Exam 3 Lectures Magnetism. Definitions Magnetic field—The vector field a magnet produces all around itself to interact with its environment (produces.

Exam 3 Lectures

Magnetism

Definitions• Magnetic field—The vector field a magnet

produces all around itself to interact with its environment (produces force)

• Tesla—the unit of the magnetic field

• Permanent magnets—have a permanent magnetic field without outside influences

• Electromagnets—magnets, which have a magnetic field in the presence of a current but not without it

AmN

msCN

smC

NT

1111

More Definitions

• Monopoles—magnetic charges (not found in nature)

• North pole—end from which field lines emerge

• South pole—end where field lines enter.• Crossed fields—an E field and B field

present that are perpendicular to each other

Magnetic Fields• Produced by:

1. Moving electrically charged particles

2. Elementary particles (such as electrons) have an intrinsic magnetic field around them—basic characteristic of such particles

• On the earth the south magnetic pole is close to the north geographic pole, and the north magnetic pole is close to the south geographic pole

• A C shaped magnet is used to get a uniform magnetic field in experiments

• Opposite magnetic poles attract each other

• Like magnetic poles repel each other

Magnetic Field Lines and Magnetic Fields• The direction of the magnetic B field at

any point on a B field line is in the direction of the tangent to the B field line.

• The spacing of the lines represents the magnitude of the B field. The B field is stronger where the B field lines are closer together.

• The B field lines all pass through the magnet forming closed loops: they go in one side and out the other

• Force a moving charged particle feels from an external magnetic field

• Must use the right hand rule for direction

qvBBqvqvBF

BvqF

B

B

sin

Differences Between Electric and Magnetic Forces

• The electric force acts in the direction of the E field, whereas the magnetic force acts perpendicular to the B field.

• The electric force acts on a charged particle whether or not it is moving, whereas the magnetic force acts on a charged particle only if it is moving.

• The electric force does work in displacing a charged particle, whereas the magnetic force does no work when a charged particle is displaced

Circulating Charged Particles• Important that force is perpendicular to

velocity; therefore force can change direction but not magnitude of velocity

• Particles under the influence of the magnetic force undergo uniform circular motion

qBmv

qvBmv

r

rv

mqvBF

2

2qBm

qBmv

vvr

T 222

mqB

Tf

21

mqB

f 2

Circulating Charged Particles cont• If velocity of charged particle has

component parallel to magnetic field, the particle will move in a helical path

• Parallel component – pitch of helix

• Perpendicular component – radius of helix

cos|| vv

sinvv

Circulating charges have uses and examples in nature:

1. The magnetic bottles used in some experiments2. Cyclotrons3. Synchrotrons

The van Allen belts of the earth

Current Carrying Wire• Current consists of charges moving

along the wire

B d

d

F q v B

qv B nAL

iL B

BsidFd B

baB BsdiF

same direction of force if we assume positive or negative charge carriers

Torque

Torque tends to rotate the loop so that A is rotated into the direction of B

BBAi

iABiabB

ibBa

ibBa

Frnet

sinsin

sin2

sin2

Potential EnergyElectric Magnetic

sinpEEp

sinBB

BANi

cospEEpU cosBBU

Crossed Fields – Hall Effect1. Charged particles moving through the conductor

subject to an external B field – get a magnetic force acting on them.

2. Magnetic force deflects the charged particles in the direction of the force and makes them line up on one wall of the conductor.

3. One wall of the conductor is more negative and the other is more positive charge – electric field is set up.

4. Eventually electric force balances magnetic force and the charges are allowed to continue on their way.

5. One wall will be at a higher potential than the other.

6. By looking at the potential difference between the two walls the sign the charges may be determined

a) Shows the situation for which the charge carriers are negative

b) Shows the situation for which the charge carriers are positive

Mass Spectrometer• Another use of crossed fields

mqV

vmvqV2

21 2

2

22

qB

VmmqV

qBm

qBmv

r

VBr

qm

VBqr

m

2

222

22

Calculating Fields Comparison

Electric Magnetic Symmetry

No symmetry

symmetry

rr

dqkE e ˆ2

o

encqAdE

2

ˆ4 r

rsdIB o

inoIdsB

Calculating Magnetic Field due to a Current• Magnetic fields are produced by moving

charges (currents)

