Energy Stored by a Capacitor

12 Key excerpts on "Energy Stored by a Capacitor"

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  eBook - PDF

  Dorf's Introduction to Electric Circuits

  • Richard C. Dorf, James A. Svoboda(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Eventually, the voltage across the capacitor is a constant, and the current through the capacitor is zero. The capacitor has stored energy by virtue of the separation of charges between the capacitor plates. These charges have an electrical force acting on them. The forces acting on the charges stored in a capacitor are said to result from an electric field. An electric field is defined as the force acting on a unit positive charge in a specified region. Because the charges have a force acting on them along a direction x, we recognize that the energy required originally to separate the charges is now stored by the capacitor in the electric field. The energy stored in a capacitor is w c t t vi dt Remember that v and i are both functions of time and could be written as v(t) and i(t). Because i C dv dt we have w c t vC dv dt dt C v t v v dv 1 2 Cv 2 v t v R C + – v c + – Switch closed 10 V R t = 0 C + – v c + – 10 V FIGURE 7.3-1 A circuit (a) where the capacitor is charged and v c 10 V and (b) the switch is opened at t 0. 272 CHAPTER 7 Energy Storage Elements Because the capacitor was uncharged at t , set v 0. Therefore, w c t 1 2 Cv 2 t J 7 3-1 Therefore, as a capacitor is being charged and v(t) is changing, the energy stored, w c , is changing. Note that w c t 0 for all v(t), so the element is said to be passive. Because q Cv, we may rewrite Eq. 7.3-1 as w c 1 2C q 2 t J 7 3-2 The capacitor is a storage element that stores but does not dissipate energy. For example, consider a 100-mF capacitor that has a voltage of 100 V across it. The energy stored is w c 1 2 Cv 2 1 2 0 1 100 2 500 J As long as the capacitor is not connected to any other element, the energy of 500 J remains stored. Now if we connect the capacitor to the terminals of a resistor, we expect a current to flow until all the energy is dissipated as heat by the resistor. After all the energy dissipates, the current is zero and the voltage across the capacitor is zero.

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  University Physics Volume 2

  • William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
  • 2016(Publication Date)
  • Openstax

    (Publisher)

  Chapter 8 | Capacitance 361 Figure 8.15 The capacitors on the circuit board for an electronic device follow a labeling convention that identifies each one with a code that begins with the letter “C.” The energy U C stored in a capacitor is electrostatic potential energy and is thus related to the charge Q and voltage V between the capacitor plates. A charged capacitor stores energy in the electrical field between its plates. As the capacitor is being charged, the electrical field builds up. When a charged capacitor is disconnected from a battery, its energy remains in the field in the space between its plates. To gain insight into how this energy may be expressed (in terms of Q and V), consider a charged, empty, parallel-plate capacitor; that is, a capacitor without a dielectric but with a vacuum between its plates. The space between its plates has a volume Ad, and it is filled with a uniform electrostatic field E. The total energy U C of the capacitor is contained within this space. The energy density u E in this space is simply U C divided by the volume Ad. If we know the energy density, the energy can be found as U C = u E ( Ad) . We will learn in Electromagnetic Waves (after completing the study of Maxwell’s equations) that the energy density u E in a region of free space occupied by an electrical field E depends only on the magnitude of the field and is (8.9) u E = 1 2 ε 0 E 2 . If we multiply the energy density by the volume between the plates, we obtain the amount of energy stored between the plates of a parallel-plate capacitor: U C = u E ( Ad) = 1 2 ε 0 E 2 Ad = 1 2 ε 0 V 2 d 2 Ad = 1 2 V 2 ε 0 A d = 1 2 V 2 C . In this derivation, we used the fact that the electrical field between the plates is uniform so that E = V /d and C = ε 0 A/d. Because C = Q/V , we can express this result in other equivalent forms: (8.10) U C = 1 2 V 2 C = 1 2 Q 2 C = 1 2 QV .

