Chapter 26

Capacitance and Dielectrics

Intro

• In this chapter we will introduce the first of the 3 simple electric circuit elements that we will discuss in AP Physics– Capacitor– Resistor– Inductor

• Capacitors are commonly used devices often as form of energy storage in a circuit.

26.1

• Capacitor- two conductors separated by an insulator called a dielectric.

• Often times the two conductors of a capacitor are called plates.

• If the plates carry a charges of equal magnitude and opposite sign, there will exist a potential difference (ΔV) or a voltage between the two.

26.1

• How much charge can we store on the plates? – Experiments have shown that the amount of

charge stored increases linearly with voltage between the conductors.

• We will call the constant of proportionality Capacitance .

VQ VCQ

26.1

• Capacitance is defined as the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between the conductors.

• Capacitance is a constant for a given capacitor and has units of C/V or F (farad)

V

QC

26.1

• 1 Farad is a Coulomb of charge per Volt which is a huge capacitance.

• More commonly in the 10-6 to 10-12 range or microfarads (μF) to picofarads (pF)

• Actual capacitors may often be marked mF (microfarad) or mmF (micromicrofarad)

26.1

• Consider two parallel plates attached to a battery (potential difference source)

• The battery creates an Electric field within the wires, moving electrons onto the negative plate.• This continues until the negativeterminal, wire and plate are equipotential.

26.1

• The opposite occurs with the positive terminal, pulling electrons from the plate until the plate/wire/+ terminal are equipotential.

• Example-– A 4 pF capacitor will be able to store 4 pC of charge

for every volt of potential difference between the plates.

– If we attach a 1.5 V battery, one of the conductors will have a +6 pC charge, the other will have -6 pC.

– 12 V battery?

26.1

• Quick Quiz p 797

26.2 Calculating Capacitance

• We can derive expressions for the capacitance of pairs of oppositely charged conductors by calculating ΔV using techniques from the previous chapters.

• The calculations are generally straightforward for simple capacitors with symmetrical geometry.

26.2

• A single conductor – Sphere (w/infinite imaginary shell)– Since at the sphere, V = kQ/R, and at ∞, V = 0

V

QC

RQk

QC

e /

ek

RC

RC o4

26.2

• Parallel Plate Capacitors– For plates whose separation is much smaller than

their size.– From earlier (Ch 24) the E field between the plates

is

– So the potential difference iso

E

A

QE

o

A

QdEdV

o

26.2

– And the capacitance is therefore

– The capacitance is proportional to area and inversely proportional to the plate separation.

– True in the middle of the plates, but not near the edges.

V

QC

AQd

QC

o/

d

AC o

26.2

• The capacitor stores electrical potential energy as well as charge due to the separation of the positive and negative charges on the plates.

Quick Quiz p. 800Examples 26.1-26.3

26.2

• Example 26.2 Cylindrical Capacitor

)/ln(2 abkC

e

26.2

• Example 26.3 Spherical Capacitor

)( abk

abC

e

26.3 Combinations of Capacitors

• Often two or more capacitors are combined in electric circuits.

• We can calculate the equivalent capacitance of a circuit, based on how the capacitors are connected.

• We will use circuit diagrams (schematics) as pictorial representations of the circuit.

26.3

• Connecting wires- straight lines• Capacitors- parallel lines of equal length• Batteries- parallel lines of unequal length• Switch- swinging line representing “open” or “closed” circuits

26.3

• Parallel Combination- two capacitors connected with their own conducting path to the battery.

• The potential difference across each capacitor is the same, and its equal to the potential across thecombination.

26.3

• When the battery is connected electrons are removed from the positive plates and deposited on the negative plates.

• This flow of charge ceases when the potential across the plates reaches that of the battery.

• The capacitors are then at maximum charge Q1 and Q2, with a total charge given by

21 QQQ

26.3

• Because the voltages across the capacitors are the same the charges are

• If we wanted to replace the two capacitors with a single equivalent capacitor the total charge stored must be

VCQ 11VCQ 22

VCQ eq

26.3

• Therefore the equivalent capacitance must be

• The equivalent capacitance for any number of parallel combination of capacitors is

21 CCCeq

...321 CCCCeq

26.3

26.3

• Series Combination- two or more capacitors connected along the same conducting path

26.3

• As the battery charges the capacitors the electrons leaving the positive plate of C1 end up on the negative plate of C2.

