Capacitance And The Storage Of Electric Energy

CAPACITANCE and The Storage of Electric Energy

PARALLEL PLATE CAPACITORS

Outline

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Objectives

1. Capacitors and Capacitance 2 The 2. Th Combination C bi i off Capacitors 3. Energy Storage in Capacitors 4. Capacitors and Dielectrics

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At the end of this chapter, you should be able to:

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1. Explain how electrostatic energy is stored; 2. Define capacitance; 3. Define a dielectric and explain how dielectrics affect the energy stored in a capacitor; and 4. Solve problems involving capacitors.

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Chapter Two

The First of Three y

The Capacitor is one of the three simple circuit elements that can be connected with wires to form an electric circuit!

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Capacitors have varietyy of uses, ranging g g from: radio fine tuner circuits to camera flashes to defibrillators

1. Definittion of Capacitancce

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Capacitor

◦ is a system y composed p of two conductors (plates) with equal and opposite pp charges g on them!

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ΔV

◦ exists between the plates because of the f ld between field b them h

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Question???:

◦ What determines the amount of charge Q on the th plates l t att a given V?

Experiment shows that the amount Q is proportional to ΔV!

I. Deefinition o of Caapacittancee

D fi iti Definition of C Amount of Charge Stored in a given Potential

C = Q/ΔV

Unit

S Some Notes

The Farad

Always positive

1 F = 1C/1V

Does not actuallyy depend on the charge or potential, but on the geometry.

The Farad is a large unit!

C is Proportionality Constant

How w Capacittors Work W k?

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Initially there are no charges at the plates.

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After making the connection, charging happens.

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Let us focus on the negative plate ◦ Charging Stops if the wire, wire plate, plate and terminal are all at the same potential!

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The positive plate also experiences a similar phenomenon!

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In the final configuration the potential difference between the plates is the same as the battery!

Cheeckpo oint 22.1

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A capacitor stores charge h Q at a potentiall difference ΔV. If the voltage applied by a battery to the capacitor is doubled to 2ΔV,

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( ) the capacitance (a) p falls to half its initial value and the charge remains the same (b) the capacitance and the charge both fall to half their initial values (c) the capacitance and the charge both double (d) the capacitance remains the same and the charge doubles.

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1. Thhe Capacitaance of Caapacittors

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Parallel Plate Capacitors ◦ Th The Capacitance C i off PP is i related only to the area A of the plates and the separation distance d between them!

Cheeckpo oint 22.2

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Many computer keyboard buttons are constructed of capacitors, as shown in the Figure. When a key is pushed down, the soft insulator between the movable plate and the fixed plate is compressed. p When the keyy is pressed, the capacitance

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(a) increases, increases (b) decreases, or ((c)) changes g in a wayy that we cannot determine because the complicated electric circuit connected to the keyboard button may cause a change in ΔV.

A parallel-plate capacitor with air between the h plates l has h an area A = 2.00 2 00 x10 10-44 m2 andd a plate separation d = 1.00 mm. y Find its capacitance. y

Exam mple 1

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Ans: 1.77 pF

1. Caapacitancce of Capaacito ors

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Cylindrical Capacitors (Coaxial)

L is length of the conductors, b is outer radius and a is inner radius

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Spherical Capacitors (Concentric)

a is inner radius, b is outer radius. If b→ ∞ then we have an isolated conductor with a new capacitance

Exam mple: Othher Capac C citors

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An isolated spherical capacitor has a capacitance of 1F. 1F Calculate the radius of the spherical capacitor.

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You bought a 1-m coaxial cable for your TVR If it TVR. it’ss indicated that the outer radius is 2mm and the capacitance is 2μF, find the inner radius of the coaxial cable.

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Your laboratory instructor asked you to create a spherical capacitor with p 12 p pF. The instructor ggave you y capacitance a solid sphere of radius 2.4mm, what should be the diameter of the shell enclosure?

Capaacitorrs as Circuuit Eleemennts

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As mentioned earlier li iin the h chapter, capacitors are used d in i electric l i circuits.

