Electric Fields and Forces

AP Physics B

Electric Charge“Charge” is a property of subatomic particles.

Facts about charge: There are 2 types basically, positive (protons)

and negative (electrons) LIKE charges REPEL and OPPOSITE

charges ATTRACT Charges are symbolic of fluids in that they

can be in 2 states, STATIC or DYNAMIC.

Electric Charge – The specificsSome important

constants:

•The symbol for CHARGE is “q”•The unit is the COULOMB(C), named after Charles Coulomb•If we are talking about a SINGLE charged particle such as 1 electron or 1 proton we are referring to an ELEMENTARY charge and often use, e , to symbolize this.

Particle Charge Mass

Proton 1.6x10-19 C 1.67 x10-27 kg

Electron 1.6x10-19 C 9.11 x10-31 kg

Neutron 0 1.67 x10-27 kg

Charge is “CONSERVED”

Charge cannot be created or destroyed only transferred from one object to another. Even though these 2 charges attract initially, they repel after touching. Notice the NET charge stays the same.

Conductors and InsulatorsThe movement of charge is limited by the substance

the charge is trying to pass through. There are generally 2 types of substances.

Conductors: Allow charge to move readily though it.

Insulators: Restrict the movement of the charge

Conductor = Copper WireInsulator = Plastic sheath

Charging and Discharging

There are basically 2 ways you can charge something.

1. Charge by friction

2. Induction

“BIONIC is the first-ever ionic formula mascara. The primary ingredient in BIONIC is a chain molecule with a positive charge. The friction caused by sweeping the mascara brush across lashes causes a negative charge. Since opposites attract, the positively charged formula adheres to the negatively charged lashes for a dramatic effect that lasts all day.”

Induction and GroundingThe second way to charge something is via

INDUCTION, which requires NO PHYSICAL CONTACT.

We bring a negatively charged rod near a neutral sphere. The protons in the sphere localize near the rod, while the electrons are repelled to the other side of the sphere. A wire can then be brought in contact with the negative side and allowed to touch the GROUND. The electrons will always move towards a more massive objects to increase separation from other electrons, leaving a NET positive sphere behind.

Electric ForceThe electric force between 2 objects is symbolic of the

gravitational force between 2 objects. RECALL:

MmFg 2

1

rFg

LawsCoulombr

qqkF

C

Nmx.k

kr

qqF

rFqqF

E

EEE

'

10998 constant Coulomb

alityproportion ofconstant

1

221

2

29

221

221

Electric Forces and Newton’s LawsElectric Forces and Fields obey Newton’s Laws.

Example: An electron is released above the surface of the Earth. A second electrondirectly below it exerts an electrostatic force on the first electron just great enough to cancel out the gravitational force on it. How far below the first electron is the second?

e

e

mg

Fe

r = ?

)8.9)(1011.9(

)106.1()99.8(

31

2199

212

21

x

x

mg

qqkrmg

r

qqk

mgFE

5.1 m

Electric Forces and VectorsElectric Fields and Forces are ALL vectors,

thus all rules applying to vectors must be followed.

Consider three point charges, q1 = 6.00 x10-9 C (located at the origin),q3 = 5.00x10-9 C, and q2 = -2.00x10-9 C, located at the corners of a RIGHT triangle. q2 is located at y= 3 m while q3 is located 4m to the right of q2. Find the resultant force on q3.

q1

q2 q3

3m

4m

5m

q3

Which way does q2 push q3?Which way does q1 push q3?

Fon 3 due to 2

Fon 3 due to 1

= 37

= tan-1(3/4)

Example Cont’

2,3

2

999

2,3 4

)102)(100.5()1099.8(

F

xxxF

1,3

2

999

1,3 5

)105)(106()1099.8(

F

xxxF

q1

q2 q3

3m

4m

5m

q3Fon 3 due to 2

Fon 3 due to 1

= 37= tan-1(3/4)

5.6 x10-9 N

1.1x10-8 N

F3,1cos37

F3,1sin37

)(tan

)()(F

1062.6)37sin(

1018.3

)37cos(

1

22resultant

91,3

9

2,31,3

x

y

res

yx

y

x

x

F

FDirection

F

FF

NxFF

NxF

FFF

7.34x10-9 N

64.3 degrees above the +x

Electric Fields

By definition, the are “LINES OF FORCE”

Some important facts: An electric field is a

vector Always is in the direction

that a POSITIVE “test” charge would move

The amount of force PER “test” charge

If you placed a 2nd positive charge (test charge), near the positive charge shown above, it would move AWAY.