• Biot Savart Law

2

ˆ4 r

rsdIB o

AmpTm

o7104

Some Magnetic Fields

• Long Straight Wire

• Half a Long Straight Wire

• Circular Arc of Wire

Ri

B2

0

Ri

B4

0

Ri

B

40

Force Between 2 Parallel Currents

• Parallel currents attract, antiparallel currents repeldii

Ldi

LiBLiF aboaobabab

22

Ampere’s Law

• This is a line integral to be integrated around a closed loop (Amperian loop)

• To use Ampere’s Law1. First decide which type of symmetry best

complements the problem

2. Draw an Amperian loop (mathematical not real) reflecting the symmetry you chose around the current distribution through the point of interest.

inoIdsB

Long Cylindrical Conductor

INSIDElinear

OUTSIDEhyperbolic

Solenoid• Solenoid – a long tightly wound helical coil

of wire

ad

dc

cb

ba

sdBsdB

sdBsdBsdB

inB 0

Toroid• Toroid – a solenoid bent into a doughnut

shape

0

2iN

Br

Definitions• Induced current—current produced in a loop by a

changing magnetic field• Induced emf—work done per unit charge in producing

a current.• Induction—process of producing current and emf• Faraday’s law of induction—an emf is induced in a

loop when the number of magnetic field lines that pass through the loop changes.

• Lenz’s law—an induced current has a direction such that the magnetic field due to the current opposes the change in magnetic field that induces the current. The direction of the induced emf is the direction of the induced current

First of Two Experiments

• Three discoveries:1. Current appears only if there is relative

motion between the loop and magnet

2. Faster motion between the loop and magnet produces greater current

3. Opposite motion of the magnet produces opposite direction of current

Second of Two Experiments

• Current in a wire produces a magnet field• As the current increases the magnetic field

increases• As the magnetic field increases a current

appears.

Magnetic Flux

• Units of magnetic flux is the Weber

• Three terms and therefore three ways that flux can change with time

coscos BABdAAdBB

dtd

dtdA

dtdB cos

,,

211 TmWb

Induction – Faraday’s Law• Faraday’s Law – an emf is induced in a loop

when the number of magnetic field lines passing through the loop changes

• The negative only tells us direction and we will ignore it (unless I have a reason not to)

• The number of field lines doesn’t matter, just the rate of change of the number of field lines determines the induced emf and current

cosBAdtd

AdBdtd

dtd B

Induction cont• Notice there are three terms and therefore

three ways that flux can change: 1. The magnitude of the magnetic field can

change

2. The area of the coil can change

3. The angle between the direction of the magnetic field and the coil can change

dtdB

NAdtd

N B cos

dtdA

NBdtd

N B cos

dtd

NBAdtd

N B cos

Lenz’s Law• An induced current has a direction such that the

magnetic field due to the current opposes the change in the magnetic flux that induces the current

• The direction of the induced emf is the direction of the induced current

Solving Problems1. First know the direction of the external changing B

field

2. Next note how the external B field is changing

3. Use the two rules below to determine the direction the induced B field must have

4. Test with to determine the direction of the induced current to give the appropriate direction of the induced B field

5. It must follow the two rules below:a) If the external B field is increasing, the induced B field is

in the opposite direction of the external B field

b) If the external B field is decreasing, the induced B field is in the same direction of the external B field

rsd ˆ

Energy Transfer• During induction thermal

energy is produced by the work done to the system

RBLv

Ri

BLvdtdx

BLdtdA

Bdtd

RRvLB

FvP

RvLB

LBRBLv

iLBF

2222

22

Electric Field Induction

a) Ring with current induced

b) No ring but E field induced

c) E field lines of induced E field

d) Four closed paths of identical areas: 1 & 2 have same emf, 3 smaller emf, 4 no emf

dtd

sdE B

E Field Induction cont• The electric field lines for induced

electric fields always produce closed loops

• The electric field lines for static charges start at positive charges and end in negative charges

• The electric potential only has meaning for electric fields produced by static charges—it has no meaning for electric fields produced by induction

dtd

sdEsdE Bfinalinitial

0

Definitions• Inductor—a device used in a circuit to

produce a desired magnetic field, usually a solenoid

• Inductance— where N is the number

of turns, i is the current in windings, and is

the magnetic flux.

