Electromagnetism

Zhu Jiongming

Department of Physics

Shanghai Teachers University

Electromagnetism

Chapter 1 Electric Field

Chapter 2 Conductors

Chapter 3 Dielectrics

Chapter 4 Direct-Current Circuits

Chapter 5 Magnetic Field

Chapter 6 Electromagnetic Induction

Chapter 7 Magnetic Materials

Chapter 8 Alternating Current

Chapter 9 Electromagnetic Waves

Chapter 5 Magnetic Field

§1. Introduction to Basic Magnetic Phenomena

§2. The Law of Biot and Savart

§3. Magnetic Flux

§4. Ampere’s Law

§5. Charged Particles Moving in a Magnetic Field

§6. Magnetic Force on a Current-Carrying Conductor

§7. Magnetic Field of a of a Current Loop

§1. Basic Magnetic Phenomena

Comparing with Electric Fields ：E ： charge electric field charge

（ produce ） （ force ）M ：

Permanent Magnets Magnetic Effect of Electric Currents Molecular Current

Movingcharge

Movingcharge

magnetic field

Permanent Magnets

Two kinds of Magnets ： natural 、 manmade Two Magnetic Poles ： south S 、 north N Force on each other ： repel （ N-N, S-S ） attract （ N-S ） Magnetic Monopole ？

Magnetic Effect of Electric Currents

Experiments Show Straight Line Current

I

S

N

I

N

S

Molecular Current—— Ampere’s Assumption

Two Parallel Lines Circular Current Solenoid and Magnetic Bar

Magnetic Field B

Experiment ： Helmholts coils in a

hydrogen bulb ， an electron gun

I I

M M’ Conclusion ： moving charge

F = q v B （ Definition of B ） ( Electric Field ： F = qE ) Unit ： Tesla )Gauss10(

m/s1C

1NTesla 1 4

Magnetic Field Lines ：（ curve with a direction ） Tangent at any point on a line is in the direction of t

he magnetic field at that point

● Number of field lines through unit area perpendicular to B equals the magnitude of B

§2. The Law of Biot and Savart

1. The Law of Biot and Savart

2. Magnetic Field of a Long Straight Line Current

3. Magnetic Field of a Circular Current Loop

4. Magnetic Field on the Axis of a Solenoid

5. Examples

1. The Law of Biot and Savart

The field of a current element Idl

dB Idl ， 1/r2 ， sin r ： Idl P ： angle between Idl and r Proportionality constant ： 0/4 = 10-7

20 sind

4d

r

lIB

20 ˆd

4d

r

I rlB

2

0 ˆd

4d

r

I rlBB

dB

Idl

r

P

rEE ˆd

4

1d

20

r

q

Direction ： dBIdl ， dB r

Integral ：

Compare with ：

2. Field of a Long Straight Line Current

current I ， distance a

all dB in same direction

20 ˆd

4 r

I rlB

2

0 sind

4 r

lIB

2

1

dsin1

40

a

I

)cos(cos4 21

0

a

I

r

2

1

a PO

I

dllsin = a/r

ctg = - l/a

r = a/sin l = - a ctg

dl= ad/sin2Infinite long ： 1= 0 ， 2= ，

a

IB

2

0 Direction ： right hand rule

3. Magnetic Field of a Circular Current

current I ， radius R ，P on axis , distance a

20 ˆd

4d

r

I rlB

zoR

r

a

dB

P

dl

I

cos

sind

4 20

r

lIB

= 90o

cos = R/r

r2 = R2 + a2

lr

RId

4 30

2/322

20

)(2 aR

IR

Rl 2d

R

IBa

20,Oat)1( 0

：3

20

2,far)2(

a

RIBRa

：

component dB||= dBcossymmetry ， dB cancel ， B = 0

4. Field on the Axis of a Solenoid

current I, radius R, Length L,

n turns per unit length

dB at P on axis caused by nIdl

（ as circular current ）

I

PR l

dl

L

2/322

20

)(

d

2d

lR

lnIRB

d)sin(

20 nI

1

2

d)sin(20

nIB

)cos(cos2 210

nI

ctg = l/R

l = R ctg

dl= - Rd /sin2R2+ l2 = R2/sin2

Field on the Axis of a Solenoid

Direction ：right hand rule

)cos(cos2 210

nI

B

PR

L

2

1

B

I

B

O L

(1) center （ or R << L ） 1= 0 ， 2= ， B = 0nI

(2) ends （ Ex. ： left ） 1= 0 ， 2= /2 ， B = 0nI /2

(3) outside, cos1 、 cos2 same sign, minus, B small

inside, opposite sign, plus ， B large

Example （ p.345/5 - 3 -11）Uniform ring with current ， find B at the center.Sol. ：

I

I

O

B

C

12

I1

I2

20 ˆd

4d

r

I rlB

0ˆd rlI

R

IB

20

2

2

210

1

R

IB

22

202 R

IB B1= B2

opposite direction

B = 0

2

2

1

R

R

21 )2( II

Straight lines ： （ circular current ： ）

arc 1 ：

arc 2 ：

parallel ： I1R1 = I2R2

Exercises

p.212 / 5-2- 3, 8, 12, 13, 16

§3. Magnetic Flux

1. Magnetic Flux

2. Magnetic Flux on Closed Surface

3. Magnetic Flux through Closed Path

Flux on area element dS

dB = B · dS = B dS cos Flux on surface S ( integral ) if B and dS in same direction ( = 0 ), write dS