If you placed that same charge near the negative charge shown above it would move TOWARDS.

Electric Fields and Newton’s Laws Once again, the equation for

ELECTRIC FIELD is symbolic of the equation for WEIGHT just like coulomb’s law is symbolic of Newton’s Law of Gravitation.

The symbol for Electric Field is, “E”. And since it is defined as a force per unit charge he unit is Newtons per Coulomb, N/C.

NOTE: the equations above will ONLY help you determine the MAGNITUDE of the field or force. Conceptual understanding will help you determine the direction.

The “q” in the equation is that of a “test charge”.

ExampleAn electron and proton are each placed at rest in an external

field of 520 N/C. Calculate the speed of each particle after 48 ns

What do we know

me=9.11 x 10-31 kg

mp= 1.67 x10-27 kg

qboth=1.6 x10-19 C

vo = 0 m/s

E = 520 N/C

t = 48 x 10-9 s

E

EE

F

x

F

q

FE

19106.1520

axamF

axamF

FFmaF

pE

eE

NetENet

)1067.1(

)1011.9(27

31

)1048(

)1048(9

9

xav

xav

atvv

pp

ee

o

8.32 x10-19 N

9.13x1013 m/s/s

4.98 x1010 m/s/s

4.38 x106 m/s

2.39 x103 m/s

An Electric Point ChargeAs we have discussed, all charges exert forces on other charges

due to a field around them. Suppose we want to know how strong the field is at a specific point in space near this charge the calculate the effects this charge will have on other charges should they be placed at that point.

POINT CHARGE

2chargepoint

2

2

r

kQE

r

QqkEq

EqFq

FE

r

QqkF E

EE

TEST CHARGE

ExampleA -4x10-12C charge Q is placed at the origin. What is the

magnitude and direction of the electric field produced by Q if a test charge were placed at x = -0.2 m ?

dir

mag

E

E

xx

r

kQE

2

129

2 2.

)104(1099.8

0.899 N/C

Towards Q to the right

Remember, our equations will only give us MAGNITUDE. And the electric field LEAVES POSITIVE and ENTERS NEGATIVE.

-Q

0.2 m E

E

E

E

Electric Field of a ConductorA few more things about electric fields, suppose you bring a conductor

NEAR a charged object. The side closest to which ever charge will be INDUCED the opposite charge. However, the charge will ONLY exist on the surface. There will never be an electric field inside a conductor. Insulators, however, can store the charge inside.

There must be a positive charge on this side

There must be a negative charge on this side OR this side was induced positive due to the other side being negative.

Electric Circuits

AP Physics B

Potential Difference =Voltage=EMFIn a battery, a series of chemical

reactions occur in which electrons are transferred from one terminal to another. There is a potential difference (voltage) between these poles.

The maximum potential difference a power source can have is called the electromotive force or (EMF), . The term isn't actually a force, simply the amount of energy per charge (J/C or V)

A Basic CircuitAll electric circuits have three main parts

1. A source of energy 2. A closed path 3. A device which uses the energy

If ANY part of the circuit is open the device will not work!

Electricity can be symbolic of Fluids Circuits are very similar to water flowing through a pipe

A pump basically works on TWO IMPORTANT PRINCIPLES concerning its flow

• There is a PRESSURE DIFFERENCE where the flow begins and ends

• A certain AMOUNT of flow passes each SECOND.

A circuit basically works on TWO IMPORTANT PRINCIPLES

• There is a "POTENTIAL DIFFERENCE aka VOLTAGE" from where the charge begins to where it ends

• The AMOUNT of CHARGE that flows PER SECOND is called  CURRENT.

CurrentCurrent is defined as the rate at which charge

flows through a surface.

