Capacitance Contents 1 Capacitance 1 1.1 Capacitors ............................................... 1 1.1.1 Voltage-dependent capacitors ................................. 2 1.1.2 Frequency-dependent capacitors ............................... 2 1.2 Capacitance matrix .......................................... 3 1.3 Self-capacitance ............................................ 4 1.4 Stray capacitance ........................................... 4 1.5 Capacitance of simple systems .................................... 4 1.6 Capacitance of nanoscale systems ................................... 4 1.6.1 Single-electron devices .................................... 5 1.6.2 Few-electron devices ..................................... 5 1.7 See also ................................................ 6 1.8 References ............................................... 6 1.9 Further reading ............................................ 7 2 Dielectric 8 2.1 Terminology .............................................. 8 2.2 Electric susceptibility ......................................... 8 2.2.1 Dispersion and causality ................................... 9 2.3 Dielectric polarization ......................................... 9 2.3.1 Basic atomic model ...................................... 9 2.3.2 Dipolar polarization ...................................... 10 2.3.3 Ionic polarization ....................................... 10 2.4 Dielectric dispersion .......................................... 10 2.5 Dielectric relaxation .......................................... 10 2.5.1 Debye relaxation ....................................... 11 2.5.2 Variants of the Debye equation ................................ 11 2.6 Paraelectricity ............................................. 11 2.7 Tunability ............................................... 11 2.8 Applications .............................................. 12 2.8.1 Capacitors ........................................... 12 2.8.2 Dielectric resonator ...................................... 12 2.9 Some practical dielectrics ....................................... 12 i ii CONTENTS 2.10 See also ................................................ 13 2.11 References ............................................... 13 2.12 Further reading ............................................ 14 2.13 External links ............................................. 14 3 Permittivity 15 3.1 Explanation .............................................. 15 3.2 Vacuum permittivity .......................................... 15 3.3 Relative permittivity .......................................... 16 3.4 Dispersion and causality ........................................ 16 3.4.1 Complex permittivity ..................................... 16 3.4.2 Tensorial permittivity ..................................... 18 3.4.3 Classification of materials ................................... 18 3.4.4 Lossy medium ........................................ 18 3.4.5 Quantum-mechanical interpretation .............................. 18 3.5 Measurement ............................................. 19 3.6 See also ................................................ 19 3.7 References ............................................... 19 3.8 Further reading ............................................ 20 3.9 External links ............................................. 20 4 Electric susceptibility 21 4.1 Definition of volume susceptibility .................................. 21 4.2 Molecular polarizability ........................................ 21 4.3 Nonlinear susceptibility ........................................ 22 4.4 Dispersion and causality ........................................ 22 4.5 See also ................................................ 22 4.6 References ............................................... 22 5 Polarization density 23 5.1 Definition ............................................... 23 5.2 Other Expressions ........................................... 23 5.3 Gauss’s Law for the Field of P ..................................... 23 5.3.1 Differential Form ....................................... 23 5.4 Relationship between the fields of P and E .............................. 24 5.4.1 Homogeneous, Isotropic Dielectrics ............................. 24 5.4.2 Anisotropic Dielectrics .................................... 24 5.5 Polarization density in Maxwell’s equations .............................. 25 5.5.1 Relations between E, D and P ................................. 25 5.5.2 Time-varying Polarization Density .............................. 25 5.6 Polarization ambiguity ......................................... 25 5.7 See also ................................................ 26 CONTENTS iii 5.8 References and notes ......................................... 26 5.9 Text and image sources, contributors, and licenses .......................... 27 5.9.1 Text .............................................. 27 5.9.2 Images ............................................ 28 5.9.3 Content license ........................................ 28 Chapter 1 Capacitance Capacitance is the ability of a body to store an electri- plate to the other against the potential difference V = q/C cal charge. Any object that can be electrically charged requires the work dW: exhibits capacitance. A common form of energy stor- age device is a parallel-plate capacitor. In a parallel plate q capacitor, capacitance is directly proportional to the sur- dW = dq face area of the conductor plates and inversely propor- C tional to the separation distance between the plates. If where W is the work measured in joules, q is the charge the charges on the plates are +q and −q respectively, and measured in coulombs and C is the capacitance, measured V gives the voltage between the plates, then the capaci- in farads. tance C is given by The energy stored in a capacitor is found by integrating this equation. Starting with an uncharged capacitance (q q C = : = 0) and moving charge from one plate to the other until V the plates have charge +Q and −Q requires the work W: which gives the voltage/current relationship Z Q 2 q 1 Q 1 1 2 dV (t) Wcharging = dq = = QV = CV = Wstored: I(t) = C : 0 C 2 C 2 2 dt The capacitance is a function only of the geometry of the design (area of the plates and the distance between them) 1.1 Capacitors and the permittivity of the dielectric material between the plates of the capacitor. For many dielectric materials, the Main article: Capacitor permittivity and thus the capacitance, is independent of the potential difference between the conductors and the The capacitance of the majority of capacitors used in total charge on them. electronic circuits is generally several orders of magni- The SI unit of capacitance is the farad (symbol: F), tude smaller than the farad. The most common sub- named after the English physicist Michael Faraday. A 1 units of capacitance in use today are the microfarad farad capacitor, when charged with 1 coulomb of electri- (µF), nanofarad (nF), picofarad (pF), and, in mi- cal charge, has a potential difference of 1 volt between its crocircuits, femtofarad (fF). However, specially made plates.[1] Historically, a farad was regarded as an inconve- supercapacitors can be much larger (as much as hundreds niently large unit, both electrically and physically. Its sub- of farads), and parasitic capacitive elements can be less divisions were invariably used, namely the microfarad, than a femtofarad. nanofarad and picofarad. More recently, technology has Capacitance can be calculated if the geometry of the con- advanced such that capacitors of 1 farad and greater (so- ductors and the dielectric properties of the insulator be- called 'supercapacitors') can be constructed in a structure tween the conductors are known. A qualitative explana- little larger than a coin battery. Such capacitors are prin- tion for this can be given as follows. cipally used for energy storage replacing more traditional Once a positive charge is put unto a conductor, this charge batteries. creates an electrical field, repelling any other positive The energy (measured in joules) stored in a capacitor is charge to be moved onto the conductor. I.e. increasing equal to the work required to push the charges into the the necessary voltage. But if nearby there is another con- capacitor, i.e. to charge it. Consider a capacitor of ca- ductor with a negative charge on it, the electrical field pacitance C, holding a charge +q on one plate and −q on of the positive conductor repelling the second positive the other. Moving a small element of charge dq from one charge is weakened (the second positive charge also feels 1 2 CHAPTER 1. CAPACITANCE the attracting force of the negative charge). So due to the in charging such a capacitor the differential increase in second conductor with a negative charge, it becomes eas- voltage with charge is governed by: ier to put a positive charge on the already positive charged first conductor, and vice versa. I.e. the necessary voltage is lowered. dQ = C(V ) dV As a quantitative example consider the capacitance of a where the voltage dependence of capacitance, C(V), sug- capacitor constructed of two parallel plates both of area gests that the capacitance is a function of the electric field A separated by a distance d: strength, which in a large area parallel plate device is given by ε = V/d. This field polarizes the dielectric, which A polarization, in the case of a ferroelectric, is a nonlinear C = " " r 0 d S-shaped function of the electric field, which, in the case of a large area