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  Physics, Volume 2

  • David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 30 679 CHAPTER 30 CAPACITANCE I n many applications of electric circuits, the goal is to store electrical charge or energy in an electrostatic field. A device that stores charge is called a capacitor, and the property that determines how much charge it can store is its capacitance. We shall see that the ca- pacitance depends on the geometrical properties of the device and not on the electric field or the potential. In this chapter we define capacitance and show how to calculate the capacitance of a few simple de- vices and of combinations of capacitors. We study the energy stored in capacitors and show how it is re- lated to the strength of the electric field. Finally, we investigate how the presence of a dielectric in a capaci- tor enhances its ability to store electric charge. 30-1 CAPACITORS A capacitor* is a device that stores energy in an electrosta- tic field. A flashbulb, for example, requires a short burst of electric energy that exceeds what a battery can generally provide. A capacitor can draw energy relatively slowly (over several seconds) from the battery, and it then can re- lease the energy rapidly (within milliseconds) through the bulb. Much larger capacitors are used to produce short laser pulses in attempts to induce thermonuclear fusion in tiny pellets of hydrogen. In this case the power level during the pulse is about 10 14 W, about 200 times the entire electrical generating capacity of the United States, but the pulses typ- ically last only for 10 9 s. Capacitors are also used to produce electric fields, such as the parallel-plate device that gives the very nearly uni- form electric field that deflects beams of electrons in a TV or oscilloscope tube. In circuits, capacitors are often used to smooth out the sudden variations in line voltage that can damage computer memories. In another application, the tuning of a radio or TV receiver is usually done by varying the capacitance of the circuit.

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  Fundamentals of Physics, Extended

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  25.4.3 For a capacitor, apply the relationship between the potential energy, the internal volume, and the internal energy density. 25.4.4 For any electric field, apply the relationship between the potential energy density u in the field and the field’s magnitude E. 25.4.5 Explain the danger of sparks in airborne dust. Figure 25.3.4 A potential difference V 0 is applied to capacitor 1 and the charging battery is removed. Switch S is then closed so that the charge on capacitor 1 is shared with capacitor 2. S C 2 C 1 q 0 After the switch is closed, charge is transferred until the potential differences match. Additional examples, video, and practice available at WileyPLUS 771 25.4 ENERGY STORED IN AN ELECTRIC FIELD Key Ideas ● The electric potential energy U of a charged capacitor, U = q 2 ___ 2c = 1 _ 2 CV 2 , is equal to the work required to charge the capacitor. This energy can be associated with the capacitor’s electric field E → . ● Every electric field, in a capacitor or from any other source, has an associated stored energy. In vacuum, the energy density u (potential energy per unit volume) in a field of magnitude E is u = 1 _ 2 ε 0 E 2 . Energy Stored in an Electric Field Work must be done by an external agent to charge a capacitor. We can imagine doing the work ourselves by transferring electrons from one plate to the other, one by one. As the charges build, so does the electric field between the plates, which opposes the continued transfer. So, greater amounts of work are required. Actually, a battery does all this for us, at the expense of its stored chemical energy. We visual- ize the work as being stored as electric potential energy in the electric field between the plates. Suppose that, at a given instant, a charge q′ has been transferred from one plate of a capacitor to the other. The potential difference V′ between the plates at that instant will be q′/C.

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  Fundamentals of Physics, Extended

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Here, just after switch S is closed, the only applied electric potential is that of capacitor 1 on capacitor 2, and that poten- tial is decreasing. Thus, the capacitors in Fig. 25-11 are not connected in series; and although they are drawn parallel, in this situation they are not in parallel. 728 CHAPTER 25 CAPACITANCE Energy Stored in an Electric Field Work must be done by an external agent to charge a capacitor. We can imagine doing the work ourselves by transferring electrons from one plate to the other, one by one. As the charges build, so does the electric field between the plates, which opposes the continued transfer. So, greater amounts of work are required. Actually, a battery does all this for us, at the expense of its stored chemical energy. We visualize the work as being stored as electric potential energy in the electric field between the plates. Additional examples, video, and practice available at WileyPLUS 25-4 ENERGY STORED IN AN ELECTRIC FIELD Learning Objectives After reading this module, you should be able to . . . 25.16 Explain how the work required to charge a capaci- tor results in the potential energy of the capacitor. 25.17 For a capacitor, apply the relationship between the potential energy U, the capacitance C, and the poten- tial difference V. 25.18 For a capacitor, apply the relationship between the potential energy, the internal volume, and the internal energy density. 25.19 For any electric field, apply the relationship between the potential energy density u in the field and the field’s magnitude E. 25.20 Explain the danger of sparks in airborne dust. Key Ideas ● The electric potential energy U of a charged capacitor, U = q 2 2c = 1 2 CV 2 , is equal to the work required to charge the capacitor. This energy can be associated with the capacitor’s electric field E → . ● Every electric field, in a capacitor or from any other source, has an associated stored energy.