The current is in the same direction as the flow of positive charge (for this course)

Note: The “I” standsfor intensity

There are 2 types of CurrentDC = Direct Current - current flows in one direction

Example: Battery

AC = Alternating Current- current reverses direction many times per second. This suggests that AC devices turn OFF andON. Example: Wall outlet (progress energy)

Ohm’s Law“The voltage (potential difference, emf) is directly

related to the current, when the resistance is constant”

IR

IRV

R

R

IV

Resistance

alityproportion ofconstant

Voltage vs. Current

0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1

Current(Amps)

Vo

ltag

e(V

)

Voltage(V)

Since R=V/I, the resistance is the SLOPE of a V vs. I graph

R= resista

nce = slo

pe

ResistanceResistance (R) – is defined as the restriction of electron

flow. It is due to interactions that occur at the atomic scale. For example, as electron move through a conductor they are attracted to the protons on the nucleus of the conductor itself. This attraction doesn’t stop the electrons, just slow them down a bit and cause the system to waste energy.

The unit for resistance is the OHM,

Electrical POWERWe have already learned that POWER is the rate at which work

(energy) is done. Circuits that are a prime example of this as batteries only last for a certain amount of time AND we get charged an energy bill each month based on the amount of energy we used over the course of a month…aka POWER.

POWERIt is interesting to see how certain electrical

variables can be used to get POWER. Let’s take Voltage and Current for example.

Other useful power formulas

These formulas can also be used! They are simply derivations of the POWER formula with different versions of Ohm's law substituted in.

Ways to Wire CircuitsThere are 2 basic ways to wire a circuit. Keep in

mind that a resistor could be ANYTHING ( bulb, toaster, ceramic material…etc)

Series – One after anotherParallel – between a set of junctions and parallel to each other

Schematic SymbolsBefore you begin to understand circuits you need to be able to

draw what they look like using a set of standard symbols understood anywhere in the world

For the battery symbol, the LONG line is considered to be the POSITIVE terminal and the SHORT line , NEGATIVE.

The VOLTMETER and AMMETER are special devices you place IN or AROUND the circuit to measure the VOLTAGE and CURRENT.

The Voltmeter and AmmeterThe voltmeter and ammeter cannot be just placed anywhere in the circuit. They must be used according to their DEFINITION.

Since a voltmeter measures voltage or POTENTIAL DIFFERENCE it must be placed ACROSS the device you want to measure. That way you can measure the CHANGE on either side of the device.

Voltmeter is drawn ACROSS the resistor

Since the ammeter measures the current or FLOW it must be placed in such a way as the charges go THROUGH the device.

Current goes THROUGH the ammeter

Simple CircuitWhen you are drawing a

circuit it may be a wise thing to start by drawing the battery first, then follow along the loop (closed) starting with positive and drawing what you see.

Series CircuitIn in series circuit, the resistors

are wired one after another. Since they are all part of the SAME LOOP they each experience the SAME AMOUNT of current. In figure, however, you see that they all exist BETWEEN the terminals of the battery, meaning they SHARE the potential (voltage).

321)(

321)(

VVVV

IIII

Totalseries

Totalseries

Series Circuit

321)(

321)(

VVVV

IIII

Totalseries

Totalseries

As the current goes through the circuit, the charges must USE ENERGY to get through the resistor. So each individual resistor will get its own individual potential voltage). We call this VOLTAGE DROP.

is

series

seriesTT

Totalseries

RR

RRRR

RIRIRIRI

IRVVVVV

321

332211

321)(

)(

;Note: They may use the terms “effective” or “equivalent” to mean TOTAL!

Example A series circuit is shown to the left. a) What is the total resistance?

b) What is the total current?

c) What is the current across EACH resistor?  

d) What is the voltage drop across each resistor?( Apply Ohm's law to each resistor separately)

R(series) = 1 + 2 + 3 = 6

V=IR 12=I(6) I = 2A

They EACH get 2 amps!

V12 V V3=(2)(3)= 6V V2=(2)(2)= 4V

Notice that the individual VOLTAGE DROPS add up to the TOTAL!!

Parallel CircuitIn a parallel circuit, we have

multiple loops. So the current splits up among the loops with the individual loop currents adding to the total current

It is important to understand that parallel circuits will all have some position where the current splits and comes back together. We call these JUNCTIONS.

The current going IN to a junction will always equal the current going OUT of a junction.

Junctions OUTIN

Totalparallel

II

IIII

:Junctions Regarding

321)(

Parallel CircuitNotice that the JUNCTIONS both touch the POSTIVE and NEGATIVE terminals of the battery. That means you have the SAME potential difference down EACH individual branch of the parallel circuit. This means that the individual voltages drops are equal.