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In circuit analysis, we study pictorial representations of circuits known as circuit diagrams

2 Combinations of Capacitors 2. y

A. B.

There are two types yp of circuit element combination: Series Parallel

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It’s series if the elements are connected from endto-end.

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It’s parallel if the elements are connected at common ends.

2.1 P Parallel Comb C binatiion

With parallel: the capacitors are at a common potential!

The individual charges can be found by QN = CNΔV

The equivalent capacitance is Ceq which is the sum of the individual capacitances

2.2 SSeriees Co ombin natio on

With series: the same charge Q is stored among all the capacitors! The equivalent capacitance can be found byy takingg the reciprocal p of the sum of the reciprocal capacitances.

The individual voltage across each capacitor can be found by ΔVN = Q/CN

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In eeach circuit, ffind the equiivalentt capaacitancce and d the chargess storeed in eeach caapacito or!

Exam mple: Seriies ies--P Paralleel!

V = 18 Volts

Vab = 15 Volts

QUIIZ #55 (1/2 2 sheeet, 110/10 0)

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Find the equivalent capacitance. Note that each h capacitor i has h the h same capacitance. i

3. Electro ostattic Fieeld Energ E gy

3. Electro ostattic Fieeld Energ E gy

3. Electro ostattic Fieeld Energ E gy

Figure to the right shows the linear relationship between Q and ΔV. The Energy U can be computed by taking the area under the curve! The energy density uE (Energy/Volume) ◦ This is the energy that is stored in an electric field, regardless of the configuration!

Cheeckpo oint 22.3

You have three capacitors and a battery. In which hi h off the h ffollowing ll i combinations bi i off the h three capacitors will the maximum possible energy be b stored d when h the h combination bi i is i attached to the battery? (a) series (b) parallel (c) Both combinations will store the same gy amount of energy. Ans: (b) for a given voltage, voltage capacitances add up when in parallel and U = ½ C(ΔV)2

Cheeckpo oint 22.4

You charge a parallel-plate capacitor, remove it from the battery, and prevent the wires connected to the plates from touching each other. When you pull the plates apart to a larger separation do the following quantities increase, separation, increase decrease, or stay the same? ((a)) C;; (b) Q; (c) E between the plates; (d) ΔV (e) Energy stored (U) in the capacitor. Ans: C decreases, Q stays the same, E remains constant,V increases, U increases because U = ½ QΔV

4. Dielecctrics and Cappacito ors

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Dielectrics are insulators!

◦ Some examples of dielectrics are air, paper, wax rubber, wax, rubber and glass glass. ◦ Characterized by the dielectric constant κ (>1) which modifies the permittivity of free space ε0!

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“When dielectrics occupy the space between the plates of a capacitor, the

capacitance increases!”

The Effecct off the Batteery!

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WARNING! ◦ Before analyzing and solving for the effect of the dielectric on electrical properties such as charge and potential… we have to ask…

“IS THE CAPACITOR CONNECTED TO A BATTERY?” CASE 1: If it was connected then removed before dielectric was inserted! • The charge on the capacitor remain the same

CASE 2: If it remains connected when the dielectric was connected • The voltage across the capacitor remains the same

Inserrting a dielectric into oa chargged capaci c itor!

Inserrting a dielectric into oa chargged capaci c itor!

When the dielectric is inserted, the charge remains the same, voltage drops by:

If voltage drops, then capacitance p increases by:

Insertiing the dielecttric while the ccapacito or is still co onnecteed to th he batteery

When the dielectric is inserted, the voltage remains the same, to accomplish this battery must supply additional charge h so charge h increases by:

g increases on If charge the plates, capacitance also increases by:

Exam mples:

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1. A capacitor is to be constructed by making circular parallel plates (radius 1cm) and with separation of 2.5mm with a paper dielectric between the plates (κ = 3.7). Find the capacitance of this capacitor.

