chapter 26 capacitance and dielectrics. intro in this chapter we will introduce the first of the 3...
TRANSCRIPT
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