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  eBook - ePub

  Electrochemical Supercapacitors for Energy Storage and Delivery

  Fundamentals and Applications

  • Aiping Yu, Victor Chabot, Jiujun Zhang(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  :

  (1.31)

  The total work performed, and thus the total potential energy stored in the capacitor, is

  (1.32a)

  Note that the capacitance C in Equation (1.32a) is independent of charge and can be taken out of the integral [2 ]. Combining Equation (1.11) with (1.32a) , a more familiar form of the energy stored in a capacitor can be obtained:

  (1.32b)

  Ideally, the energy stored in a capacitor and capacitive charge stored by the capacitor do not leak or dissipate and are retained indefinitely until discharged [1 ]. However, in practice, due to the leaking of dielectric material, the self-discharge rate of the capacitor is faster relative to batteries.

  1.6 Capacitor Containing Electrical Circuits and Corresponding Calculation

  In general, all electric circuits are driven by an external power source such as a portable battery, supercapacitor, stationary electric generator, solar cell, or thermopile. Despite their distinct modes of operation, they all have the same principal function: performing work on charge carriers while keeping a potential difference between their connected terminals.

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  eBook - ePub

  College Physics Essentials, Eighth Edition

  Electricity and Magnetism, Optics, Modern Physics (Volume Two)

  • Jerry D. Wilson, Anthony J. Buffa, Bo Lou(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  −10 m. Is it feasible to build this capacitor?

  Deriving an expression for energy storage by a capacitor is based on the fact that the charge and the voltage across the plates are proportional since Δ V = (1/C )Q ∝ Q . Thus the work done by the battery W batt (and the energy stored in the capacitor U C ) can be found by considering the battery as transferring the total charge Q , across an average voltage

  Δ V

  ¯

  . Because the voltage varies linearly with charge, assuming the capacitor initially uncharged,

  Δ V

  ¯

  is

  Δ V

  ¯

  =

  Δ

  V final

  + Δ

  V initial

  2

  =

  Δ V + 0

  2

  =

  Δ V

  2

  Since U C is equal to W batt and, by definition,

  W batt

  = Q

  Δ V

  ¯

  , an expression for U C is

  U C

  =

  W batt

  = Q   ⋅  

  Δ V

  ¯

  =

  1 2

  Q   ⋅   Δ V

  Since Q = C · Δ V , this can be rewritten in several equivalent forms:

  U C = 1 2 Q   ⋅   Δ V = Q 2 2 C = 1 2 C ( Δ V ) 2   (energy   storage   in   a   capacitor )	(16.13)

  Typically, the form

  U C

  =

  1 2

  C

  ( Δ V )

  2

  is the most practical, since capacitance and voltage are usually the known quantities. A very important medical application of energy storage in a capacitor is the cardiac defibrillator , discussed in Example 16.7.

  Example 16.7: Capacitors to the Rescue – Energy Storage in a Cardiac Defibrillator

  During a heart attack, the heart can beat in an erratic fashion, called fibrillation . One way to get it back to normal rhythm is to shock it with electrical energy supplied by a cardiac defibrillator (▼ Figure 16.19 ). About 300 J of energy is required to produce the desired effect. Typically, a defibrillator stores this energy in a capacitor charged by a 5000-V power supply. (a) What capacitance is required? (b) What is the charge on the capacitor’s plates?

  ▲ Figure 16.19 Capacitors in action: The defibrillator A flow of charge (and thus energy) from a discharging capacitor through the heart muscle may restore a normal heartbeat during a bout of cardiac fibrillation.

  Thinking It Through. (a) To find the capacitance, solve for C

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  eBook - PDF

  College Physics Essentials, Eighth Edition (Two-Volume Set)

  • Jerry D. Wilson, Anthony J. Buffa, Bo Lou(Authors)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  Although the symbol for a capacitor is similar, its lines are of equal length. 428 College Physics Essentials discussed until Chapter 17 , all you need to know now is that a battery removes electrons from the positive plate and transfers, or “pumps,” them through a wire to the negative plate. (Take note of Figure 16.18b – our first electric circuit diagram – for future reference.) In the process of doing work, the battery loses some of its stored chemical energy. Of primary interest here is the result: a separation of charge and the creation of an electric field between the plates of the capacitor. The battery will continue to transfer charge until the potential difference between the plates is the same as the terminal voltage of the bat- tery. When the capacitor is disconnected from the battery, the energy remains stored on it until the charge is removed. Let us use Q be the magnitude of the charge on either plate (of course the net charge on the capacitor is zero). For a capacitor, Q is proportional to the voltage (electric potential difference) applied the plates, or Q ∝ ΔV . This proportionality can be made into an equation by using a constant, C, called capacitance: Q C V C Q V = ⋅ = Δ Δ or (definition of capacitance) (16.9) SI unit of capacitance is the coulomb per volt (C/V) or the farad (F) The coulomb per volt is named the farad after Michael Faraday, a famous physicist of the nineteenth century, thus 1 C/V = 1 F. The farad is a large unit (see Example 16.6), so the microfarad (1 μF = 10 −6 F), the nanofarad (1 nF = 10 −9 F), and the picofarad (1 pF = 10 −12 F) are commonly used. Capacitance represents the charge Q that can be stored per volt. If a capacitor has a large capacitance value, this means it is capable of holding a large charge per volt compared to one of smaller capacitance. Thus if you connect the same battery to two different capacitors, the one with the larger capacitance stores more charge and energy.

