Chapter 29 Chapter 29 -- Magnetic FieldsMagnetic FieldsA PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

© 2007

Objectives: Objectives: After completing this After completing this module, you should be able to:module, you should be able to:

•• Define the Define the magnetic field,magnetic field, discussing discussing magnetic polesmagnetic poles and flux lines.and flux lines.

•• Solve problems involving the Solve problems involving the magnitude and direction of magnitude and direction of forces on forces on chargescharges moving in a magnetic field.moving in a magnetic field.

•• Solve problems involving the magnitude Solve problems involving the magnitude and direction of and direction of forces on currentforces on current carrying conductorscarrying conductors in a Bin a B--field.field.

MagnetismMagnetismSince ancient times, certain materials, called Since ancient times, certain materials, called magnetsmagnets, have been known to have the property of , have been known to have the property of attracting tiny pieces of metal. This attractive attracting tiny pieces of metal. This attractive property is called property is called magnetismmagnetism..

NS

Bar Magnet

N

S

Magnetic PolesMagnetic PolesThe The strengthstrength of a magnet is of a magnet is concentrated at the ends, concentrated at the ends, called north and south called north and south ““polespoles”” of the magnet.of the magnet.

A suspended magnet: A suspended magnet: NN--seeking end and seeking end and SS--seeking end are seeking end are NN and and SS polespoles..

NNSS

N

E

W

SNN

CompassCompassBar magnetBar magnet

S

N

Iron filings

Magnetic AttractionMagnetic Attraction--RepulsionRepulsion

NS

NN

S

S

NSNS

Magnetic Forces: Magnetic Forces: Like Poles RepelLike Poles Repel Unlike Poles AttractUnlike Poles Attract

Magnetic Field LinesMagnetic Field Lines

N S

We can describe We can describe magnetic field linesmagnetic field lines by imagining a tiny by imagining a tiny compass placed at compass placed at nearby points.nearby points.

The The directiondirection of the of the magnetic field magnetic field BB at at any point is the same any point is the same as the direction as the direction indicated by this indicated by this compass. compass.

Field Field BB is is strong strong where where lines are lines are densedense and weak and weak where lines are sparse.where lines are sparse.

Field Lines Between MagnetsField Lines Between Magnets

N S

N N

Unlike poles

Like poles

Leave N and enter S

Attraction

Repulsion

The Density of Field LinesThe Density of Field Lines

Magnetic Field B is sometimes called the flux density in Webers per square meter (Wb/m2). Magnetic Field B is sometimes called the flux density in Webers per square meter (Wb/m2).

N

NEA

Line density

A

Electric field

B

A

Line density

A

Magnetic field flux lines

NS

Magnetic Flux DensityMagnetic Flux Density

Magnetic Flux density:

ABA

•• Magnetic flux lines are Magnetic flux lines are

continuous and closed.continuous and closed.

•• Direction is that of the B Direction is that of the B vector at any point.vector at any point.

•• Flux lines are Flux lines are NOTNOT in in direction of force but direction of force but ..

; = B BAA

When area A is perpendicular to flux:

When area A is perpendicular to flux:

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

Calculating Flux Density When Calculating Flux Density When Area is Not PerpendicularArea is Not Perpendicular

The flux penetrating the The flux penetrating the area area AA when the normal when the normal vector vector nn makes an angle makes an angle of of

with the with the BB--fieldfield is:is:

cosBA

The angle The angle is the complement of the angle a that the is the complement of the angle a that the plane of the area makes with the B field.plane of the area makes with the B field. (Cos (Cos

= Sin = Sin

nA

B

Origin of Magnetic FieldsOrigin of Magnetic FieldsRecall that the strength of an Recall that the strength of an electric field Eelectric field E was was defined as the electric force per unit charge.defined as the electric force per unit charge.

Since Since no isolated magnetic poleno isolated magnetic pole has ever been has ever been foundfound, we can, we can’’t define the magnetic field t define the magnetic field B B in in terms of the terms of the magnetic force per unit north polemagnetic force per unit north pole..

We will see instead that magnetic fields result from charges in motion—not from stationary charge or poles. This fact will be covered later.

We will see instead that magnetic fields result from charges in motion—not from stationary charge or poles. This fact will be covered later.

+ E

+ B vv

Magnetic Force on Moving ChargeMagnetic Force on Moving Charge

N S

B

N

Imagine a tube that Imagine a tube that projects charge projects charge +q+q with velocity with velocity vv into into perpendicular perpendicular BB field.field.

