charge density

• 電荷密度

関

density of electron charge、electric charge density

WordNet

1. set or ask for a certain price; "How much do you charge for lunch?"; "This fellow charges $100 for a massage"
2. the quantity of unbalanced electricity in a body (either positive or negative) and construed as an excess or deficiency of electrons; "the battery needed a fresh charge" (同)electric_charge
3. financial liabilities (such as a tax); "the charges against the estate"
4. the price charged for some article or service; "the admission charge"
5. heraldry consisting of a design or image depicted on a shield (同)bearing, heraldic bearing, armorial bearing
6. a quantity of explosive to be set off at one time; "this cartridge has a powder charge of 50 grains" (同)burster, bursting charge, explosive charge
7. an impetuous rush toward someone or something; "the wrestlers charge carried him past his adversary"; "the battle began with a cavalry charge"
8. request for payment of a debt; "they submitted their charges at the end of each month" (同)billing
9. (criminal law) a pleading describing some wrong or offense; "he was arrested on a charge of larceny" (同)complaint
10. a person committed to your care; "the teacher led her charges across the street"
11. blame for, make a claim of wrongdoing or misbehavior against; "he charged the director with indifference" (同)accuse
12. to make a rush at or sudden attack upon, as in battle; "he saw Jess charging at him with a pitchfork" (同)bear_down
13. demand payment; "Will I get charged for this service?"; "We were billed for 4 nights in the hotel, although we stayed only 3 nights" (同)bill
14. direct into a position for use; "point a gun"; "He charged his weapon at me" (同)level, point
15. file a formal charge against; "The suspect was charged with murdering his wife" (同)lodge, file
16. impose a task upon, assign a responsibility to; "He charged her with cleaning up all the files over the weekend" (同)saddle, burden
17. cause formation of a net electrical charge in or on; "charge a conductor"
18. energize a battery by passing a current through it in the direction opposite to discharge; "I need to charge my car battery"
19. enter a certain amount as a charge; "he charged me $15"
20. fill or load to capacity; "charge the wagon with hay"
21. instruct (a jury) about the law, its application, and the weighing of evidence
22. instruct or command with authority; "The teacher charged the children to memorize the poem"
23. lie down on command, of hunting dogs
24. make an accusatory claim; "The defense attorney charged that the jurors were biased"
25. pay with a credit card; pay with plastic money; postpone payment by recording a purchase as a debt; "Will you pay cash or charge the purchase?"
26. place a heraldic bearing on; "charge all weapons, shields, and banners"
27. saturate; "The room was charged with tension and anxiety"
28. burn to charcoal; "Without a drenching rain, the forest fire will char everything" (同)coal
29. burn slightly and superficially so as to affect color; "The cook blackened the chicken breast"; "The fire charred the ceiling above the mantelpiece"; "the flames scorched the ceiling" (同)blacken, sear, scorch
30. a charred substance
31. any of several small trout-like fish of the genus Salvelinus (同)charr
32. of a particle or body or system; having a net amount of positive or negative electric charge; "charged particles"; "a charged battery"
33. fraught with great emotion; "an atmosphere charged with excitement"; "an emotionally charged speech" (同)supercharged
34. capable of producing violent emotion or arousing controversy; "the highly charged issue of abortion"
35. the amount per unit size (同)denseness

PrepTutorEJDIC

1. 〈C〉(…の)『費用』;料金,請求代金《+『for』+『名』(do『ing』)》 / 〈U〉(…の)『監督』,管理,保護,世話《+『of』+『名』》;〈C〉監督(保護)される人(物) / 〈C〉(…に対する)『非難』,告発,告訴《+『against』+『名』》 / 〈C〉突撃;突撃の合図 / 〈U〉〈C〉(弾薬などの)装てん,装薬;充電《+『of』+『名』》 / 〈C〉(…の)命令,訓令《+『of』+『名』》 / 〈C〉責任,義務 / 〈料金・支払い〉'を'〈人〉‘に'『請求する』,〈税など〉'を'課する / 〈買物など〉'を'『つけにする』,掛け売りにする / (…を)〈物・容器〉‘に'詰める,入れる(fill)《+『名』+『with』+『名』》 / …'を'『非難する』,責める(blame),告訴する(accuse) / (義務・仕事・責任などを)〈人〉‘に'ゆだねる,委託する《+『名』+『with』+『名』》 / 《『charge』+『名』+『to』 do》(…するように)〈人〉‘に'命ずる,訓示する / …‘に'突進する / (…の)値をいう,支払いを請求する《+『for』+『名』》 / (…に)突撃する,突進する《+『at』(『on』)+『名』》
2. …'を'焦がす / (焼いて)…'を'炭にする / こげる
3. (時間ぎめや日雇いで)〈婦人が〉雑役をする
4. =chatwoman
5. =tea
6. 強く気持ちを動かされた / 人の気持ちを強く動かす,論争を巻き起こす
7. 〈U〉『密集』;『密度』,濃さ / 〈U〉〈C〉(物質の)密度,濃度,比重 / 〈U〉《話》頭の鈍さ,愚鈍さ

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/02/06 12:32:51」(JST)

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Charge distribution redirects here. This article is about the physical quantity in electromagnetism. For other uses see charge and density (disambiguation).