• Henry—the unit of inductance

iN

L

ATm

H2

11

More Definitions• Magnetic flux linkage— since N windings

are “linked” by shared flux

• Self-induction—a time varying current in a circuit produces in the circuit an induced emf opposing the emf, which initially set up the time varying current

• Mutual induction—the process by which a coil produces an induced emf in another coil and visa versa, the mutual interaction (induction) of 2 coils

N

Inductance• Distinction between source emf and

induced emf – The source emf and current is caused by a

physical source such as a battery.– The induced emf and current is caused by

a changing magnetic field

• self-induced emf or back emf

NLiiN

L

dtdi

LdtLid

dtNd

L

Inductance of SolenoidnlN

inABA o

AnlL

lAniinAnl

iN

L

o

oo

2

2

•Inductance depends only on the geometry of the device•L is a constant of proportionality between the

magnetic flux and the current in the circuit

Groups of Inductors• Series

• Parallel

• Note we must be careful, this is true only if there is a large distance between the inductors; otherwise we will have a mutual inductance problem

321 LLLLequivalent

321

1111LLLL eequivalenc

Energy Stored in a Magnetic Field

• This holds for all magnetic fields not just solenoids even though we used a solenoid to calculate it

dtdi

LiRii 2

2

21LiU

LididU

B

B

dtdU

timeenergy

dtdi

LiP B

o

ooBB

B

B

inAAin

Ai

lL

Al

Li

AlU

VU

u

2

222222

21

21

2212

1

Self and Mutual Induction• Magnetic flux through coil 2 associated with

a current in coil 1 which links the N2 turns of

coil 2 is

• Mutual inductance M21 of coil 2 wrt coil 1

• M12 = M21 = M

21

Self Induction Mutual Induction

NLiiN

L

dtdi

LL

2121211

21221 NiM

iN

M

dtdi

M 1212

RL Circuits• RL circuit – another circuit in which the

current varies with time

L

t

eR

i 1

L

t

eR

i

Charging Discharging

dtdi

LiR

RL

L

L

t

L e

Magnetic Materials• The simplest magnetic structure that can exist

is a magnetic dipole

• Gauss’s Law for magnetic fields

since there are no magnetic monopoles

• Electrons have two types of magnetic dipole moments:

1. Spin magnetic dipole moment associated with its spin

2. Orbital magnetic dipole moment associated with its orbiting about the nucleus

0AdBB

Magnetism of the Earth• The earth is a huge magnet and has

the magnetic field of a huge dipole• Remember field lines enter in

magnetic north pole; therefore the geomagnetic south pole is near the geographic north pole

• Field declination – the angle (left or right) between geographic north and the horizontal component of the earth’s field

• Field inclination – the angle (up or down) between a horizontal plane and the earth’s field direction

Diamagnetism

• Diamagnetism – (exhibited by all common materials) weak magnetic dipole moments are produced in the atoms of the material placed in an external B field. The combination of all these induced dipole moments give the material only a feeble net B field which disappears when the external B field is removed

Paramagnetism• Paramagnetism – (exhibited by materials

containing transition elements, rare earth elements, and actinide elements) each atom of material has a permanent magnetic dipole moment which is random and the net magnetic field is approximately 0. An external magnetic field may align material moments giving a net magnetic field which disappears when the external B field is removed

Ferromagnetism

• Ferromagnetism – (exhibited by iron, nickel, and certain other elements) some electrons in these materials align their resultant magnetic dipole moments to produce regions with strong magnetic dipole moments. External magnetic fields align these magnetic moments producing a strong B field which partially persists when the external B field is removed

Ferromagnetism

The arrows show the directions of magnetization in the given domains. Domains that are magnetized in the direction of the applied magnetic field grow stronger.

