MagneticField

7/30/2019 MagneticField

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Objectives:After completing thismodule, you should be able to:

Define the magnetic field, discussingmagnetic poles and flux lines.

Solve problems involving themagnitude and direction offorces oncharges moving in a magnetic field.

Solve problems involving the magnitudeand direction offorces on currentcarrying conductors in a B-field.


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Magnetism

Since ancient times, certain materials, calledmagnets, have been known to have the property ofattracting tiny pieces of metal. This attractive

property is called magnetism.

NS

Bar Magnet

N

S


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Magnetic Poles

The strength of a magnet isconcentrated at the ends,called north and south

poles of the magnet.

A suspended magnet:N-seeking end andS-seeking end are Nand Spoles.

NS

N

E

W

S

N

CompassBar magnet

S

N

Ironfilings


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Magnetic Attraction-Repulsion

N

S N N

S

S

NSNS

Magnetic Forces:Like Poles Repel Unlike Poles Attract


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Magnetic Field Lines

N S

We can describemagnetic field linesby imagining a tinycompass placed atnearby points.

The direction of themagnetic field B atany point is the sameas the directionindicated by this

compass.

Field B is strong wherelines are dense and weakwhere lines are sparse.


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Field Lines Between Magnets

N S

N N

Unlikepoles

Like poles

Leave Nand enter S

Attraction

Repulsion


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The Density of Field Lines

Magnetic Field B is sometimes called the flux

density in Webers per square meter (Wb/m2).

DN

NE

A

DD

Line density

DA

Electric field

DfB

A

D

D

Line density

DA

Magnetic field flux lines f

NS


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Magnetic Flux DensityDf

Magnetic Fluxdensity:

DAB A

Magnetic flux lines arecontinuous and closed.

Direction is that of the Bvector at any point.

Flux lines are NOT indirection of force but ^.

; =B BAA

When area A is

perpendicular to flux:

The unit of flux density is the Weber per square meter.


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Calculating Flux Density When

Area is Not PerpendicularThe flux penetrating theareaAwhen the normalvector n makes an angle

ofq with the B-field is:

cosBA q

The angle q is the complement of the angle a that theplane of the area makes with the B field. (Cos q = Sin a)

n

A q

a

B


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Origin of Magnetic Fields

Recall that the strength of an electric field E wasdefined as the electric force per unit charge.

Since no isolated magnetic polehas ever been

found, we cant define the magnetic field B interms of the magnetic force per unit north pole.

We will see instead that

magnetic fields result fromcharges in motionnot fromstationary charge or poles.This fact will be covered later.

+E

+ B vv


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Magnetic Force on Moving Charge

N S

B

N

Imagine a tube thatprojects charge +qwith velocity vintoperpendicular Bfield.

Upward magnetic force Fon charge moving in B field.

v

F

Experiment shows:

F qvB

Each of the following results in a greater magneticforce F: an increase in velocityv, an increase inchargeq, and a larger magnetic field B.


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Direction of Magnetic Force

B

v

F

N SN

The right hand rule:With a flat right hand,point thumb in directionof velocity v, fingers indirection ofB field. Theflat hand pushes in thedirection offorce F.

The force is greatest when the velocity visperpendicular to the B field. The deflectiondecreases to zero for parallel motion.

B

v

F


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Force and Angle of Path

SNN

SNN

SNN

Deflection force greatestwhen path perpendicular

to field. Least at parallel.

sinF v q

B

v

Fv sin q

vq


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Definition of B-fieldExperimental observations show the following:

sin or constantsin

FF qv

qvq

q

By choosing appropriate units for the constant of

proportionality, we can now define the B-field as:

or sinsin

FB F qvB

qvq

q

Magnetic FieldIntensity B:

Amagnetic field intensity of one tesla (T) exists in aregion of space where a charge ofone coulomb(C)moving at 1 m/s perpendicular to the B-field willexperience a force of one newton (N).


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Example 1.A2-nC charge is projected withvelocity 5 x 104 m/s at an angle of300 with a 3mT magnetic field as shown. What are the

magnitude and direction of the resulting force?

v sin f v30

0

B

v

FDraw a rough sketch.

q= 2 x 10-9 C

v= 5 x 104 m/sB= 3 x 10-3 Tq= 300

Using right-hand rule, the force is seen to be upward.