This junction touches the POSITIVE terminal

This junction touches the NEGATIVE terminal

V

iP

P

ParallelT

T

Totalparallel

Totalparallel

RR

RRRR

R

V

R

V

R

V

R

V

IRVIIII

VVVV

11

1111

)(

;

321

3

3

2

2

1

1

321)(

321)(

Example

454.0

1454.0

1

9

1

7

1

5

11

Pp

P

RR

R

)(8 RI

IRV

To the left is an example of a parallel circuit. a) What is the total resistance?  

b) What is the total current?  

c) What is the voltage across EACH resistor?   d) What is the current drop across each resistor? (Apply Ohm's law to each resistor separately)

2.20

3.64 A

8 V each!

9

8

7

8

5

8975 III

IRV

1.6 A 1.14 A 0.90 A

Notice that the individual currents ADD to the total.

Compound (Complex) CircuitsMany times you will have series and parallel in the SAME circuit.

Solve this type of circuit from the inside out.

WHAT IS THE TOTAL RESISTANCE?

3.1133.3380

3.33;50

1

100

11

s

PP

R

RR

Compound (Complex) Circuits

3.1133.3380

3.33;50

1

100

11

s

PP

R

RR

T

T

TTT

I

I

RIV

)3.113(120

Suppose the potential difference (voltage) is equal to 120V. What is the total current?

1.06 A

What is the VOLTAGE DROP across the 80 resistor?

80

80

808080

)80)(06.1(

V

V

RIV

84.8 V

Compound (Complex) Circuits

AI

VV

AI

VV

R

T

T

T

06.1

8.84

06.1

120

3.113

80

80

3&2

3&2

3&21)(

32)(

8.84120

V

V

VVV

VVV

seriesT

parallelT

What is the VOLTAGE DROP across the 100 and 50 resistor?

35.2 V Each!

50

2.35100

2.35

50

100

3&21)(

32)(

I

I

III

III

seriesT

parallelT

What is the current across the 100 and 50 resistor?

0.352 A

0.704 A

Add to 1.06A

Electrical Energy, Potential and Capacitance

AP Physics B

Electric Fields and WORK

In order to bring two like charges near each other work must be done.   In order to separate two opposite charges, work must be done.  Remember that whenever work gets done, energy changes form.

As the monkey does work on the positive charge, he increases the energy of that charge.  The closer he brings it, the more electrical potential energy it has.   When he releases the charge, work gets done on the charge which changes its energy from electrical potential energy to kinetic energy.  Every time he brings the charge back, he does work on the charge.  If he brought the charge closer to the other object, it would have more electrical potential energy.  If he brought 2 or 3 charges instead of one, then he would have had to do more work so he would have created more electrical potential energy.  Electrical potential energy could be measured in Joules just like any other form of energy.

Electric Fields and WORKConsider a negative charge moving

in between 2 oppositely charged parallel plates initial KE=0 Final KE= 0, therefore in this case Work = PE

We call this ELECTRICAL potential energy, UE, and it is equal to the amount of work done by the ELECTRIC FORCE, caused by the ELECTRIC FIELD over distance, d, which in this case is the plate separation distance.

Is there a symbolic relationship with the FORMULA for gravitational potential energy?

Electric Potential

Edq

W

qEdWU

dxh

Eg

qm

WorUU

mghU

E

Eg

g

)(

)(

Here we see the equation for gravitational potential energy.

Instead of gravitational potential energy we are talking about ELECTRIC POTENTIAL ENERGY

A charge will be in the field instead of a mass

The field will be an ELECTRIC FIELD instead of a gravitational field

The displacement is the same in any reference frame and use various symbols

Putting it all together!

Question: What does the LEFT side of the equation mean in words? The amount of Energy per charge!

Energy per chargeThe amount of energy per charge has a specific

name and it is called, VOLTAGE or ELECTRIC POTENTIAL (difference). Why the “difference”?

q

mv

q

K

q

WV

2

21

Understanding “Difference”Let’s say we have a proton placed

between a set of charged plates. If the proton is held fixed at the positive plate, the ELECTRIC FIELD will apply a FORCE on the proton (charge). Since like charges repel, the proton is considered to have a high potential (voltage) similar to being above the ground. It moves towards the negative plate or low potential (voltage). The plates are charged using a battery source where one side is positive and the other is negative. The positive side is at 9V, for example, and the negative side is at 0V. So basically the charge travels through a “change in voltage” much like a falling mass experiences a “change in height. (Note: The electron does the opposite)

BEWARE!!!!!!W is Electric Potential Energy (Joules)

is notV is Electric Potential (Joules/Coulomb)

a.k.a Voltage, Potential Difference

The “other side” of that equation?