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2. A 10nF capacitor is charged with a 12V battery. After fully charging the capacitor, capacitor the battery was removed and a dielectric (κ = 2) was inserted between the plates of the capacitor. Find the f ll i ((a)) iinitial following: i i l and d fifinall charges, h (b) final fi l voltage, l (c) final capacitance

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3. A 4.7nF capacitor is charged with a 9V battery. Then a glass dielectric was inserted between the plates of the capacitor. Find the following: (a) initial and final charges, (b) final voltage, (c) final capacitance

CHAPTTER TH HREE: D DIRECT C CURREN NT CIRC CUITS

This Chapter is divided into two parts: CIRCUIT ELEMENTS 1.Resistors 2. Batteries 3. Combination of Resistors DC CIRCUITS 1 1.

Analysis

2.

RC Circuits

CHAPTER OBJECTIVES 1. Define steady y state currents and its relation to a material’s resistance. | 2. Relate voltage, current and resistance of resistors. i t | 3. Differentiate a real from an ideal battery and monitor the energies of the circuital parameters | 4. Compute for the equivalent resistance of a network of resistors | 5. Analyze Direct Current Circuits using Kirchhoff’s Rules. | 6. 6 Analyze the behavior of RC Circuits. Circuits |

Electric Current

2.

Resistance and Resistors: Ohm’s Law

CIRCUITT ELEM MENTTS

1.

3.

EMF Sources: Real and Ideal B tt i Batteries

4.

Combination of Resistors

5.

Energy in Electric Circuits

ELECTRIC CURRENT |

IIs the h rate off flow fl off charges h per unit time

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SI Unit: Ampere (A) after Andre Marie Ampere p

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1 A = 1C/1s

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The direction of current is the direction of flow of positive charges Many types: 1. 2. 3.

Electron in Hydrogen Atom Electron Beam in Cathode Ray Tubes in TV’s Electricity in Wires

ELECTRIC CURRENT AND THE ELECTRIC FIELD |

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Since the direction of current is the direction of flow of positive charges: The direction of the electric field is in the same direction as the electric current!

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E OHM’s LAW: Current is related to the Electric Field via the conductivity σ of the and cross sectional area A of the material:

ELECTRICAL RESISTANCE |

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When a current passes through a material it encounters a potential drop! This potential drop is related to the current via the Ohm’s Relation! R is the proportionality constant called “RESISTANCE OF THE MATERIAL”

High V

Low V I E

Ohmic Non Oh i Ohmic

ELECTRICAL RESISTANCE AND RESISTIVITY |

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The Unit for Electrical R i Resistance iis the h Ohm Oh (Ω) ( ) after Georg Ohm Just lik J like capacitance, i resistance is not dependent on V or I, it’s dependent on the geometry and the kind of material we have.

ρ iss tthe e resistivity es st v ty oof tthe e material in (Ω -m), L is the length (m) and A(m2) is the cross sectional area of the material!

Examples: A Nichrome wire ( ρ=10-6 Ω-m) has a radius of 0.65mm. What length of wire is needed to obtain a resistance of 2.0 Ω)?

RESISTANCE AND RESISTORS Schematic Symbol for Resistors |

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Resistors are devices that provide resistance in a circuit. Resistors and the resistance they carry have many purposes and applications in a variety circuits. circuits

(ELECTROMOTIVE FORCE) EMF SOURCES |

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An emf source is a device th t elevates that l t th the potential of a charge across its terminal.

Positive Terminal

The potential gain is the emf of the battery! It serves as the source or the pump of current in the circuit! A very good example of emf source is the b battery! !

+ Negative Terminal

REAL AND IDEAL BATTERIES |

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The distinction Th di i i between an ideal and real battery y is in their terminal voltages (TV). Ideal Battery:

Real Battery: (Ideal yp plus a small Battery internal resistance r) IDEAL

REAL

BATTERIES…

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Thus,, the TV of a real battery is always less than an EMF. EMF Malfunctioning batteries have very large internal resistances.