= Magnitude of B

Unit ： 1 Web = 1 T · m2

define number of B lines through dS = B · dS = dB

then line density =

1. Magnetic Flux

dS

B

SB B

d

dB ： Flux per unit area perpendicular to B

BS

B

d

d

Show ： (1) dB of current element Idl

B lines are concentric circles

these circles and the surface S either not intercross （ no contribution to flux ） or intercross 2 times （ in/out ， flux +/- ）

2. Magnetic Flux on Closed Surface

)surfaceclosedany(0d SS

SB

dB

11111 dddddin SBSB：

22222 dddddout SBSB：21 dd BB

21 dd SS

21 dd 0dd 21 0d

S

Idl

Magnetic Flux on Closed Surface

Show ： (2) magnetic field of any currents

superposition ： B = B1 + B2 + …

0dddd 21 SSSSSB

B lines are continual ， closed ， or

—— called The field without sources

Compare with ： E lines from +q or ， into -q or

—— called The field with sources

Turn the normal vector of S1

opposite, same as that of S2

then

3. Magnetic Flux through Closed Path

Any surfaces bounded by the closed path L have the same fluxShow ：

L

S1 S2

n

n

0ddd21

SSSSBSBSB

21

ddSS

SBSB

21

ddSS

SBSB

—— called Magnetic Flux through Closed Path L

Exercises

p.214 / 5-3- 1, 3

§4. Ampere’s Law

1. Ampere’s Law

2. Magnetic Field of a Uniform Long Cylinder

3. Magnetic Field of a Long Solenoid

4. Magnetic Field of a Toroidal Solenoid

1. Ampere’s Law

Ampere’s Law : L ： any closed loop I ： net current enclosed by L

Three steps to show the law ： L encloses a Long Straight Current I L encloses no Currents L encloses Several Currents

IL 0d lB

I

L

L Encloses a Long Straight Current I

Field of a long line current I ： I

La

IB

2

0

sBlB dcosdd lB

d

20 aa

I

d2

0I

II

LL 00 d

2d

lBI

ds dld

LB

（ direction: tangent ）

L Encloses no Currents

Current I is outside L

21

dddLLL

lBlBlBI L2L1

)dd(2 21

0 LL

I

0)(2

0

I

L Encloses Several Currents

L encloses several currents

Principle of superposition ： B = B1 + B2 + …

LL

lBBlB d)(d 21

I is algebraic sum of the currents enclosed by L direction of Ii with direction of L （ integral ）：

right hand rule ， take positive sign

)( 210 II I0

I0

2. Field of a Uniform Long Cylinder

radius R ， current I （ outgoing ），find B at P a distance r from the axis

concentric circle L with radius r ，symmetry ： same magnitude of B on L ，

direction： tangent

P

L

rBL

2d lB

)(2

0 Rrr

IB

rBL

2d lB

)(2 2

0 RrR

IrB

0 r

B

R

outside ：

inside ： 2

2

0 R

rI

radius R ， current I （ outgoing ），field B at P a distance r from the axis

symmetry ： B in direction of tangent

Direction is along the Tangent

P

B

3. Magnetic Field of a Long Solenoid

Field inside is along axis

Show ： turn 180 o round zz’ ： B B’

I opposite ： B’ B’’

B’’ should coincide with B

a

d

d

c

c

b

b

aLlBlBlBlBlB ddddd

nIllBlB cdab 000 nIBab 0

0 cdB

nIllBlBab 0outin 00axison not if ：nIB 0in

B’B

B’’

z’

z

ab

dc

direction ：right hand rule

4. Field of a Toroidal Solenoid

Symmetry ： B on the circle L

magnitude ： same

direction ： tangent

（ L >> r ， N turns ）in ：

out ：

NILBL 0ind lB

nIL

NIB 0

0in

0d out LBL

lB

0out B

direction ： right hand rule

if L ， becomes a long solenoid

Surface current （ width l ， thickness d ）

5. Field of a Uniform Large Plane

dB

l

Jdl

Jld

l

I

llBlB zzL012d lB z

20

12

zz BB

012 2

nn EE比较电场：

Direction ： parallel opposite on two sides

( right hand rule )

Exercises

p.215 / 5-4- 2, 3, 4, 5

§5. Charged Particles Moving in B

1. Motion of Charged Particles in a Magnetic Field

2. Magnetic Converging

3. Cyclotrons

4. Thomson’s e/m Experiment （ skip ）5. The Hall Effect

1. Motion of Charged ParticlesLorents Force ： F = q ( E + v B )if E = 0 ， F = q v B

if v B ， q moves in a circle

with constant speed

Centripetal force ：qvB

R

mvF

2

C

OR

v

F

m, q=- e

Radius ： R = mv/qB Period ： T = 2R/v = 2 m/qB Frequency ： f = 1/ T = qB/ 2 m Ratio of charge to mass ： q/ m= v/BR = /B

