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EMLAB

Magnetic field

2

EMLAB

qI

Generation of magnetic field

• A charged particle in motion generates magnetic field nearby.

• In the same way, currents generate mag-netic field nearby.

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EMLAB

Magnetic field due to currents or magnet

B due to magnetic moment of electron

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EMLAB

Forces on charges due to magnetic field (Lorentz force)

B

B

BvF q

B

Electron beams are deflected by Lorentz force

Horizontal and vertical deflection yoke control the path of electron beams.

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EMLAB

Force and Torque on a closed circuit

BvF q

CId BrF

FrT

Current loops in a magnetic field experi-ence torque, and are rotated until the plane of loops are perpendicular to the applied B.

F

FB

F

Angular acceleration is proportional to the applied torque.

Torque is proportional to the product of radius and force.

If the sum of torques due to A and B has nonzero value, the seesaw is rotated.

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EMLAB

1r

Relative permeability

Magnetic flux density

]H/m[104 70

Magnetic field and magnetic flux density

HHMHB rm 000 )1()(

• Arrows represent magnetic field due to orbiting elec-trons.

• The orbits of electrons are aligned due to external magnetic field.

extH

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EMLAB

(a) Hard disk tracks. (b) Sketch of qualitative shapes of hysteresis curves required for the head and track magnetic materials.

The magnetic head aerodynamically flies over the disk sur-face at a distance above it of only about 1mm while following the surface profile. In the figure, the surface profile is shown as ideally flat, which in practice is not the case.

Hard disk application

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EMLAB

Electromagnetic forces on a charge

1) Electric force

EF q

2) Magnetic force

BvF q

F

E

F

B

v

(Lorentz force)

)(total BvEF q

(Coulomb force)

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EMLAB

Prediction of magnetic field : Biot-Savart law

24

ˆ

R

Idd

Rs

H

rrR r

r

sId

Direction of H-field

Current segment

The magnetic field can be predicted by Biot-Savart’s law with known current distribution.

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EMLAB

Ampere’s law

path closed

IdsH

• Ampere law facilitates calculation of mangetic field like the Gauss law for electric field..

• Unlike Gauss’ law, Ampere’s law is related to line integrals.

• Ampere’s law is discovered experimentally and states that a line integral over a closed path is equal to a current flowing through the closed loop.

• In the left figure, line integrals of H along path a and b is equal to I because the paths enclose cur-rent I completely. But the integral along path c is not equal to I because it does not encloses com-pletely the current I.

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EMLAB

SCdId aJsH

From the integral form, we will derive the differential form of Ampere’s law.

JH

n

CndrH

C

drH

Line integrals from these adjacent currents add up to zero.

nC

Line integrals over a closed path is equal to the sum of line integrals over infinitesimally small loops.

Differential form of Ampere’s law

nn S nC

dd aJrH

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EMLAB

Example- Coaxial cable

I

I

a b cI

I

002

)4(

2ˆˆ

1

ˆˆ

ˆ2

)3(2

ˆˆ2

)2(2

ˆˆ

ˆ

2ˆ

)1(

22

22

22

22

0

2

0

2

0

0

2

0

2

2

2

0

2

0

2

0

H

H

zJzJ

zJ

H

H

zJaJ

Hs

rH

cr

bc

rc

r

IH

Ibc

br

dddd

ddrH

crbr

IHIrH

braa

rIH

Ia

rddd

rHrddH

ar

a r

b

outin

r

in

r

in

S

C

zJ ˆa

I2in

zJ ˆ)bc(

I22out

H

•The direction of magnetic fields can be found from right hand rule.

• The currents flowing through the inner conductor and outer sheath should have the same magnitude with different polar-ity to minimize the magnetic flux leakage

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EMLAB

Example : Surface current

nK

x

xH

s

ˆ20z

2

Kˆ

0z2

Kˆ

KL)L)(H(LHdH x

C

x

•The direction of magnetic field con be conjectured from the right hand rule.

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EMLAB

Example : Solenoid

)0H(

ˆK

KL)L(HLHdH

out

out

C

in

zH

s

•The direction of magnetic field con be conjectured from the right hand rule.

• If the length of the solenoid becomes in-finite, H field outside becomes 0.

d

I

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EMLAB

Example : Torus

2

NIKHNId aC

rH

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EMLAB

NS

]V[dt

dV

Electromotive force (emf)

• (-) sign explains the emf is induced across the terminals of the coil in such a way that hinders the change of the magnetic flux nearby.

1. A time-varying flux linking a stationary circuit.

2. A constant magnetic flux with a moving circuit

3. Combination of the above two cases

Situations when EMF is generated

Faraday’s law

1) Faraday experiment

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EMLAB

+V-

SC

ddt

d

dt

ddV aBrE

C

B

t

dt

dSS

B

E

aB

aE

(1) A time-varying flux linking a stationary circuit.

