Electric capacity

Electric capacity
Electric capacity
Dimension	L -2 M -1 T 4 I 2
Units
SI	farad
GHS	centimeter

Electric capacitance - a characteristic of a conductor , a measure of its ability to accumulate an electric charge . In the theory of electrical circuits, capacitance refers to the mutual capacitance between two conductors; parameter of the capacitive element of the electric circuit, presented in the form of a two-terminal device. Such a capacitance is defined as the ratio of the electric charge to the potential difference between these conductors ^[1] .

In the International System of Units (SI), capacity is measured in farads , in the GHS system - in centimeters .

For a single conductor, the capacitance is equal to the ratio of the charge of the conductor to its potential under the assumption that all other conductors are infinitely distant and that the potential of the infinitely distant point is taken equal to zero. In mathematical form, this definition has the form

C={\frac {Q}{\varphi }},

{\ displaystyle C = {\ frac {Q} {\ varphi}},}

C = {\ frac {Q} {\ varphi}},

Where $Q$ ${\ displaystyle Q}$ $Q$ - charge $\varphi$ ${\ displaystyle \ varphi}$ $\ varphi$ - the potential of the conductor.

The capacity is determined by the geometric dimensions and shape of the conductor and the electrical properties of the environment (its dielectric constant) and does not depend on the material of the conductor. For example, the capacity of a conducting ball (or sphere) of radius R is (in the SI system):

C=4\pi \varepsilon _{0}\varepsilon _{r}R,

{\ displaystyle C = 4 \ pi \ varepsilon _ {0} \ varepsilon _ {r} R,}

C = 4 \ pi \ varepsilon _ {0} \ varepsilon _ {r} R,

where ε ₀ is the electric constant equal to 8.854⋅10 ⁻¹² F / m , ε _r is the relative dielectric constant .

Formula output

It is known that $\varphi _{1}-\varphi _{2}=\int _{1}^{2}E\,dl\Rightarrow \varphi =\int _{R}^{\mathcal {\infty }}E\,dl={\frac {1}{4\pi \varepsilon _{r}\varepsilon _{0}}}\int _{R}^{\mathcal {\infty }}{\frac {q}{r^{2}}}\,dr={\frac {1}{4\pi \varepsilon \varepsilon _{0}}}{\frac {q}{R}}.$ ${\ displaystyle \ varphi _ {1} - \ varphi _ {2} = \ int _ {1} ^ {2} E \, dl \ Rightarrow \ varphi = \ int _ {R} ^ {\ mathcal {\ infty} } E \, dl = {\ frac {1} {4 \ pi \ varepsilon _ {r} \ varepsilon _ {0}}} \ int _ {R} ^ {\ mathcal {\ infty}} {\ frac {q } {r ^ {2}}} \, dr = {\ frac {1} {4 \ pi \ varepsilon \ varepsilon _ {0}}} {\ frac {q} {R}}.}$ ${\ displaystyle \ varphi _ {1} - \ varphi _ {2} = \ int _ {1} ^ {2} E \, dl \ Rightarrow \ varphi = \ int _ {R} ^ {\ mathcal {\ infty} } E \, dl = {\ frac {1} {4 \ pi \ varepsilon _ {r} \ varepsilon _ {0}}} \ int _ {R} ^ {\ mathcal {\ infty}} {\ frac {q } {r ^ {2}}} \, dr = {\ frac {1} {4 \ pi \ varepsilon \ varepsilon _ {0}}} {\ frac {q} {R}}.}$

Because $C={\frac {q}{\varphi }}$ ${\ displaystyle C = {\ frac {q} {\ varphi}}}$ $C = {\ frac {q} {\ varphi}}$ then substituting the found here $\varphi$ ${\ displaystyle \ varphi}$ $\ varphi$ we get that $C=4\pi \varepsilon _{0}\varepsilon _{r}R.$ ${\ displaystyle C = 4 \ pi \ varepsilon _ {0} \ varepsilon _ {r} R.}$ $C = 4 \ pi \ varepsilon _ {0} \ varepsilon _ {r} R.$

The concept of capacitance also refers to a system of conductors, in particular, to a system of two conductors separated by a dielectric or vacuum , a capacitor . In this case, the capacitance (mutual capacitance) of these conductors (capacitor plates) will be equal to the ratio of the charge accumulated by the capacitor to the potential difference between the plates. For a flat capacitor, the capacity is:

C=\varepsilon _{0}\varepsilon _{r}{\frac {S}{d}},

{\ displaystyle C = \ varepsilon _ {0} \ varepsilon _ {r} {\ frac {S} {d}},}

C = \ varepsilon _ {0} \ varepsilon _ {r} {\ frac Sd},

where S is the area of one plate (it is assumed that the plates are the same), d is the distance between the plates, ε _r is the relative dielectric constant of the medium between the plates.