Electromagnetism
Electricity Magnetism
Electrostatics Electric charge Static electricity Electric field Conductor Insulator Triboelectricity Electrostatic discharge Induction Coulomb's law Gauss's law Electric flux / potential energy Electric dipole moment Polarization density
Magnetostatics Ampère's law Magnetic field Magnetization Magnetic flux Biot–Savart law Magnetic dipole moment Gauss's law for magnetism
Electrodynamics Lorentz force law Electromagnetic induction Faraday's law Lenz's law Displacement current Magnetic potential Maxwell's equations Electromagnetic field Electromagnetic radiation Maxwell tensor Poynting vector Liénard–Wiechert potential Jefimenko's equations Eddy current London equations
Electrical network Electric current Electric potential Voltage Resistance Ohm's law Series circuit Parallel circuit Direct current Alternating current Charged current Neutral current Electromotive force Capacitance Inductance Impedance Resonant cavities Waveguides
Covariant formulation Electromagnetic tensor (stress–energy tensor) Four-current Electromagnetic four-potential
Scientists Ampère Coulomb Faraday Gauss Heaviside Henry Hertz Lorentz Maxwell Tesla Volta Weber Ørsted

In electromagnetism, charge density is a measure of electric charge per unit volume of space, in one, two or three dimensions. More specifically: the linear, surface, or volume charge density is the amount of electric charge per unit length, surface area, or volume, respectively. The respective SI units are C·m⁻¹, C·m⁻² or C·m⁻³.

Like any density, charge density can depend on position, but charge and thus charge density can be negative. It should not be confused with the charge carrier density, the number of charge carriers (e.g. electrons, ions) in a material per unit volume, not including the actual charge on the carriers

In chemistry, it can refer to the charge distribution over the volume of a particle; such as a molecule, atom or ion. Therefore, a lithium cation will carry a higher charge density than a sodium cation due to the lithium cation's having a smaller ionic radius, even though sodium has more protons (11) than lithium (3).

Definitions

Continuous charges

Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P. Position vector r is a point to calculate the electric field; r′ is a point in the charged object.

Following are the definitions for continuous charge distributions.

The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element,

similarly the surface charge density uses a surface area element dS

and the volume charge density uses a volume element dV

Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λ_q(r) over a line or 1d curve C,

similarly a surface integral of the surface charge density σ_q(r) over a surface S,

and a volume integral of the volume charge density ρ_q(r) over a volume V,

where the subscript q is to clarify that the density is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ, σ, ρ in electromagnetism for wavelength, electrical resistivity and conductivity.

Within the context of electromagnetism, the subscripts are usually dropped for simplicity: λ, σ, ρ. Other notations may include: ρ_ℓ, ρ_s, ρ_v, ρ_L, ρ_S, ρ_V etc.

Average charge densities

The total charge divided by the length, surface area, or volume will be the average charge densities:

Free, bound and total charge

In dielectric materials, the total charge of an object can separate into "free" and "bound" charges.

Bound charges set up electric dipoles in response to an applied electric field E, and polarize other nearby dipoles tending to line them up, the net accumulation of charge from the orientation of the dipoles is the bound charge. They are called bound because they cannot be removed: in the dielectric material the charges are the electrons bound to the nuclei.

Free charges are the excess charges which can move into electrostatic equilibrium, i.e. when the charges are not moving and the resultant electric field is independent of time, or constitute electric currents.

Total charge densities

In terms of volume charges densities, the total charge density is:

as for surface charge densities:

where subscripts "f" and "b" denote "free" and "bound" respectively.

Bound charge

The bound surface charge is the charge piled-up at the surface of the dielectric, given by the dipole moment perpendicular to the surface:

where s is the separation between the point charges constituting the dipole. Taking infinitesimals:

and dividing by the differential surface element dS gives the bound surface charge density:

where P is the polarization density, i.e. density of electric dipole moments within the material, and dV is the differential volume element.

Using the divergence theorem, the bound volume charge density within the material is

hence:

The negative sign arises due to the opposite signs on the charges in the dipoles, one end is within the volume of the object, the other at the surface.

A more rigorous derivation is given below.

