quad \ Leftrightarrow \ quad Q = \ int _ {V} \ rho ({\ vec {r}}) \, \ mathrm {d} V}$,

where Q is the electrical charge and V is the volume.

For the area and line charge density , the following is derived from the area or length : ${\ displaystyle A}$${\ displaystyle l}$

${\ displaystyle \ sigma ({\ vec {r}}) = {\ frac {\ mathrm {d} Q} {\ mathrm {d} A}} \ quad \ Leftrightarrow \ quad Q = \ int _ {A} \ sigma ({\ vec {r}}) \, \ mathrm {d} A}$
${\ displaystyle \ lambda ({\ vec {r}}) = {\ frac {\ mathrm {d} Q} {\ mathrm {d} l}} \ quad \ Leftrightarrow \ quad Q = \ int _ {l} \ lambda ({\ vec {r}}) \, \ mathrm {d} l.}$

Discrete charge distribution

If the charge in a volume consists of discrete charge carriers (such as electrons), the charge density can be expressed using the delta distribution : ${\ displaystyle N}$

${\ displaystyle \ rho ({\ vec {r}}) = \ sum _ {i = 1} ^ {N} q_ {i} \ cdot \ delta ({\ vec {r}} - {\ vec {r} } _ {i})}$

With

• the cargo and${\ displaystyle q_ {i}}$
• the location of the -th load carrier.${\ displaystyle {\ vec {r}} _ {i}}$${\ displaystyle i}$

If all charge carriers carry the same charge (with electrons the same as the negative elementary charge :) , the above formula can be simplified with the help of the charge carrier density : ${\ displaystyle q}$${\ displaystyle q = -e}$${\ displaystyle n ({\ vec {r}})}$

{\ displaystyle {\ begin {aligned} \ rho ({\ vec {r}}) & = q \ cdot \ sum _ {i = 1} ^ {N} \ delta ({\ vec {r}} - {\ vec {r}} _ {i}) \\ & = q \ cdot n ({\ vec {r}}). \ end {aligned}}}

Electrical potential

The electrical potential depends on the Poisson equation of electrostatics

${\ displaystyle \ Delta \ Phi ({\ vec {r}}) = - {\ frac {\ rho ({\ vec {r}})} {\ varepsilon}}}$

only depends on the charge density. Here denotes the permittivity . ${\ displaystyle \ varepsilon}$

Point of zero charge

The point of zero charge (PZC) is reached when the charge density of a surface is zero. This concept comes from physical chemistry and is relevant for the adsorption of substances or particles on surfaces.

For particles in suspension, the PZC is the point at which the zeta potential is zero. This can be the case for a certain pH value , for example . Aside from the PZC, the particles are charged and therefore repel each other electrically and are less likely to clump into flakes or aggregates. The lack of charge on the PZC also leads to a reduction in solubility / hydration in water.

Knowledge of the PZC is useful for assessing the mobility of dissolved substances or particles, which among other things can play a role in the risk assessment of pollutants.

A similar concept is the isoelectric point .