Lesson 7Lesson 7

Magnetic FieldsMagnetic FieldsMagnetic Force on moving Magnetic Force on moving chargechargeMotion of charge in Magnetic Motion of charge in Magnetic FieldFieldMagnetic Force on current Magnetic Force on current carrying conductorcarrying conductorTorque on current carrying Torque on current carrying looploopHall EffectHall EffectMagnetic DevicesMagnetic Devices

Some metallic objects cause other

metallic objects to accelerateThus must be producing a forceThis force is different to Gravitational

Force and Electric ForceCan plot the directions of lines force

#Lines of motion of effected objectsMAGNETIC FORCEMAGNETIC FORCE

Magnetic Force

Magnetic Field Lines

Properties of Magnetic FieldsProperties#Moving charge (current) is effected

Force proportional ³Q and v

FB = 0 if v parallel to Magnetic

Field BFB perpendicular to v and BFB on positive charge opposite to

that on negative charge FB proportional to Sin of the

angle v makes with B

Putting these properties together one sees

FB Qv B

compare this to

FE QE

Mathematical Expression

Vector Product

A Ax i Ay j Az k

B Bx i By j Bz k

A B i j k

Ax Ay Az

Bx By Bz

Ay Az

By Bzi

Ax Az

Bx Bzj

Ax Ay

Bx Byk

Ay Bz Az By i Ax Bz Az Bx j Ax By Ay Bx k

Vector Product I

A B BA

A A 0

A B 0 A parallel to B

A B (AB)Sin A B AB if A is perpendicular to B

A B is perpendicular to A and B

Vector Product II

i j k

i k j

jk i

Unit vector products

Work done by Magnetic Field�

Work Done by Magnetic Force Work Done by Magnetic Force on chargeon charge

dWB FB ds Qv B ds 0

As v B is perpendicular to v and B

and v is parallel to ds

Compare to Electric ForceCompare to Electric Force

dWE FE ds QE ds 0

unless E is perpendicular to path of motion

which is never the case if E causes the motion.

Work done by Electric Field

ImplicationsThus Kinetic Energy of charge is not changed by Thus Kinetic Energy of charge is not changed by FFBB

Potential Energy is not changedPotential Energy is not changed

K 1

2m v 2 0

v 0

i.e. speed is constant under influence of B

but v can change direction

FB Q v B

B FB Q v

N

Cm

s

N

AmT (Tesla)

SI units

Magnetic FluxMagnetic Flux B

For constant field over flat area perpendicular

with area vector A to magnetic field B

B B A

in general

B B d

surface A

B Tm2 Wb (Weber )

Right Hand Rule

FB Qv B Q vy j Bz k

Qv y Bz j k Qv y Bz

i j k

0 1 0

0 0 1

Qv y Bz i

kk

iijj

vv

FFBB

QQ

B

Magnetic Force on Moving Charge

FB t Qv t BMathematical expression

Uniform Circular Motion of Charge

Uniform Circular Motion

i

j k

Mathematical analysis

ac v2

r F

mQvBm

mv2

rQvB

r mvQB

radius of motion

v = r

=BQm

angular speed =angular freqency ( rads )

f

2 BQ

2 m frequency (cps = Hz )

T 1f2 mBQ

period of the motion .

Magnetic BottlesIf initial v has a component in k direction

v t 1

mFBt v 0

vx t i vy t j vz 0 k

magnetic bottles

Vector Integral I

a

b

rb

ra

I

ds

B

vector integral

Vector Integral II

a

brb

ra

I

ds

Brb- ra

dsSum

Vector Integral III

Magnetic Force on Current

Force on current carrying conductor in

a uniform magnetic field :

FB I d s B

a

b

Il B

where l ra rb

dQ v dQ ds

dtˆ s dQ

dtds

s Ids s Id s

Magnetic Force on Current Loop I

I

Force on Current Loop

B

k

ji

Magnetic Force on Current Loop II

FB IlB0

l r a ra 0

Total Force on Loop

Torque on Current Loop I

I

y

z

B

Torque on Current Loop

k

ji

Torque

r F

Torque on Current Loop II

Torque on Current Loop IIIForce on segment 1

F1 IzBk j IzBi

Force on segment 3

F3 IzBk j IzBi

Forces on segment 2 and 4 are 0 as

magnetic field and wire are parallel

F Iyjj 0

Torque about central axis

= 1 2 IzB y k IzB k IzBy k IABk

A Area of loop

Magnetic Moment

Torque on Current Loop in yz plane in

a uniform magnetic field in positive y direction

= IABk

In general the torque on a current loop in magnetic field is

= M B

M IA, where A is the area vector of the loop,

with the orientation given by the current flow.

Oriented Area Vector

I

k

ji

A=Ai: Area vector with orientation

A

Hall EffectHall EffectHall Effect II

BB

vvdd

FFEE FFBB

EE

d

+ -

AA

t

V

Mathematical analysisAt Equlibrium

F E F B

E V

dF E eE F B vdeB

vd E

B

Remember vd J

ne I

neA

IneA

EB

n BIEAe

BIV

ddte

BIVte

can measure vd by measuring

the PD V.

Measurement

Lorentz Force

F FE FB QE Qv B

Lorentz Force