Edq

W

qEdWU

dxh

Eg

qm

WorUU

mghU

E

Eg

g

)(

)(

Since the amount of energy per charge is called Electric Potential, or Voltage, the product of the electric field and displacement is also VOLTAGE

This makes sense as it is applied usually to a set of PARALLEL PLATES.

V=Ed

E dV

ExampleA pair of oppositely charged, parallel plates are separated by

5.33 mm. A potential difference of 600 V exists between the plates. (a) What is the magnitude of the electric field strength between the plates? (b) What is the magnitude of the force on an electron between the plates?

Cxq

EE

EVV

EdVmd

e

19106.1

?

)0053.0(600600

00533.0

e

ee

F

Cx

F

q

FE

19106.1113,207.55 N/C 1.81x10-14 N

ExampleCalculate the speed of a proton that is accelerated

from rest through a potential difference of 120 V

?

120

1067.1

106.1

27

19

v

VV

kgxm

Cxq

p

p

27

19

2

1067.1

)120)(106.1(22

21

x

x

m

Vqv

qmv

q

K

q

WV

1.52x105 m/s

Electric Potential of a Point ChargeUp to this point we have focused our attention solely to

that of a set of parallel plates. But those are not the ONLY thing that has an electric field. Remember, point charges have an electric field that surrounds them.

So imagine placing a TEST CHARGE out way from the point charge. Will it experience a change in electric potential energy? YES!

Thus is also must experience a change in electric potential as well.

Electric PotentialLet’s use our “plate” analogy. Suppose we had a set of parallel plates symbolic of being “above the ground” which has potential difference of 50V and a CONSTANT Electric Field.

+++++++++++

----------------

Ed 0.5d, V= 25 V

0.25d, V= 12.5 V

1

2 3

4

V = ? From 1 to 2

V = ? From 2 to 3

V = ? From 3 to 4

V = ? From 1 to 4

25 V

0 V

12.5 V

37.5 V

Notice that the “ELECTRIC POTENTIAL” (Voltage) DOES NOT change from 2 to 3. They are symbolically at the same height and thus at the same voltage. The line they are on is called an EQUIPOTENTIAL LINE. What do you notice about the orientation between the electric field lines and the equipotential lines?

Equipotential LinesSo let’s say you had a positive

charge. The electric field lines move AWAY from the charge. The equipotential lines are perpendicular to the electric field lines and thus make concentric circles around the charge. As you move AWAY from a positive charge the potential decreases. So V1>V2>V3.

Now that we have the direction or visual aspect of the equipotential line understood the question is how can we determine the potential at a certain distance away from the charge?

r

V(r) = ?

Electric Potential of a Point Charge

r

Qk

qr

QqrkV

r

QqkF

q

rFV

rxq

xF

q

WV

EE

E

2

2;

;

Why the “sum” sign?

Voltage, unlike Electric Field, is NOT a vector! So if you have MORE than one charge you don’t need to use vectors. Simply add up all the voltages that each charge contributes since voltage is a SCALAR.

WARNING! You must use the “sign” of the charge in this case.

Potential of a point chargeSuppose we had 4 charges

each at the corners of a square with sides equal to d.

If I wanted to find the potential at the CENTER I would SUM up all of the individual potentials.

2

2

:equal will

center thecorner to a

from distance theThus

dr

2d

dk

d

qk

r

qkV

r

q

r

q

r

q

r

qkV

r

QkV

center

center

center

25)

2

10()

5(

)532

(

Electric field at the center? ( Not so easy) If they had asked us to find the

electric field, we first would have to figure out the visual direction, use vectors to break individual electric fields into components and use the Pythagorean Theorem to find the resultant and inverse tangent to find the angle

So, yea….Electric Potentials are NICE to deal with!