-9 4 -3 0sin (2 x 10 C)(5 x 10 m/s)(3 x 10 T)sin30F qvB q

Resultant Magnetic Force: F =1.50 x 10-7 N, upward

B


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Forces on Negative Charges

Forces on negative charges are opposite to those onpositive charges. The force on the negative chargerequires a left-hand rule to show downward force F.

N SN N SN

B

v

FRight-handrule for

positive q F

Bv

Left-handrule for

negative q


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Indicating Direction of B-fields

One way of indicating the directions of fields perpen-dicular to a plane is to use crosses X and dots :

X X X XX X X XX X X XX X X X

A field directed into the paperis denoted by a cross X like

the tail feathers of an arrow.

A field directed out of the paperis denoted by a dot like thefront tip end of an arrow.


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Practice With Directions:

X X X X

X X X XX X X XX X X X

X X X X

X X X XX X X XX X X X

What is the direction of the force F on the charge ineach of the examples described below?

-v

-

v

+

v

v+

Up

F

LeftF

FRight

UpF

negative q


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Crossed E and B Fields

The motion of charged particles, such as electrons, canbe controlled by combined electric and magnetic fields.

x x x xx x x x

+

-

e-

v

Note:FEon electron

is upward andopposite E-field.

But, FBon electron is

down (left-hand rule).

Zero deflectionwhen FB= FE

B

v

FE

E e--

B

vFB

-


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The Velocity SelectorThis device uses crossed fields to select only thosevelocities for which FB = FE. (Verify directions for +q)

x x x xx x x x

+

-

+q

v

Source

of +q

Velocity selector

When FB = FE :

qvB qE

Ev

B

By adjusting the E and/or B-fields, a person canselect only those ions with the desired velocity.


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Example 2. A lithium ion, q= +1.6 x 10-16 C,is projected through a velocity selector where

B = 20 mT. The E-field is adjusted to select avelocity of1.5 x 106 m/s. What is the electricfield E?

x x x xx x x x

+

-

+qv

Sourceof +q

V

EvB

E = vB

E =(1.5 x 106 m/s)(20 x 10-3 T); E= 3.00 x 104 V/m


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Circular Motion in B-fieldThe magnetic force F on a moving charge is alwaysperpendicular to its velocity v. Thus, a charge movingin a B-field will experience a centripetal force.

X X X X X X

X X X X X X

X X X X X XX X X X X X

X X X X X X+

+

+

+

Centripetal Fc = FB

R

Fc

2

; ;C Bmv

F F qvBR

2mv

qvB

R

C BF F

The radius ofpath is:

mvR

qB


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Mass Spectrometer

+q

R

Ev

B

+-

x x x x x x x x xx x x x x x x x

x x x x x x xx x x x x x

x x x x

x xx xx x

x x

Photographicplate

m1

m2

slit

Ions passed through avelocity selector atknown velocity emergeinto a magnetic field asshown. The radius is:

The mass is found by

measuring the radius R:

mvR

qB

qBRm

v

2mv

qvBR


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Example 3.A Neon ion, q = 1.6 x 10-19 C, followsa path of radius 7.28 cm. Upper and lower B =0.5 T and E = 1000 V/m. What is its mass?

mvR

qB

qBR

mv

1000 V/m

0.5 T

Ev

B

v =2000 m/s

-19(1.6 x 10 C)(0.5 T)(0.0728 m)

2000 m/sm m =2.91 x 10-24 kg

+q

R

Ev

B

+-x xx xx x

x x

Photographicplate

m

slitx x x x x x xx x x x x x xx x x x x x xx x x x x x

x x x x


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Summary

N SN

B

v

FRight-handrule for

positive q

N SN

F

Bv

Left-handrule for

negative q

The direction of forces on a charge moving in an electricfield can be determined by the right-hand rule for positivecharges and by the left-hand rule for negative charges.


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Summary (Continued)

B

v

F

v sin qv

q

For a charge moving in aB-field, the magnitude ofthe force is given by:

F = qvB sinq


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Summary (Continued)

mvR

qB

qBRm

v

x x xx x xx x

+

-

+qv

V

Ev

B

The velocityselector:

+q

R

Ev

B+-

x xx x

x xx x

m

slitx x x x x x xx x x x x x xx x x x x x x

x x x x x

The massspectrometer:


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CONCLUSION: Chapter 29

Magnetic Fields

MagneticField

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