Derivation of bound surface and volume charge densities from internal dipole moments (bound charges)

The electric potential due to a dipole moment d is: For a continuous distribution, the material can be divided up into infinitely many infinitesimal dipoles where dV = d3r′ is the volume element, so the potential is the volume integral over the object: Since where ∇′ is the gradient in the r′ coordinates, integrating by parts using the divergence theorem: which separates into the potential of the surface charge (surface integral) and the potential due to the volume charge (volume integral): that is

Free charge density

The free charge density serves as a useful simplification in Gauss's law for electricity; the volume integral of it is the free charge enclosed in a charged object - equal to the net flux of the electric displacement field D emerging from the object:

See Maxwell's equations and constitutive relation for more details.

Homogeneous charge density

For the special case of a homogeneous charge density ρ₀, independent of position i.e. constant throughout the region of the material, the equation simplifies to:

The proof of this is immediate. Start with the definition of the charge of any volume:

Then, by definition of homogeneity, ρ_q(r) is a constant denoted by ρ_{q, 0} (to differ between the constant and non-constant densities), and so by the properties of an integral can be pulled outside of the integral resulting in:

so,

The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.

Discrete charges

For a single point charge q at position r₀ inside a region of 3d space R, like an electron, the volume charge density can be expressed by the Dirac delta function:

where r is the position to calculate the charge.

As always, the integral of the charge density over a region of space is the charge contained in that region. The delta function has the sifting property for any function f:

so the delta function ensures that when the charge density is integrated over R, the total charge in R is q:

This can be extended to N discrete point-like charge carriers. The charge density of the system at a point r is a sum of the charge densities for each charge q_i at position r_i, where i = 1, 2, ..., N:

The delta function for each charge q_i in the sum, δ(r − r_i), ensures the integral of charge density over R returns the total charge in R:

If all charge carriers have the same charge q (for electrons q = −e, the electron charge) the charge density can be expressed through the number of charge carriers per unit volume, n(r), by

Similar equations are used for the linear and surface charge densities.

Charge density in special relativity

Further information: classical electromagnetism and special relativity and relativistic electromagnetism

In special relativity, the length of a segment of wire depends on velocity of observer because of length contraction, so charge density will also depend on velocity. Anthony French has described how the magnetic field force of a current-bearing wire arises from this relative charge density. He used (p 260) a Minkowski diagram to show "how a neutral current-bearing wire appears to carry a net charge density as observed in a moving frame." When a charge density is measured in a moving frame of reference it is called proper charge density.

It turns out the charge density ρ and current density J transform together as a four current vector under Lorentz transformations.

Charge density in quantum mechanics

Main article: quantum mechanics

In quantum mechanics, charge density ρ_q is related to wavefunction ψ(r) by the equation

where q is the charge of the particle and |ψ(r)|² = ψ*(r)ψ(r) is the probability density function i.e. probability per unit volume of a particle located at r.

When the wavefunction is normalized - the average charge in the region r ∈ R is

where d³r is the integration measure over 3d position space.

Application

The charge density appears in the continuity equation for electric current, also in Maxwell's Equations. It is the principal source term of the electromagnetic field, when the charge distribution moves this corresponds to a current density.

See also

• Continuity equation relating charge density and current density
• Ionic potential
• charge density wave

References

• A. Halpern (1988). 3000 Solved Problems in Physics. Schaum Series, Mc Graw Hill. ISBN 978-0-07-025734-4. 
• G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2. 
• P. A. Tipler, G. Mosca (2008). Physics for Scientists and Engineers - with Modern Physics (6th ed.). Freeman. ISBN 978-0-7167-8964-2. 
• R.G. Lerner, G.L. Trigg (1991). Encyclopaedia of Physics (2nd ed.). VHC publishers. ISBN 978-0-89573-752-6. 
• C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). VHC publishers. ISBN 978-0-07-051400-3. 

External links

• [1] - Spatial charge distributions

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Related Pictures

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★リンクテーブル★

リンク元	「電荷密度」「density of electron charge」
拡張検索	「electric charge density」
関連記事	「charge」「char」「charging」

「電荷密度」

　　[★]

英

charge density、electric charge density、density of electron charge

「density of electron charge」

　　[★]

• 電荷密度

関

charge density、electric charge density

「electric charge density」

　　[★]

• 電荷密度

関

charge density、density of electron charge

「charge」

　　[★]

• n.

• 電荷、負荷、帯電、チャージ

• v.

• 荷電する、チャージする、負担させる

関

burden、charging、electric charge、electrification、load、remuneration

　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

「char」

　　[★]

• n.

• 木炭、(イワナなどの淡水魚)チャー、

• v.

• 黒焦げにする

関

charcoal、Salvelinus、trout

　 　 　 　 　 　 　 　

「charging」

　　[★]

• 荷電

関

charge、electric charge、load