Eresultant

ExampleAn electric dipole consists of two charges q1 = +12nC and q2 =

-12nC, placed 10 cm apart as shown in the figure. Compute the potential at points a,b, and c.

a

a

aaa

V

xxxV

r

q

r

qkV

)04.0

1012

06.0

1012(1099.8

)(

999

21

-899 V

Example cont’

c

b

b

bbb

V

V

xxxV

r

q

r

qkV

)14.0

1012

04.0

1012(1099.8

)(

999

21

1926.4 V

0 V

Since direction isn’t important, the electric potential at “c” is zero. The electric field however is NOT. The electric field would point to the right.

Applications of Electric PotentialIs there any way we can use a set of plates with an electric

field? YES! We can make what is called a Parallel Plate Capacitor and Store Charges between the plates!

Storing Charges- CapacitorsA capacitor consists of 2 conductors of any shape placed near one another without touching. It is common; to fill up the region between these 2 conductors with an insulating material called a dielectric. We charge these plates with opposing charges toset up an electric field.

Capacitors in Kodak CamerasCapacitors can be easily purchased at a

local Radio Shack and are commonly found in disposable Kodak Cameras. When a voltage is applied to an empty capacitor, current flows through the capacitor and each side of the capacitor becomes charged. The two sides have equal and opposite charges. When the capacitor is fully charged, the current stops flowing. The collected charge is then ready to be discharged and when you press the flash it discharges very quickly released it in the form of light.

Cylindrical Capacitor

CapacitanceIn the picture below, the capacitor is symbolized by a set of parallel

lines. Once it's charged, the capacitor has the same voltage as the battery (1.5 volts on the battery means 1.5 volts on the capacitor) The difference between a capacitor and a battery is that a capacitor can dump its entire charge in a tiny fraction of a second, where a battery would take minutes to completely discharge itself. That's why the electronic flash on a camera uses a capacitor -- the battery charges up the flash's capacitor over several seconds, and then the capacitor dumps the full charge into the flash tube almost instantly

Measuring CapacitanceLet’s go back to thinking about plates!

V

QC

CVQ

C

VQ

ThereforeQE

difE

EdV

eCapacitancC

alityproportion ofcontant

constant,V

,

The unit for capacitance is the FARAD, F.

Capacitor GeometryThe capacitance of a

capacitor depends on HOW you make it.

d

AC

Nm

Cx

d

AC

A

CAC

o

o

2

212

o

o

1085.8

constantty permittivi vacuum

alityproportion ofconstant

platesbeteween distance d

plate of aread

1

Capacitor ProblemsWhat is the AREA of a 1F capacitor that has a plate

separation of 1 mm?

Sides

A

Ax

D

AC o

001.01085.81 12

1.13x108 m2

10629 m

Is this a practical capacitor to build?

NO! – How can you build this then?

The answer lies in REDUCING the AREA. But you must have a CAPACITANCE of 1 F. How can you keep the capacitance at 1 F and reduce the Area at the same time?

Add a DIELECTRIC!!!

DielectricRemember, the dielectric is an insulating material placed

between the conductors to help store the charge. In the previous example we assumed there was NO dielectric and thus a vacuum between the plates.

Dielectric

kd

AkC o

All insulating materials have a dielectric constant associated with it. Here now you can reduce the AREA and use a LARGE dielectric to establish the capacitance at 1 F.

Using MORE than 1 capacitorLet’s say you decide that 1

capacitor will not be enough to build what you need to build. You may need to use more than 1. There are 2 basic ways to assemble them together

Series – One after another

Parallel – between a set of junctions and parallel to each other.

Capacitors in SeriesCapacitors in series each charge each other by INDUCTION. So

they each have the SAME charge. The electric potential on the other hand is divided up amongst them. In other words, the sum of the individual voltages will equal the total voltage of the battery or power source.

Capacitors in ParallelIn a parallel configuration, the voltage is the same

because ALL THREE capacitors touch BOTH ends of the battery. As a result, they split up the charge amongst them.

Capacitors “STORE” energyAnytime you have a situation where energy is “STORED” it is called

POTENTIAL. In this case we have capacitor potential energy, Uc

Suppose we plot a V vs. Q graph. If we wanted to find the AREA we would MULTIPLY the 2 variables according to the equation for Area.

A = bh

When we do this we get Area = VQ

Let’s do a unit check!

