class 6 - byu department of physics and astronomy · class 6 physics 106 learning outcomes...

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors

Capacitors inCircuitsCapacitors and Energy

Dielectrics

Voltage, Charge,and Current

Class 6Capacitance and Capacitors

Physics 106

Winter 2019

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Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Learning Outcomes

From last time:I The field lines are related to the field as follows:I What is the electric potential?I How are the electric field and the electric potential

related?I How can we find the electric field and the electric

potential?I How are electric fields and electric potentials used in

practical applications?I The electric potential is the potential energy divided

by the chargeI The electric potential is also called the voltageI Applying fields to a CRT

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Learning Outcomes

Today we will discuss:I CapacitanceI CapacitorsI Capacitors in series and parallel circuitsI Dielectrics

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitance

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitance

I A capacitor consists of two conductors, one with acharge +Q and one with a charge −Q.

I Often the conductors are parallel plates.I The voltage difference between the conductors is

∆V .I Out of tradition and laziness, we usually write the

voltage difference as just V .

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitance

I C ≡ QV

I Units: Farad (F)I 1 F = 1 C / 1 VI A farad is very largeI Often will see µF or pF

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitance

I Q = CVI A big capacitor holds a large charge at a small

voltage.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitors

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitors

I First developed by Pieter van Musschenbroek inLeyden in 1746

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Parallel-Plate Capacitor

I The capacitance of a device depends on thegeometric arrangement of the conductors

I For a parallel-plate capacitor whose plates areseparated by air:

C = ε0Ad

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I The capacitorconsists of twoparallel plates

I Each have area AI They are separated

by a distance dI The + charge on one

plate holds the −charge on the otherplate in place.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I If the plates arelarge, the capacitorcan hold morecharge.

I If the plates arecloser together, thecapacitor can holdmore charge,because the +charge attracts the −charge morestrongly.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Parallel Plate Capacitor

I Consists of two conducting plates, one positive andone negative

I Charge is pulled to the inside surface of either plateI The field outside either plate is zero

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Electric Field in a Parallel-Plate Capacitor

I The electricfield betweenthe plates isquite uniform

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Example 1: Derive the Parallel-PlateCapacitor Equations

I Consider a parallel-plate capacitor with plates ofarea A separated by distance d .

I The electric field in a capacitor is

E =σ

ε0

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I The surface charge density is the total charge of a

plate divided by its total area:

σ =QA

I This gives:

E =σ

ε0=

Qε0A

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I The electric field can be given in terms of the voltage:

E = −∆V∆x

I The − sign gives the field direction, but we’reonly interested in magnitude, so we ignore it.

I We also write ∆V as V , giving us:

E =Vd

I So the voltage is

V = Ed =Qdε0A

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I The capacitance can then be found:

C =QV

= Qε0AQd

C =ε0Ad

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I The capacitance can then be found:

C =QV

=

Q

ε0A

Q

d

C =ε0Ad

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


A Short Problem:

I You have two square plates 1.00 m on each side andyou wish to make a 1.00 F capacitor. (That’s a hugecapacitace!) If there is air between the plates, what isthe separation distance?

C =ε0Ad⇒ d =

ε0AC

= 8.85× 10−12m

That’s much smaller than one atom!

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitors in Circuits

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I The simplest capacitor circuit is a capacitorconnected to a battery with a switch to allow currentto flow.

I This is schematicallyrepresented as:

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I When we close the switch, charge flows from thebattery into capacitor.

I As the capacitor charges, itpushes charges in the wirein opposition to the battery.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I When the voltage on the capacitor matches thevoltage of the battery, current ceases to flow.

I The charge on thecapacitor is Q = CV whereV is the voltage of thebattery.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Series and Parallel

The most common ways of connecting multiple circuitelements are in "series" and "parallel"

Two capacitors joined inseries.

Two capacitors joined inparallel.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitors in Parallel

I The total charge isequal to the sum ofthe charges on thecapacitors

I Qtotal = Q1 + Q2I The voltage across

each capacitor isthe same − andthe same as thevoltage of thebattery

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Combining Capacitors in ParallelI The capacitors

are equivalentto a singlecapacitor witha capacitanceof Ceq

Qeq = Q1 + Q2

CeqV =

C1V + C2V

Ceq = C1 + C2

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Combining Capacitors in Parallel

I Addingcapacitance inparallel isanalogous toincreasing thearea of acapacitor.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Combining Capacitors in Parallel

I Ceq = C1 + C2 + · · ·I The equivalent capacitance of a parallel

combination of capacitors is greater than any ofthe individual capacitors

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitors in Series

Consider the two capacitors shown below.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics




I As + charge enters on the left, it drives charge fromthe right plate of the left capacitor to the left plate ofthe right capacitor.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics




I The charge on each capacitor is the same.