Rate and Energy Stored:

Rate: 1 Ah = 3600 C

Totall Energy T E Stored W = Qξξ

ANALYZING A BASIC CIRCUIT

ENERGY IN ELECTRIC CIRCUITS The Unit of Power is Watts.

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Power delivered by a battery

Powered dissipated across a resistor i t

ideal

real

If V and R given are g If I and R are given

Example: 1. A 12-Ω resistor carries a current of 3 A. Find the power dissipated in this resistor. 2. A wire of resistance 5 Ω carries a current of 3A for 6s. (a) How much power is put into the wire? (45W) (b) How much thermal energy is produced? (270 J)

EXAMPLE: |

An 11-Ω resistor is connected across a battery y of emf 6V and internal resistance 1 Ω.

Find the following (a) The current (b) The terminal voltage g of the battery y (c) The power delivered by the emf source (d) The power delivered to the external resistor (e) The Th power dissipated di i t d by b the th b battery’s tt ’ iinternal t l resistance (f) If the battery is rated at 150 A•h, how much energy does it store? |

COMBINATIONS OF RESISTORS |

Resistors are also known as “Loads”

SERIES

PARALLEL

RESISTORS IN SERIES

Req

oYou can replace R1 and R2 with a single g resistor with a resistance Req. oFor series connection, the current is i the h same across each capacitor but there is a potential drop across each resistor!

RESISTORS IN PARALLEL

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Req

You can replace R1 and R2 with a resistor with resistance Req. For parallel connection the voltage across each resistor is the same but the current splits along the junctions.

EXAMPLE: For the circuit that appears below find the following: a) I, I1 and I2 b) Req c) Voltage drop across each resistor

1. Kirchhoff’s

Rules

CIRCUIIT ANA ALYSIIS

2 RC 2.

Circuits

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KIRCHHOFF’S RULES I1 |

Junction Rule: All currents in and all currents out the junction are equal y Iin = Iout y

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Loop Rule: In a single loop, all voltage gain is equal to all voltage drop y Vgain = Vdrop y It is important to take note of the loop direction y

I4 = I1+I2+I3 I4

Current I Current,

Loop direction DROP

GAIN

Loop Direction GAIN

DROP

ANALYSIS OF CIRCUITS |

1 Si 1. Single l Loop L y

Find the current in this circuit

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2 M 2. Multiloop l il Circuits Ci i y

Find the currents I1, I2, and I3.

MORE KIRCHHOFF’S |

Find all the currents through g jjunction b

RC CIRCUITS | |

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Contains C i a resistor i and da capacitor. I flows in a single direction but its magnitude varies with time. RC Circuit “charges” g and “discharges” For charging: g g we p put in the maximum amount of charge possible in the capacitor over a time constant For discharging: we drain the charge until it it’ss value is negligible!

CHARGING RC |

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We assume that W h the h capacitor is initially uncharged. g Charge will increase in the capacitor, however, current decreases. Charge in the capacitor at some time later, will reach its maximum value of Q = Cξ when the current I equals zero.

Qf is the maximum charge that can be stored in a capacitor I0 is the initial current in the circuit

DISCHARGING RC |

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Discharge happens because when the switch is closed at t = 0, there is a potential drop across the resistor, meaning there is current in it. After some time, Af i the h charge on the capacitor is reduced, hence the current is also reduced! (Why is this happening?) This happens pp again g and again, until at some time, the charge and the current are both negligible hence “discharged”

Qo is the initial charge that is stored in a capacitor I0 is the initial current in the circuit

EXAMPLES: |

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1. An uncharged capacitor and a resistor are connected in series to a battery. If ξ = 12 0 V 12.0 V, C = 5.00 5 00 μF, μF and R = 8.00 x 105 Ω, find the time constant of the circuit, the maximum charge on the capacitor, it and d the th maximum current in the circuit. 2. Consider a capacitor of capacitance C that is being discharged through a resistor of resistance R, as shown in the figure. After how many time constants is th charge the h g on th the capacitor it one-fourth its initial value?