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  Halliday's Fundamentals of Physics, 1st Australian & New Zealand Edition

  • David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  The first electron is easy to transfer because the plates are initially neutral, but thereafter, the work required of us progressively increases. 1. The charges on the plates increase. 2. The electric potential V between the plates increases. 3. The magnitude of the electric field  E in the volume between the plates increases. 4. The electric force  F opposing each additional transfer increases. 5. The work W required of us for each additional transfer increases. When we finally stop, we say that all our work W is then stored as an electric potential energy U in the capacitor. We can also say that the potential energy is stored in the electric field that we have set up in the volume between the plates. Our work went to producing that field in that volume. Of course, we cannot ourselves transfer the electrons but a battery can do the work for us. In charging a capacitor, the battery transfers some of its stored chemical energy to the energy of the electric field in the volume inside the capacitor. Let’s relate the final charge q to the stored electric potential energy U by transferring a charge increment dq ′ when the charge already on the plates is q ′ and the potential difference is already V ′ . That is the incremental work required for that particular transfer, when the capacitor happens to have charge q ′ and potential difference V ′ . To find the total work W required to reach the final charge q, we must sum all the incremental works, which increases as the existing charge q ′ increases. We sum by integration: W = ∫ dW = ∫ q 0 q ′ C dq ′ = 1 C ∫ q 0 q ′ dq ′ = 1 2C [ ( q ′ ) 2 ] q 0 = 1 2C [ q 2 − 0 ] = q 2 2C . (25.29) That work is stored as electrical potential energy and thus U = q 2 2C . (25.30) Using our basic capacitor equation, q = CV, we can also write U = (CV) 2 2C , Pdf_Folio:560 560 Fundamentals of physics and then U = 1 2 CV 2 . (25.31) These two equations for a capacitor’s potential energy hold no matter what the geometry of the capacitor is.

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  eBook - ePub

  Energy Storage

  • Gerard M Crawley(Author)
  • 2017(Publication Date)
  • WSPC

    (Publisher)

  Chapter 5

  Capacitive Energy Storage

  Wentian Gu* , Lu Wei

  * ,†

  and Gleb Yushin

  * ,‡

  * School of Materials Science and EngineeringGeorgia Institute of TechnologyRoom 288, 771 Ferst Drive NWAtlanta, GA 30332-0245, USA† School of Materials Science and EngineeringHuazhong University of Science and TechnologyWuhan, Hubei 430074, China‡ [email protected]

  Capacitors are electrical devices for electrostatic energy storage. There are several types of capacitors developed and available commercially. Conventional dielectric and electrolytic capacitors store charge on parallel conductive plates with a relatively low surface area, and therefore, deliver limited capacitance. However, they can be operated at high voltages. As an alternative, electrochemical capacitors (ECs) (also called supercapacitors) store charge in electric double layers or at surface reduction–oxidation (Faradaic) sites. Thanks to the large surface area of the electrode and the nanoscale charge separation, electrochemical capacitors provide much higher capacitance, filling in the gap in the energy and power characteristics between batteries and conventional capacitors. However, they offer a lower energy density than batteries and commonly lower power than traditional capacitors. In the past decade, intensive research on ECs brought about the discovery of new electrode materials and in-depth understanding of ion behavior in small pores, as well as the design of new hybrid systems combining Faradaic and capacitive electrodes, which are essential for the enhancement of the performance of ECs.

  This chapter presents the classification, construction, performance, advantages, and limitations of capacitors as electrical energy storage devices. The materials for various types of capacitors and their current and future applications are also discussed.

  1Introduction

  Capacitors are passive electrical devices which store electrostatic energy in an electric field. The basic form of a capacitor (previously called a “condenser”) consists of two parallel conductors separated by a dielectric. The ability to store charge can be characterized by a single quantity, the capacitance, with the unit Farad (F), which is defined as the ratio of the charge on one of the conductors divided by the voltage applied across the dielectric that induces this charge.