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



Veq = V1 + V2

QCeq

=QC1

+QC2

1Ceq

=1

C1+

1C2

I An equivalent capacitor can be found that performsthe same function as the series combination

I The potential differences add up to the batteryvoltage

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics



I Adding capacitors in series is analogous toincreasing the distance between capacitor plates.

I The equivalent capacitance of a series combinationis always less than any individual capacitor in thecombination

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Energy Stored in a Capacitor

I Energy stored is U = 12 CV 2

I From the definition of capacitance, this can berewritten in different forms:

U =12

CV 2 =Q2

2C=

12

QV

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Dielectrics

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitors with Dielectrics

I A dielectric is an insulating material placed betweenthe plates of a capacitor

I Dielectrics increase capacitanceI C = κC0 = κε0(A/d) where κ is the dielectric

constantI Dielectrics include rubber, plastic, or waxed paper

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Dielectric Strength

I For any given plate separation, there is a maximumelectric field that can be produced in the dielectricbefore it breaks down and begins to conduct

I This maximum electric field is called the dielectricstrength

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Capacitors with Dielectrics

Adding a dielectric between charged capacitor platesreduces the voltage. Why?

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


An Atomic Description of Dielectrics

I Polarization occurs when there is a separationbetween the negative charge and the positive chargeof the dielectric

I The dielectric becomes polarized because it is in anelectric field of the plates

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Adding a Dielectric to a Capacitor withFixed Charge

I The charge on thedielectric creates afield that opposesthe field of theplates

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Adding a Dielectric to a Capacitor withFixed Charge

I The charge on thedielectric creates afield that opposesthe field of theplates

I This reduces thetotal electric fieldand the voltage

I The capacitancetherefore increases

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Voltage, Charge, andCurrent

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


We are going to make a human model ofcircuits

I Traditionally, we think of positive charge as moving ina circuit. You will be the positive charge.

I A few of you will be a neutral wire. Hold your handsup and repel each other.

I Now be a positively-charged wire.I Now be a negatively-charged wire.I If you were charges on a real wire, where would you

go?

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


A Battery

I A battery pushes charges onto one end of a wire andpulls charges off the other end. A few of you will be abattery behind the stand.

I Now make a current flow around the stand.I Where is the wire positive, neutral, negative?I Where is the energy of positive charges highest

along the wire? What kind of energy is it?I Where is the voltage highest in the wire?I How do you think electrons actually move in a wire?

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


A Resistor

I A few of you will be a resistor.I Charges collide with atoms in the resistor and

change their direction of flow.I Be a large resistor.I Be a small resistor.I How does resistance affect current.?I What could we do to get more current?

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


A Resistor, Part B

I A few more students will make the resistor longer.I What happens to the current?I Now make the resitor shorter and wider.I What happens to the current?I What can you say about the voltage on the wire?

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


A Capacitor

I The battery is now disconnected. – But don’t goaway.

I A few of you will make a capacitor in front of thestand. Remember that you are replled by positivecharges on the other plate, but attracted to the"negative protons" in your own metal plate.

I Now we’ll hook up the battery.I What happens to the total charge?I What happens to the charge on each plate?I What happens to the current?I What happens to the voltage?

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Parallel Capacitors

I Disconnect the battery again.I Now split the capacitor into two separate capacitors,

side by side. You will also need some extra wire.I These capacitors are in parallel.I Reconnect the battery. What can you say about the

voltages across the two capacitors?I What can you say about the charges on the

capacitors?I What happens if one capacitor is bigger than the

other?

Class 6

Physics 106

LearningOutcomes

Capacitance

Capacitors


Dielectrics


Series Capacitors

I Disconnect the batteryI Now connect the capacitors in series, one after the

other.I Reconnect the battery.I What happens to the charges between the two

capacitors?I What can you say about the charge on each

capacitor?I What can you say about the voltages on the two

capacitors?

class 6 - byu department of physics and astronomy · class 6 physics 106 learning outcomes...

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