• The electrons from the positive plate of C2 move to the negative plate of C1.

• All capacitors hold the same charges.QQQ 21

26.3

• The voltage of the battery is split across the capacitors.

• The total potential difference across a series combination of capacitors is the sum of the potential difference across each individual capacitor.

21 VVV

26.3

• If we wanted to find one capacitor equivalent to the series combination, the total potential difference is

• And each individual is

eqC

QV

22 C

QV

11 C

QV

26.3

• So from

• We get

• And finally

21 VVV

21 C

Q

C

Q

C

Q

eq

...1111

321 CCCCeq

26.3

• The inverse of the equivalent capacitance is equal to the sum of the inverses of the individual capacitances in series combination.

26.3

• Quick Quizzes p. 805• Example 26.4 p. 806

26.4 Energy Stored in a Charged Capacitor

• To determine the energy in a capacitor, we’re going to look at an atypical charging process.

• We’re going to imagine moving the charge from one plate to the other mechanically, through the space in between.

26.4

• Assume we currently have a charge q on our capacitor, giving the current potential difference to be ΔV = q/C.

• The work it will take to move a small increment of charge across the gap is

dqC

qVdqdW

26.4

• The total work W, required to charge the capacitor from q = 0 to q = Q is

dqC

qW

Q

0

dqqC

WQ

0

1

C

QW

2

2

26.4

• The work done in charging the capacitor is the Electrical Potential Energy stored and applies to any capacitor regardless of geometry.

• Energy increases as both Charge and Voltage increase, within a limit. At high enough potential difference, discharge will occur between the plates.

221

21

2

)(2

VCVQC

QU

26.4

• We can describe the energy as being stored in the electric field between the plates.

• For parallel plate caps

• Therefore

EdV d

AC o

221 EAdU o

26.4

• We use this expression to derive a new quantity called Energy Density (uE)

• Since the volume occupied by the field is Ad, the energy U per unit volume is (U/Ad)

• The energy density of an E-field is proportional the square of the magnitude of the E-field at a given point.

221 Eu oE

26.4

• Quick Quizzes p 808• Example 26.5

26.4

• Defibrillation- Capacitors store 360 J of energy at a potential difference that will deliver the energy in a time of 2 ms.

• Circuitry allows the capacitor to be charged (to a much higher voltage than the battery) over several seconds.

• Similar technology to camera flashes

26.5 Capacitors and Dielectrics

• A dielectric is a non-conducting material (rubber/glass/waxed paper) that can be placed between the plates of a capacitor to increase its capacitance.

• If the space is entirely filled with the dielectric material, C will increase by a dimensionless factor κ, the dielectric constant of the material.

26.5

• The new capacitance voltage and charge will be given by

(for constant charges)

(for constant voltage)

• Or specifically for a parallel plate capacitor

oCC

d

AC o

oVV

oQQ

26.5

26.5

• Again we see that capacitance still increases with decreasing d.

• In practice though, there is a lower limit to d before discharge across the plates will occur for a given voltage.

• So for any given capacitor of separation d, there is a maximum voltage limit.

26.5

• This limit depends on a factor called the Dielectric Strength.

• This is the maximum Electric Field (V/m) that the material can withstand before its insulating properties break down and the material becomes a conductor.

• Similar in concept to the spark touching a doorknob and also corona discharge.

26.5

• For parallel plates, the maximum voltage (AKA “working voltage,” “breakdown voltage,” and “rated voltage” is determined by

• Where Emax is the dielectric strength

dEV maxmax

26.5• Table of Dielectric Constants/Strengths p. 812

26.5

• Types of Commercial Capacitors– Tubular Capacitors- metallic foil interlaced with

wax paper/mylar, rolled into a tube.

26.5

– High voltage Capacitors- interwoven metallic plates immersed in an insulating (silicon) oil.

26.5

– Electrolytic Capacitors• Designed for large charges at low voltages.• One conducting foil immersed in an conducting fluid

(electrolyte)• The metal forms a thin insulating oxide layer when

voltage is applied.

26.5

– Variable Capacitors- interwoven sets of plates with one set fixed and one set able to be rotated.• Typical used for tuning dial circuits (radios, power

supplies etc)

26.5

• Quick Quizzes p 813-814• Examples 26.6, 26.7