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  Emerging Trends in Energy Storage Systems and Industrial Applications

  • Dr. Prabhansu, Nayan Kumar, Prabhansu(Authors)
  • 2022(Publication Date)
  • Academic Press

    (Publisher)

  A conventional capacitor or dielectric capacitor consists of two conducting electrodes separated by a dielectric insulating medium. When a voltage is applied to a capacitor, opposite charges accumulated at the surface of each electrode. The charges are kept separated by the insulator, thereby the electric field produced lets the capacitor store the energy. Different types of capacitors are reported in the literature. Capacitors connected to electronic and power circuits are of electrolyte type or dielectric type and are usually solid-state devices. These capacitors have properties like rapid response time and a long-life period. However, these devices store less energy mostly lower than 0.1 Wh/kg. Therefore, these are not commonly used for bulk energy storage.

  4.2.2 Electrochemical capacitors

  Another type of capacitor is the electrochemical capacitor [16] or supercapacitors (SCs). Here electrodes of high-surface-area and electrolytes of acetonitrile or sulfuric acid are commonly used for bulk energy storage. According to current R&D activities, ECs are categorized as electrostatic double-layer capacitors, pseudocapacitors, and hybrid capacitors as shown in Fig. 4.5 .

  Figure 4.5 Classification of supercapacitor [17] .

  4.2.2.1 Electrostatic double-layer capacitors

  The energy is stored at the surface of the electrostatic double-layer capacitors (EDLCs) (Fig. 4.6 ). As compared to solid-state capacitors, bulk power can be stored across these capacitors and can store bulk energy. However, the stored energy is still less than 10 Wh/kg. The cell voltages are limited to prevent the decomposition of the liquid electrolytes: less than 1 V for water-based and around 3 V for non-aqueous electrolytes. These capacitors are commonly used for short periods of the energy demand of the range from 0.01 to 100 seconds. These are more reliable and less expensive and used in trucks and cranes.

  Figure 4.6 Electrostatic double-layer capacitors [18] .

  4.2.2.2 Pseudocapacitors

  Another capacitor bank called pseudo capacitors (Fig. 4.7 ) can be designed by hybridization of double-layer capacitors and batteries where the materials' surface rials play a major role [19] . Therefore, these capacitor banks can handle a large amount of energy as compared to the surface capacitors. However, these banks face several challenges like reliability and technical issues like the selection of dielectric medium [20] . The materials especially transition-metal oxides like manganese or vanadium oxides are commonly used in the construction of pseudocapacitors [21] . Generally, many crystalline materials with flat discharge levels like vanadium pentoxide react with lithium and are converted into amorphous substances [22]

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  Electrochemical Components

  • Marie-Cécile Pera, Daniel Hissel, Hamid Gualous, Christophe Turpin, Marie-Cécile Pera, Daniel Hissel, Hamid Gualous, Christophe Turpin(Authors)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  This relation shows that the value of the “nominal” capacitance is higher when the nominal voltage is greater too. Thus, for a supercapacitor, two capacitances can be distinguished: a charge capacitance, and a differential capacitance linking the current to the charge voltage.

  The energy E stored in a supercapacitor is given by the following expression:

  If we compare this expression with that of the energy of a capacitor (1/2 CU2 ), we can define an energy capacitance for the supercapacitor by:

  [4.7]

  4.3. Supercapacitor module sizing

  The sizing of a supercapacitor module consists of determining the number of elementary cells needing to be connected in a series and/or in parallel to satisfy the technical specifications. The design needs to take account of the maximum power and the energy needed to serve the requirements of the intended application.

  Generally, a supercapacitor module is designed to provide a large amount of power for a very limited period of time (typically less than ten seconds). We can study the problem in two different, but closely linked, ways. The first is based on the power and the second uses the stored energy.

  4.3.1. Power-based design

  A supercapacitor module is sized according to a set of specifications established on the basis of the power required and the duration for which the module provides the power. The design method is illustrated in Figure 4.9 . This method consists of:

  – setting the levels of the nominal voltages and currents; – determining the total capacitance of the supercapacitor module to be used; – determining the number of elements needing to be connected in a series and/or in parallel. We define the following parameters:

  – P the power prescribed by the technical specifications;

  – Δt the time for which the supercapacitor module provides the required power (discharge time);

  – U

  M max

  : maximum voltage of the supercapacitor module;

  – U M min: minimum voltage of the supercapacitor module. In general, UM min = UM max /2, because the supercapacitor module discharges between UM max and UM max

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