2. Magnetic Converging

v making an angle with B ： v || = v cos

v = v sin

Helical pathradius ： sin

qB

mv

qB

mvR

cos

2|| qB

mvTvh

）（ small ~sin vv

)small (~cos vv

R

h

P P’

pitch ：

Magnetic Converging ： different R ， same h from P to P’ distance h

Take period of emf same as that of qaccelerated 2 times per revolution

v r ( ), T not changed

3. Cyclotrons

Principle ： uniform field ， outward

2 Dees ， alternating emf

q accelerated as crossing the gap

qB

mT

2 （ not depend on v, r ）

qB

mvr

Application ： accelerating proton 、 etc. to slam into a solid target to learn it’s structure

Ex. ： deuteron q/ m ~ 10 7 ， B ~ 2 ， R ~ 0.5

need U ~ 10 7 （ volts ） frequency of emf f = qB/ 2m ~ B magnetic field relativity ： v m f varying frequency —— Synchrotrons

Cyclotrons Compare with straight line accelerator

Str. ：Cyc. ：

qUmv 2

2

1

m

RBq

m

qBRmmv

2)(

2

1

2

1 22222

m

RqBU

2

22

22

0

/1 cv

mm

To gain the same v ， need

Exp. carrier q ， force fL = q v B

q > 0 , v - positive x ， fL - positive z

q < 0 , v - opposite x ， fL - positive z direction

A’

5. The Hall Effect A conducting strip of width l, thickness d x - current ， y - magnetic field z - voltage UAA’

x

yz

I

BA

l

d

fL

fe

for q > 0 ， positive charges pile up on side A ， negative on A’ produce an electric field Et

fe = qEt opposite to fL slow down qEt = qvB

stop piling q moves along x （ as without B ） the Hall potential difference ： UAA’ = Et l = vBl

The Hall Constant

I = q n ( vld )

v = I/qnld

UAA’ = IB/qnd

write ： UAA’ = K IB/d

proportional to IB/d （ macroscopic ）Hall constant ： K = 1/qn （ microscopic ）

determined by q 、 n q > 0 , K > 0 UAA’ > 0

q < 0 , K < 0 UAA’ < 0

（ A - negative charges ， A’ - positive ）

Exercises

p.216 / 5-5- 1, 3, 4, 5, 6

§6. Magnetic Force on a Conductor

1. Ampere Force

2. Rectangular Current Loop in a Uniform

Magnetic Field

3. The Principle of a Galvanometer

1. Ampere Force

current carriers magnetic force on conductor

electron ： f = - ev B

current ： j = - env

force on current element Idl ： dF = N (- ev B )

= n dS dl (- ev B )

= dS dl ( j B )

= Idl B

Ampere force ：

L

I0

d BlF

I

B

dldS

dl and j in same direction

j and dS in same direction

I = j dS = j dS

2. Rectangular Loop in Magnetic Field

Normal vector n and current I

------ right-hand ruleu① ：

d③ ：l ② ：r ④ ：

II

B

n

①

②③

④

l2

l1

)90sin(d o

01

1 l

lIBF

cos1IBl （ up ）cos13 IBlF （ down ）

2

02 dl

lIBF 2IBl （ ⊙ ）

24 IBlF （ ）F1 ， F3 cancel out

B

n

l1

F2

F4

F2 ， F4 produce a net torque ： T = F2l1sin = IBl2l1sin = ISBsin

The Magnetic Dipole Moment

Torque on a current carrying rectangular loop ：T = ISBsin （ direction ： n B ）

Definition ：Magnetic Dipole Moment of a

current carrying rectangular loop

pm = IS n

then the torque

T = pm B

B

pm

I

T

（ Comparison ： in an electric field

p = ql ， T = p E ）

Magnetic Moment of Any Loop

Divided into many small rectangular loops

outline ~ the loop ， inner lines cancel out

dT = dpm B = IdS n B

all dT in the same direction

T = dT = IdSnB = InBdS

= IS n B = pm B

Definition ： Magnetic Dipole Moment of Any Loop pm = IS n

no matter what shape

（ same form as that of a rectangular loop ）

n

BI

S

Magnetic Dipole Moment of Any Loop

pm making an angle with B

maximum T for = /2 T = 0 for = 0

equilibrium, stable

lowest energy T = 0 for =

equilibrium, unstable

highest energy

n

BI

3. The Principle of a Galvanometer

n turns ： T = nISB

countertorque by springs

T’ = kwhen in balance

= nISB/ k I

（ = 0 for I = 0 ）

NS

Exercises

p.217 / 5-6- 1, 5, 8

§7. Field of a Current Loop

Circular loop of radius R ， current I ， on axis

o

R

a PI2/322

20

)(2 aR

IRB

3

20

2for

a

RIBRa

：

ISpRS m2

3m0

2 a

pB

)2

1 withcompare(

30 a

pE

：

pm = IS n is important

• torque exerted by magnetic field• produce magnetic field

B

Exercises

p.219 / 5-6- 11