Upward magnetic force F on charge moving in B field.

vv

FF

Experiment shows:Experiment shows:

F qvB

Each of the following results in a greater magnetic Each of the following results in a greater magnetic force Fforce F: an increase in : an increase in velocityvelocity vv, an increase in , an increase in chargecharge qq, and a larger , and a larger magnetic field Bmagnetic field B..

Direction of Magnetic ForceDirection of Magnetic Force

Bvv

FF

N SN

The right hand ruleThe right hand rule::With a flat With a flat rightright hand, hand, point point thumbthumb in direction in direction of velocity of velocity vv, , fingersfingers in in direction of direction of BB field. The field. The flat flat handhand pushes in the pushes in the direction of direction of force Fforce F..

The force is greatest when the velocity v is perpendicular to the B field. The deflection decreases to zero for parallel motion.

The force is greatest when the velocity v is perpendicular to the B field. The deflection decreases to zero for parallel motion.

Bvv

FF

Force and Angle of PathForce and Angle of Path

SNN

SNN

SNN

Deflection force greatest Deflection force greatest when path perpendicular when path perpendicular to field. Least at parallel.to field. Least at parallel.

sinF v

B

vv

FF

v sin v sin vv

Definition of BDefinition of B--fieldfieldExperimental observations show the following:Experimental observations show the following:

sin or constantsinFF qv

qv

By choosing appropriate units for the constant of By choosing appropriate units for the constant of proportionality, we can now define the proportionality, we can now define the BB--fieldfield as:as:

or sinsinFB F qvB

qv

Magnetic Field

Intensity B:

A magnetic field intensity of one tesla (T) exists in a region of space where a charge of one coulomb (C) moving at 1 m/s perpendicular to the B-field will experience a force of one newton (N).

A A magnetic field intensitymagnetic field intensity of one of one teslatesla (T)(T) exists in a exists in a region of space where a charge of region of space where a charge of one coulombone coulomb (C)(C) moving at moving at 1 m/s1 m/s perpendicular to the Bperpendicular to the B--field will field will experience a force of one experience a force of one newton (N).newton (N).

Example 1.Example 1. A A 22--nCnC charge is projected with charge is projected with velocity velocity 5 x 105 x 1044 m/sm/s at an angle of at an angle of 303000 with a with a 3 3 mTmT magnetic field as shown. What are the magnetic field as shown. What are the magnitude and direction of the resulting force? magnitude and direction of the resulting force?

v sin v sin vv

B

vv

FFDraw a rough sketch.Draw a rough sketch.qq = 2 x 10= 2 x 10--99 C C vv = 5 x 10= 5 x 1044 m/s m/s B B = 3 x 10= 3 x 10--33 T T = 30= 3000

Using rightUsing right--hand rule, the force is seen to behand rule, the force is seen to be upwardupward..

-9 4 -3 0sin (2 x 10 C)(5 x 10 m/s)(3 x 10 T)sin 30F qvB

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

B

Forces on Negative ChargesForces on Negative ChargesForces on negative charges are opposite to those on positive charges. The force on the negative charge requires a left-hand rule to show downward force F.

Forces onForces on negativenegative charges are opposite to those on charges are opposite to those on positive charges. The force on the negative charge positive charges. The force on the negative charge requires a requires a leftleft--hand rulehand rule to show to show downwarddownward force force FF..

N SN N SN

Bvv

FFRight-hand rule for

positive q FF

Bvv

Left-hand rule for

negative q

Indicating Direction of BIndicating Direction of B--fieldsfieldsOne way of indicating the directions of fields One way of indicating the directions of fields perpenperpen-- diculardicular to a plane is to use crosses to a plane is to use crosses X X and dots and dots

:

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

A field directed into the paper is denoted by a cross “X” like the tail feathers of an arrow.

A field directed out of the paper is denoted by a dot “ ” like the front tip end of an arrow.

Practice With Directions:Practice With Directions:

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

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

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

-vv

-vv

+

vvvv

+Up

FF

LeftFF

FFRight

UpFF

negative qnegative q

Crossed E and B FieldsCrossed E and B FieldsThe motion of charged particles, such as electrons, can be controlled by combined electric and magnetic fields. The motion of charged particles, such as electrons, can The motion of charged particles, such as electrons, can be controlled by combined electric and magnetic fields.be controlled by combined electric and magnetic fields.

x x x x x x x x

+

-

e-

v

Note:Note: FFEE on electron on electron is is upwardupward and and opposite Eopposite E--field.field.