Content

Electrical capacity of some systems

The calculation of the electric capacity of the system requires the solution of the Laplace Equation ∇ ² φ = 0 with a constant potential φ on the surface of the conductors. This is trivial in cases with high symmetry. There is no solution in terms of elementary functions in more complex cases.

In quasi-two-dimensional cases, analytic functions map one situation to another; the electric capacitance does not change with such maps. See also Schwartz-Christoffel Mapping .

Electric capacitance of simple systems (GHS)
View	Capacity	Comment
Flat capacitor	${\frac {\varepsilon S}{d}}$ ${\ displaystyle {\ frac {\ varepsilon S} {d}}}$	S : Area d : distance
Coaxial cable	${\frac {2\pi \varepsilon l}{\ln \left(R_{2}/R_{1}\right)}}$ ${\ displaystyle {\ frac {2 \ pi \ varepsilon l} {\ ln \ left (R_ {2} / R_ {1} \ right)}}}$	l : Length R ₁ : Radius R ₂ : Radius
Two parallel wires ^[2]	${\frac {\pi \varepsilon l}{\operatorname {arcosh} \left({\frac {d}{2a}}\right)}}={\frac {\pi \varepsilon l}{\ln \left({\frac {d}{2a}}+{\sqrt {{\frac {d^{2}}{4a^{2}}}-1}}\right)}}$ ${\ displaystyle {\ frac {\ pi \ varepsilon l} {\ operatorname {arcosh} \ left ({\ frac {d} {2a}} \ right)}} = {\ frac {\ pi \ varepsilon l} {\ ln \ left ({\ frac {d} {2a}} + {\ sqrt {{\ frac {d ^ {2}} {4a ^ {2}}} - 1}} \ right)}}}$	a : Radius d : Distance, d> 2a
The wire is parallel to the wall ^[2]	${\frac {2\pi \varepsilon l}{\operatorname {arcosh} \left({\frac {d}{a}}\right)}}={\frac {2\pi \varepsilon l}{\ln \left({\frac {d}{a}}+{\sqrt {{\frac {d^{2}}{a^{2}}}-1}}\right)}}$ ${\ displaystyle {\ frac {2 \ pi \ varepsilon l} {\ operatorname {arcosh} \ left ({\ frac {d} {a}} \ right)}} = {\ frac {2 \ pi \ varepsilon l} {\ ln \ left ({\ frac {d} {a}} + {\ sqrt {{\ frac {d ^ {2}} {a ^ {2}}} - 1}} \ right)}}}$	a : Radius d : Distance, d> a l : Length
Two parallel coplanar stripes ^[3]	$\varepsilon l{\frac {K\left({\sqrt {1-k^{2}}}\right)}{K\left(k\right)}}$ ${\ displaystyle \ varepsilon l {\ frac {K \ left ({\ sqrt {1-k ^ {2}}} \ right)} {K \ left (k \ right)}}}$	d : distance w ₁ , w ₂ : Bandwidth k _m : d / (2w _m + d) k ² : k ₁ k ₂ K: Elliptic integral l : Length
Two concentric balls	${\frac {4\pi \varepsilon }{{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}}}$ ${\ displaystyle {\ frac {4 \ pi \ varepsilon} {{\ frac {1} {R_ {1}}} - {\ frac {1} {R_ {2}}}}}}$	R ₁ : Radius R ₂ : Radius