Time varying

E

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EMLAB

(2) A constant magnetic flux with a moving circuit

Bdvdt

dyBd

dt

dV

ByddS

emf

aB

(1) A phenomena observed by a stationary person

Direction of induced current

Due to the motion of a conducting bar, the charges in it moves in the (+y) direction. The moving charges experience Lorentz force such that

BvEEF

xzyBvF

mq

BqBqq ˆˆˆ

1. Effectively, the motion of bar gener-ates electric field which has the strength of (υ x B)

2. emf = Ed = υBd

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EMLAB

Example : Hard disk head

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EMLAB

(3) Combination of the two

rBvaB

rE ddt

demfS

)(

t

B

BvE )(

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EMLAB

Example : AC generator

tBabdS

cos aB

tBabdt

demf sin

A simple AC generator

dt

dd

dt

dd

tddemf

SSSC

aBaB

aErE )(

n̂ B

Observer’s coordinate frame is rotating with the loop.

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EMLAB

Example : Eddy current

B υ

BυE

B

υ

BυE

Relative velocity of the copper tube to the magnet.

Falling magnet inside a copper tube

Insulator tube Conductor tube Conductor tube

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EMLAB

Inductance

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EMLAB

Two important laws on magnetic field

Current generates magnetic field (Biot-Savart Law)

inducedV

Time-varying magnetic field generates induced electric field that opposes the variation. (Faraday’s law)

Current

Current

B-field Top view

Electric field

B-field

dt

dVind

i

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EMLAB

Current B i

l

NB

length

lengthl

Ni

l

iNSBSadB

lengthS

Magnetic flux :

Magnetic flux

B in a solenoid with N turn coil

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EMLAB

Concept of inductance

Current ,B

iN 00 ,Magnetic flux : i

NL

dt

diL

dt

dNL

Ф is the magnetic flux due to the coil wound N times.

Ф0 is magnetic flux due to the single turn coil.

Self inductance is proportional to the square of winding N.

The change of magnetic flux intensity due to changing current generates electro-motive force. The proportionality constant between the emf and current is called a inductance.

SNL ,2

S

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EMLAB

Mutual Inductance

dt

idM

dt

dN 1

2122

(1) When the secondary circuit is open

dt

idL

dt

dN 1

1111

1

11111 i

NL

1

21221 i

NM

1121 :: NN

The current flowing through the primary circuit generates magnetic flux, which influences the secondary circuit. Due to the magnetic flux, a repulsive voltage is induced on the secondary circuit.

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EMLAB

Work to move a current loop in a magnetic field

Idt

dt

dIdttIVW

tt )())((

rr

IBA

If we want to move a current loop with I flowing in a region with a magnetic flux density B, energy should be supplied from an external source.

The voltage induced in the current loop hinders the current flow, which should be canceled by an external source.

S

)(ti

)(tVR

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EMLAB

B

The energy is equal to assemble circuits with current Ii.

Magnetic energy : Mutual interaction

Ii

Ij

iIWi

N

n

n

iintotal

n

iin

ii

IW

IW

IW

IW

2

1

1

1

1n

2

133

122

1

1 1

11-N

322

211

)(

)(

)(

N

n

N

niintotal

NN

N

ii

N

ii

IW

IW

IW

IW

N

n

N

nii

intotal IW1 12

1

•Energy needed to as-semble I1, I2~IN in a

free space.

•Energy needed to dis-integrate I1, I2,~,In.

Magnetic material

S

daB

N

i

N

jjiij

N

i

N

jjitotal IILIW

1 11 1 2

1

2

1(Including self energy)

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EMLAB

Magnetic energy

S

)(ti

RtitS )()(

)(tLi

20

2

0

2

00

2

1)(

2

1)]([

2

1

)()()()(

00

00

LItiLdtdt

tidL

dtdt

tdiLtidttVtiW

tt

tt

R

(Initially, this circuit has a zero current flowing. Then , the current increases to I.)

(To support current i(t), the current source should provide additional voltage which cancels induced voltage by Faraday’s law.)

dt

tdiL

dt

dtVt R

)()()(

)(tVR

Self energy : The energy needed for the circuit to have a current I flow in spite of the re-pelling electromotive force from Faraday’s law.

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EMLAB

)()(2)()(2

1

)()(2)()(2

1

)()()(

)()()(

)()()()(

121112221

211

0

21222

211

0

212

2121

1

0

2211

1

1

1

tItIMtILtIL

dttitiMtiLtiLdt

d

dttidt

tdiM

dt

tdiLti

dt

tdiM

dt

tdiL

dttittitW

t

t

t

Magnetic energy : two coil system