But, But, FFBB on electron is on electron is downdown (left(left--hand rule).hand rule).

Zero deflection Zero deflection when when FFBB = F= FEE

Bvv

FFEE

E e--

B

vvFFBB

-

The Velocity SelectorThe Velocity SelectorThis device uses crossed fields to select only those velocities for which FB = FE . (Verify directions for +q) This device uses crossed fields to select only those This device uses crossed fields to select only those velocities for which Fvelocities for which FBB = F= FEE . (Verify directions for +q). (Verify directions for +q)

x x x x x x x x

+

-

+qv

Source of +q

Velocity selector

When FWhen FBB = F= FE E ::

qvB qE

EvB

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

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

Example 2.Example 2. A lithium ion, A lithium ion, qq = +1.6 x 10= +1.6 x 10--1616 CC, , is projected through a velocity selector where is projected through a velocity selector where B = 20 B = 20 mTmT. The E. The E--field is adjusted to select a field is adjusted to select a velocity of velocity of 1.5 x 101.5 x 1066 m/sm/s. What is the electric . What is the electric field E?field E?

x x x x x x x x

+

-

+qv

Source of +q

VV

EvB

E = E = vBvB

E = E = (1.5 x 10(1.5 x 1066 m/s)(20 x 10m/s)(20 x 10--33 T);T); E = 3.00 x 104 V/mE = 3.00 x 104 V/m

Circular Motion in BCircular Motion in B--fieldfieldThe magnetic force F on a moving charge is always perpendicular to its velocity v. Thus, a charge moving in a B-field will experience a centripetal force.

The magnetic force F on a moving charge is always perpendicular to its velocity v. Thus, a charge moving in 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 X X X X X X X

X X X X X X X X X X X X

X X X X X X X X X X X X

X X X X X XX X X X X X+

+

+

+

Centripetal Centripetal FFcc = F= FBB

RR

FFcc

2

; ;C BmvF F qvBR

2mv qvB

RC BF F

The radius of path is:

The radius of path is:

mvRqB

Mass SpectrometerMass Spectrometer

+q

R

EvB

+-

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x

x x x x

x x x x x x x x

Photographic plate

m1

m2

slit

Ions passed through a Ions passed through a velocity selector at velocity selector at known velocity emerge known velocity emerge into a magnetic field as into a magnetic field as shown. The radius is:shown. The radius is:

The mass is found by The mass is found by measuring the radius R:measuring the radius R:

mvRqB

qBRmv

2mv qvB

R

Example 3.Example 3. A Neon ion, A Neon ion, q = 1.6 x 10q = 1.6 x 10--19 19 CC, follows , follows a path of radius a path of radius 7.28 cm7.28 cm. Upper and lower . Upper and lower B = B = 0.5 T0.5 T and and E = 1000 V/mE = 1000 V/m. What is its mass?. What is its mass?

mvRqB

qBRmv

1000 V/m0.5 T

EvB

v = v = 2000 m/s2000 m/s

-19(1.6 x 10 C)(0.5 T)(0.0728 m)2000 m/s

m m = 2.91 x 10-24 kgm = 2.91 x 10-24 kg

+q

R

EvB

+-

x x x x x x x x

Photographic plate

m

slitx x x x x x x x x x x x x x x x x x x x x x x x x x x

x x x x

Summary Summary

N SN

Bvv

FFRight-hand rule for

positive q

N SN

FF

Bvv

Left-hand rule for

negative q

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

The direction of forces on a charge moving in an electric The direction of forces on a charge moving in an electric field can be determined by the rightfield can be determined by the right--hand rule for positive hand rule for positive charges and by the leftcharges and by the left--hand rule for negative charges.hand rule for negative charges.

Summary (Continued)Summary (Continued)

B

vv

FF

v sin v sin vv

For a charge moving in a For a charge moving in a BB--field, the magnitude of field, the magnitude of the force is given by:the force is given by:

F = qvB sin

Summary (Continued)Summary (Continued)

mvRqB

qBRmv

x x x x x x x x

+

-

+ qv

VV

EvB

The velocity The velocity selector:selector:

+q

R

EvB

+- x x x x x x x x

m

slitx x x x x x x x x x x x x x x x x x x x x

x x x x x

The mass The mass spectrometer:spectrometer:

CONCLUSION: Chapter 29CONCLUSION: Chapter 29 Magnetic FieldsMagnetic Fields