Two balls same radius ^[4] ^[5]	$2\pi \varepsilon a\sum _{n=1}^{\infty }{\frac {\sinh \left(\ln \left(D+{\sqrt {D^{2}-1}}\right)\right)}{\sinh \left(n\ln \left(D+{\sqrt {D^{2}-1}}\right)\right)}}$ ${\ displaystyle 2 \ pi \ varepsilon a \ sum _ {n = 1} ^ {\ infty} {\ frac {\ sinh \ left (\ ln \ left (D + {\ sqrt {D ^ {2} -1}} \ right) \ right)} {\ sinh \ left (n \ ln \ left (D + {\ sqrt {D ^ {2} -1}} \ right) \ right)}}}$ $=2\pi \varepsilon a\left\{1+{\frac {1}{2D}}+{\frac {1}{4D^{2}}}+{\frac {1}{8D^{3}}}+{\frac {1}{8D^{4}}}+{\frac {3}{32D^{5}}}+O\left({\frac {1}{D^{6}}}\right)\right\}$ ${\ displaystyle = 2 \ pi \ varepsilon a \ left \ {1 + {\ frac {1} {2D}} + {\ frac {1} {4D ^ {2}}} + {\ frac {1} {8D ^ {3}}} + {\ frac {1} {8D ^ {4}}} + {\ frac {3} {32D ^ {5}}} + O \ left ({\ frac {1} {D ^ {6}}} \ right) \ right \}}$ $=2\pi \varepsilon a\left\{\ln 2+\gamma -{\frac {1}{2}}\ln \left({\frac {d}{a}}-2\right)+O\left({\frac {d}{a}}-2\right)\right\}$ ${\ displaystyle = 2 \ pi \ varepsilon a \ left \ {\ ln 2+ \ gamma - {\ frac {1} {2}} \ ln \ left ({\ frac {d} {a}} - 2 \ right ) + O \ left ({\ frac {d} {a}} - 2 \ right) \ right \}}$	a : Radius d : Distance, d > 2 a D = d / 2 a γ : Euler constant
Ball near the wall ^[4]	$4\pi \varepsilon a\sum _{n=1}^{\infty }{\frac {\sinh \left(\ln \left(D+{\sqrt {D^{2}-1}}\right)\right)}{\sinh \left(n\ln \left(D+{\sqrt {D^{2}-1}}\right)\right)}}$ ${\ displaystyle 4 \ pi \ varepsilon a \ sum _ {n = 1} ^ {\ infty} {\ frac {\ sinh \ left (\ ln \ left (D + {\ sqrt {D ^ {2} -1}} \ right) \ right)} {\ sinh \ left (n \ ln \ left (D + {\ sqrt {D ^ {2} -1}} \ right) \ right)}}}$	a : Radius d : Distance, d> a D = d / a
Ball	$4\pi \varepsilon a$ ${\ displaystyle 4 \ pi \ varepsilon a}$	a : Radius
Round disc ^[6]	$8\varepsilon a$ ${\ displaystyle 8 \ varepsilon a}$	a : Radius
Thin straight wire limited length ^[7] ^[8] ^[9]	${\frac {2\pi \varepsilon l}{\Lambda }}\left\{1+{\frac {1}{\Lambda }}\left(1-\ln 2\right)+{\frac {1}{\Lambda ^{2}}}\left[1+\left(1-\ln 2\right)^{2}-{\frac {\pi ^{2}}{12}}\right]+O\left({\frac {1}{\Lambda ^{3}}}\right)\right\}$ ${\ displaystyle {\ frac {2 \ pi \ varepsilon l} {\ Lambda}} \ left \ {1 + {\ frac {1} {\ Lambda}} \ left (1- \ ln 2 \ right) + {\ frac {1} {\ Lambda ^ {2}}} \ left [1+ \ left (1- \ ln 2 \ right) ^ {2} - {\ frac {\ pi ^ {2}} {12}} \ right] + O \ left ({\ frac {1} {\ Lambda ^ {3}}} \ right) \ right \}}$	a : wire radius l : Length Λ : ln (l / a)

Elastance

The reciprocal of the capacity is called elastance (elasticity). The unit of elasticity is daraf, but it is not defined in the SI system of physical units ^[10] .