Voltage = Joules/CoulombCharge = CoulombsArea = ENERGY

Potential Energy of a Capacitor Since the AREA under the line is a

triangle, the ENERGY(area) =1/2VQ

C

QQ

C

QU

CVVCVU

V

QCVQU

C

C

C

2)(2

1

21)(2

1

21

2

2

This energy or area is referred as the potential energy stored inside a capacitor.

Note: The slope of the line is the inverse of the capacitance.

most common form

ElectroMagnetic Induction

AP Physics B

What is E/M Induction?Electromagnetic Induction is the

process of using magnetic fields to produce voltage, and in a complete circuit, a current.

Michael Faraday first discovered it, using some of the works of Hans Christian Oersted. His work started at first using different combinations of wires and magnetic strengths and currents, but it wasn't until he tried moving the wires that he got any success.

It turns out that electromagnetic induction is created by just that - the moving of a conductive substance through a magnetic field.

Magnetic InductionAs the magnet moves back and forth a current is said

to be INDUCED in the wire.

Magnetic FluxThe first step to understanding the complex nature of

electromagnetic induction is to understand the idea of magnetic flux.

Flux is a general term associated with a FIELD that is bound by a certain AREA. So MAGNETIC FLUX is any AREA that has a MAGNETIC FIELD passing through it.

AB

We generally define an AREA vector as one that is perpendicular to the surface of the material. Therefore, you can see in the figure that the AREA vector and the Magnetic Field vector are PARALLEL. This then produces a DOT PRODUCT between the 2 variables that then define flux.

Magnetic Flux – The DOT product

Wb)or Weber( Tm :Unit

cos2

BA

AB

B

B

How could we CHANGE the flux over a period of time? We could move the magnet away or towards (or the wire) We could increase or decrease the area We could ROTATE the wire along an axis that is PERPENDICULAR to the field thus changing the angle between the area and magnetic field vectors.

Faraday’s LawFaraday learned that if you change any part of the flux over time

you could induce a current in a conductor and thus create a source of EMF (voltage, potential difference). Since we are dealing with time here were a talking about the RATE of CHANGE of FLUX, which is called Faraday’s Law.

wireof turns#

)cos(

Nt

BAN

tN B

Useful ApplicationsThe Forever Flashlight uses the Faraday Principle of

Electromagnetic Energy to eliminate the need for batteries. The Faraday Principle states that if an electric conductor, like copper wire, is moved through a magnetic field, electric current will be generated and flow into the conductor.

Useful ApplicationsAC Generators use Faraday’s

law to produce rotation and thus convert electrical and magnetic energy into rotational kinetic energy. This idea can be used to run all kinds of motors. Since the current in the coil is AC, it is turning on and off thus creating a CHANGING magnetic field of its own. Its own magnetic field interferes with the shown magnetic field to produce rotation.

TransformersProbably one of the greatest inventions of all time is the

transformer. AC Current from the primary coil moves quickly BACK and FORTH (thus the idea of changing!) across the secondary coil. The moving magnetic field caused by the changing field (flux) induces a current in the secondary coil.

If the secondary coil has MORE turns than the primary you can step up the voltage and runs devices that would normally need MORE voltage than what you have coming in. We call this a STEP UP transformer.

We can use this idea in reverse as well to create a STEP DOWN transformer.

MicrophonesA microphone works when sound

waves enter the filter of a microphone. Inside the filter, a diaphragm is vibrated by the sound waves which in turn moves a coil of wire wrapped around a magnet. The movement of the wire in the magnetic field induces a current in the wire. Thus sound waves can be turned into electronic signals and then amplified through a speaker.

ExampleA coil with 200 turns of wire is wrapped on an 18.0 cm square frame.

Each turn has the same area, equal to that of the frame, and the total resistance of the coil is 2.0 . A uniform magnetic field is applied perpendicularly to the plane of the coil. If the field changes uniformly from 0 to 0.500 T in 0.80 s, find the magnitude of the induced emf in the coil while the field has changed as well as the magnitude of the induced current.

I

IIR

xt

BAN

tN B

)2(

80.0

90cos)18.018.0)(0500.0(200

cos

4.05 V

2.03 A

Why did you find the ABSOLUTE VALUE of the EMF?

What happened to the “ – “ that was there originally?