Hysteresis Loops

Induced Magnetic Fields• Faraday’s Law of Induction

• Maxwell’s Law of Induction

• Ampere – Maxwell Law

• Displacement current

dtd

sdE B

dtd

sdB E00

encE i

dtd

sdB 000

dE i

dtd

0

Maxwell’s Equations

Name Integral Form Differential Form

Gauss’s Law

Gauss’s Law

Faraday’s Law

Ampere-Maxwell’s Law

S o

QAdE

S

AdB 0

dtd

sdE B

dtd

isdB Eooo

o

E

0 B

tB

E

tE

JB ooo

Definitions

• Del—

• Gradient—(of a scalar function)

• The gradient of a function points in the direction of maximum increase of the function S. The magnitude of the gradient gives the slope or rate of increase along this maximal direction.

kz

jy

ix

ˆˆˆ

kzS

jyS

ixS

Skz

jy

ix

S ˆˆˆˆˆˆ

More Definitions• Divergence—(of a vector)

• The divergence is a measure of how much the vector spreads out or diverges from the point in question.

zV

y

V

xV

kVjViVkz

jy

ix

V

zyx

zyx

ˆˆˆˆˆˆ

More Definitions• Curl—(of a vector)

• The curl is a measure of how much the vector “curls around” the point in question

kyV

x

Vj

zV

xV

iz

V

yV

VVVzyx

kji

kVjViVkz

jy

ix

V

xyxzyz

zyx

zyx

ˆˆˆ

ˆˆˆ

ˆˆˆˆˆˆ

Conversion from Integral to Differential• Green’s Theorem

• Stoke's Theorem

• For Gauss’ Law

AdVdV

sdVdAV

o

oo

S o

E

VQ

E

QdEAdE

• Remember that we could find the electric field from the potential function

• There is also a magnetic vector potential A, such that

• The Lorentz force law

• Together with the Maxwell’s equations, this force law completely describes all classical electromagnetic interactions

sV

Es VE

AB

BvqEqF

LC Oscillations• LC Circuit – oscillating circuit

LC Oscillations cont

dtdq

Cq

dtdi

LiCqLi

dtd

dtdU

CqLi

UUU EBtotal

22

2222

22

tqq cosmax

maxmax

maxmax sinsin

qi

titqdtdq

i

LC1

C

qtt

Cq

tC

qt

Cq

UU EB

2cossin

2

cos2

sin2

2max22

2max

22max2

2max

RLC Circuit – Damped Oscillating Circuit

0

0

0

2

2

2

2

Cq

dtdq

Rdt

qdL

Cq

iRdtdi

L

RiiCq

dtdi

Li

dtdq

Cq

dtdi

LiRidtdU

teqq LRt

cos2max

2

222

4

12 L

RLCL

R

oscillates but witha decaying amplitude

RLC Circuit cont• For RLC circuits, if we put energy in at

the rate in which it is taken out by the resistor, we will stop the damping of the oscillations (forced oscillations)

• The driving angular frequency is the frequency of the external alternating emf producing the driving force

• Resonance occurs and the amplitude of the current in the circuit is a maximum if

d

CL

RL

RLCL

R 40

4

12 2

222 Criticallydamped

AC Circuits• In AC circuits the power source is

alternating and the voltage and current are sinusoidal

• Power is still lost through resistors even with an alternating power source

maxmax 707.

2i

iirms

RiRiRi

P rmsav2

2max

2max

22

0avei

Resistive Load

VR and iR are in phase

0 RV

titR

VRV

i

tVV

tV

RRR

R

RR

R

sinsin

sin

sin

maxmax

max

max

Capacitive Load

VC and iC are 90° out of phase and iC leads VC

0 CV

90sin

90sin

cos

sin

sin

sin

max

max

max

max

max

max

ti

tX

V

tCVdtdq

i

tCVCVq

tVV

tV

C

C

C

CC

C

CCC

CC

C

CXC

1

Inductive Load

VL and iL are 90 out of phase and VL leads iL

0 LV

LV

dtdi

dtdi

LtVV

tV

LLLLL

L

sin

sin

max

max

90sin90sin

cos

sin

maxmax

max

max

titX

V

tL

Vi

tL

Vdtdi

LL

L

LL

LL

LX L

Series RLC Circuit• Solve vectorally using

phasors

• Z is the impedance and is a resistance

• To get the phase we again use the phasors and get

LCR VVV

ZXXRi

CL

22max

RXX CL tan