Notes

↑ Shakirzyanov F.N. Electric capacity // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Soviet Encyclopedia , 1990. - T. 2. - S. 28-29. - 704 s. - 100,000 copies. - ISBN 5-85270-061-4 .
↑ ¹ ² Jackson, JD Classical Electrodynamics. - Wiley, 1975 .-- P. 80.
↑ Binns. Analysis and computation of electric and magnetic field problems / Binns, Lawrenson. - Pergamon Press, 1973. - ISBN 978-0-08-016638-4 .
↑ ¹ ² Maxwell, JC A Treatise on Electricity and Magnetism. - Dover, 1873. - P. 266 ff. - ISBN 0-486-60637-6 .
↑ Rawlins, AD Note on the Capacitance of Two Closely Separated Spheres (Eng.) // IMA Journal of Applied Mathematics : journal. - 1985. - Vol. 34 , no. 1 . - P. 119-120 . - DOI : 10.1093 / imamat / 34.1.119 .
↑ Jackson, JD Classical Electrodynamics. - Wiley, 1975 .-- P. 128, problem 3.3.
↑ Maxwell, JC On the electrical capacity of a long narrow cylinder and of a disk of sensible thickness (Eng.) // Proc. London Math. Soc. : journal. - 1878. - Vol. IX . - P. 94-101 . - DOI : 10.1112 / plms / s1-9.1.94 .
↑ Vainshtein, LA Static boundary problems for a hollow cylinder of finite length. III Approximate formulas (Eng.) // Zh. Tekh. Fiz. : journal. - 1962. - Vol. 32 . - P. 1165-1173 .
↑ Jackson, JD Charge density on thin straight wire, revisited (neopr.) // Am. J. Phys. - 2000. - T. 68 , No. 9 . - S. 789-799 . - DOI : 10.1119 / 1.1302908 . - .
↑ Tensor analysis of networks, 1978 , p. 509.

Literature

Borgman I.I. ,. Electric capacity // Brockhaus and Efron Encyclopedic Dictionary : 86 tons (82 tons and 4 additional). - SPb. , 1890-1907.
Saveliev I.V. Chapter X. The movement of charged particles. // Course in general physics .. - 3. - M .: Science. Ch. ed. Phys.-Math. Lit., 1988. - T. 2. - P. 87-88. - 496 p. - 220,000 copies.
G. Cron. Tensor analysis of networks. - Moscow: Sov. Radio, 1978.- 720 p.

[ФЭ2-1] Shakirzyanov F.N. Electric capacity // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Soviet Encyclopedia , 1990. - T. 2. - S. 28-29. - 704 s. - 100,000 copies. - ISBN 5-85270-061-4 .

[Jackson_1975_80-2] ¹ ² Jackson, JD Classical Electrodynamics. - Wiley, 1975 .-- P. 80.

[3] Binns. Analysis and computation of electric and magnetic field problems / Binns, Lawrenson. - Pergamon Press, 1973. - ISBN 978-0-08-016638-4 .

[Maxwell_1873_266_ff-4] ¹ ² Maxwell, JC A Treatise on Electricity and Magnetism. - Dover, 1873. - P. 266 ff. - ISBN 0-486-60637-6 .

[5] Rawlins, AD Note on the Capacitance of Two Closely Separated Spheres (Eng.) // IMA Journal of Applied Mathematics : journal. - 1985. - Vol. 34 , no. 1 . - P. 119-120 . - DOI : 10.1093 / imamat / 34.1.119 .

[Jackson_1975_128-6] Jackson, JD Classical Electrodynamics. - Wiley, 1975 .-- P. 128, problem 3.3.

[7] Maxwell, JC On the electrical capacity of a long narrow cylinder and of a disk of sensible thickness (Eng.) // Proc. London Math. Soc. : journal. - 1878. - Vol. IX . - P. 94-101 . - DOI : 10.1112 / plms / s1-9.1.94 .

[8] Vainshtein, LA Static boundary problems for a hollow cylinder of finite length. III Approximate formulas (Eng.) // Zh. Tekh. Fiz. : journal. - 1962. - Vol. 32 . - P. 1165-1173 .

[9] Jackson, JD Charge density on thin straight wire, revisited (neopr.) // Am. J. Phys. - 2000. - T. 68 , No. 9 . - S. 789-799 . - DOI : 10.1119 / 1.1302908 . - .

[_e4e0d4ac1cf3ebf3-10] Tensor analysis of networks, 1978 , p. 509.

Electric capacity
$C$ ${\ displaystyle C}$ $C$
Dimension	L ^-2 M ^-1 T ⁴ I ²
Units
SI	farad
GHS	centimeter