Lenz’s LawLenz's law gives the direction of the induced emf and current

resulting from electromagnetic induction. The law provides a physical interpretation of the choice of sign in Faraday's law of induction, indicating that the induced emf and the change in flux have opposite signs.

tN B

Lenz’s Law

In the figure above, we see that the direction of the current changes. Lenz’s Law helps us determine the DIRECTION of that current.

Lenz’s Law & Faraday’s Lawt

N B

Let’s consider a magnet with it’s north pole moving TOWARDS a conducting loop.

DOES THE FLUX CHANGE?

DOES THE FLUX INCREASE OR DECREASE?

WHAT SIGN DOES THE “” GIVE YOU IN FARADAY’S LAW?

DOES LENZ’S LAW CANCEL OUT?

What does this mean?

Yes!

Increase

Positive

NO

This means that the INDUCED MAGNETIC FIELD around the WIRE caused by the moving magnet OPPOSES the original magnetic field. Since the original B field is downward, the induced field is upward! We then use the curling right hand rule to determine the direction of the current.

Binduced

Lenz’s Law

A magnet is dropped down a conducting tube.

The magnet INDUCES a current above and below the magnet as it moves.

The INDUCED current creates an INDUCED magnetic field of its own inside the conductor that opposes the original magnetic field.

Since the induced field opposes the direction of the original it attracts the magnet upward slowing the motion caused by gravity downward.

If the motion of the magnet were NOT slowed this would violate conservation of energy!

Lenz’s Law

Let’s consider a magnet with it’s north pole moving AWAY from a conducting loop.

DOES THE FLUX CHANGE?

DOES THE FLUX INCREASE OR DECREASE?

WHAT SIGN DOES THE “” GIVE YOU IN FARADAY’S LAW?

DOES LENZ’S LAW CANCEL OUT?

What does this mean?

tN B

Yes!

Decreases

negative

yes

In this case, the induced field DOES NOT oppose the original and points in the same direction. Once again use your curled right hand rule to determine the DIRECTION of the current.

Binduced

In summaryFaraday’s Law is basically used to find the

MAGNITUDE of the induced EMF. The magnitude of the current can then be found using Ohm’s Law provided we know the conductor’s resistance.

Lenz’s Law is part of Faraday’s Law and can help you determine the direction of the current provided you know HOW the flux is changing

Motional EMF – The Rail GunA railgun consists of two parallel metal rails (hence the name) connected to an electrical power supply. When a conductive projectile is inserted between the rails (from the end connected to the power supply), it completes the circuit. Electrons flow from the negative terminal of the power supply up the negative rail, across the projectile, and down the positive rail, back to the power supply.

In accordance with the right-hand rule, the magnetic field circulates around each conductor. Since the current is in opposite direction along each rail, the net magnetic field between the rails (B) is directed vertically. In combination with the current (I) across the projectile, this produces a magnetic force which accelerates the projectile along the rails. There are also forces acting on the rails attempting to push them apart, but since the rails are firmly mounted, they cannot move. The projectile slides up the rails away from the end with the power supply.

Motional EmfThere are many situations where motional EMF can occur that are

different from the rail gun. Suppose a bar of length, L, is pulled to right at a speed, v, in a magnetic field, B, directed into the page. The conducting rod itself completes a circuit across a set of parallel conducting rails with a resistor mounted between them.

R

BlvI

IR

Blvt

Blx

t

BAt

N B

;

Motional EMFIn the figure, we are

applying a force this time to the rod. Due to Lenz’s Law the magnetic force opposes the applied force. Since we know that the magnetic force acts to the left and the magnetic field acts into the page, we can use the RHR to determine the direction of the current around the loop and the resistor.

ExampleAn airplane with a wing span of 30.0 m flies parallel to the Earth’s

surface at a location where the downward component of the Earth’s magnetic field is 0.60 x10-4 T. Find the difference in potential between the wing tips is the speed of the plane is 250 m/s.

)250)(30(1060.0 4x

Blv

0.45 V

In 1996, NASA conducted an experiment with a 20,000-meter conducting tether. When the tether was fully deployed during this test, the orbiting tether generated a potential of 3,500 volts. This conducting single-line tether was severed after five hours of deployment. It is believed that the failure was caused by an electric arc generated by the conductive tether's movement through the Earth's magnetic field.