Chapter 22

Magnetic Forces

and

Magnetic Fields

A Brief History of Magnetism 13th century BC

Chinese used a compass Uses a magnetic needle Probably an invention of Arab or Indian origin

800 BC Greeks

Discovered magnetite attracts pieces of iron

A Brief History of Magnetism, 2 1269

Pierre de Maricourt found that the direction of a needle near a spherical natural magnet formed lines that encircled the sphere

The lines also passed through two points diametrically opposed to each other

He called the points poles

A Brief History of Magnetism, 3 1600

William Gilbert Expanded experiments with magnetism to a variety of

materials Suggested the earth itself was a large permanent

magnet

1750 John Michell

Magnetic poles exert attractive or repulsive forces on each other

These forces vary as the inverse square of the separation

A Brief History of Magnetism, 4

1819 Hans Christian Oersted

Pictured, 1777 – 1851 Discovered the relationship between

electricity and magnetism An electric current in a wire

deflected a nearby compass needle André-Marie Ampère

Deduced quantitative laws of magnetic forces between current-carrying conductors

Suggested electric current loops of molecular size are responsible for all magnetic phenomena

A Brief History of Magnetism, final 1820’s

Faraday and Henry Further connections between electricity and

magnetism A changing magnetic field creates an electric

field Maxwell

A changing electric field produces a magnetic field

Electric and Magnetic Fields An electric field surrounds any

stationary electric charge The region of space surrounding a

moving charge includes a magnetic field In addition to the electric field

A magnetic field also surrounds any material with permanent magnetism

Both fields are vector fields

Magnetic Poles Every magnet, regardless of its shape,

has two poles Called north and south poles Poles exert forces on one another

Similar to the way electric charges exert forces on each other

Like poles repel each other N-N or S-S

Unlike poles attract each other N-S

Magnetic Poles, cont The poles received their names due to the

way a magnet behaves in the Earth’s magnetic field

If a bar magnet is suspended so that it can move freely, it will rotate The magnetic north pole points toward the earth’s

north geographic pole This means the earth’s north geographic pole is a

magnetic south pole Similarly, the earth’s south geographic pole is a magnetic

north pole

Magnetic Poles, final The force between two poles varies as

the inverse square of the distance between them

A single magnetic pole has never been isolated In other words, magnetic poles are always

found in pairs There is some theoretical basis for the

existence of monopoles – single poles

Magnetic Fields A vector quantity Symbolized by Direction is given by the direction a

north pole of a compass needle points in that location

Magnetic field lines can be used to show how the field lines, as traced out by a compass, would look

B

Magnetic Field Lines, Bar Magnet Example The compass can

be used to trace the field lines

The lines outside the magnet point from the North pole to the South pole

Magnetic Field Lines, Bar Magnet Iron filings are used

to show the pattern of the magnetic field lines

The direction of the field is the direction a north pole would point

Magnetic Field Lines, Unlike Poles

Iron filings are used to show the pattern of the magnetic field lines

The direction of the field is the direction a north pole would point

Compare to the electric field produced by an electric dipole

Magnetic Field Lines, Like Poles Iron filings are used to

show the pattern of the magnetic field lines

The direction of the field is the direction a north pole would point

Compare to the electric field produced by like charges

Definition of Magnetic Field The magnetic field at some point in

space can be defined in terms of the magnetic force,

The magnetic force will be exerted on a charged particle moving with a velocity,

BF

v

Characteristics of the Magnetic Force The magnitude of the force exerted on

the particle is proportional to the charge, q, and to the speed, v, of the particle

When a charged particle moves parallel to the magnetic field vector, the magnetic force acting on the particle is zero

Characteristics of the Magnetic Force, cont When the particle’s velocity vector

makes any angle 0 with the field, the magnetic force acts in a direction perpendicular to both the speed and the field The magnetic force is perpendicular to the

plane formed by andv B

Characteristics of the Magnetic Force, final The force exerted on a negative charge

is directed opposite to the force on a positive charge moving in the same direction

If the velocity vector makes an angle with the magnetic field, the magnitude of the force is proportional to sin

More About Direction

The force is perpendicular to both the field and the velocity

Oppositely directed forces exerted on oppositely charged particles will cause the particles to move in opposite directions

Force on a Charge Moving in a Magnetic Field, Formula The characteristics can be summarized

in a vector equation

is the magnetic force q is the charge is the velocity of the moving charge is the magnetic field

B q F v B

BF

v

B

Units of Magnetic Field The SI unit of magnetic field is the

Tesla (T)

The cgs unit is a Gauss (G) 1 T = 104 G

N sT

C m

Directions – Right Hand Rule #1

Depends on the right-hand rule for cross products

The fingers point in the direction of the velocity

The palm faces the field Curl your fingers in the direction of

field The thumb points in the

direction of the cross product, which is the direction of force

For a positive charge, opposite the direction for a negative charge

Direction – Right Hand Rule #2 Alternative to Rule #1 Thumb is the direction of

the velocity Fingers are in the

direction of the field Palm is in the direction of

force On a positive particle Force on a negative charge

is opposite You can think of this as

your hand pushing the particle

More About Magnitude of the Force The magnitude of the magnetic force on a

charged particle is FB = |q| v B sin is the angle between the velocity and the field The force is zero when the velocity and the field

are parallel or antiparallel = 0 or 180o

The force is a maximum when the velocity and the field are perpendicular

= 90o

Differences Between Electric and Magnetic Fields Direction of force

The electric force acts parallel or antiparallel to the electric field

The magnetic force acts perpendicular to the magnetic field

Motion The electric force acts on a charged particle

regardless of its velocity The magnetic force acts on a charged particle only

when the particle is in motion and the force is proportional to the velocity

More Differences Between Electric and Magnetic Fields Work

The electric force does work in displacing a charged particle

The magnetic force associated with a steady magnetic field does no work when a particle is displaced

This is because the force is perpendicular to the displacement

Work in Fields, cont The kinetic energy of a charged particle

moving through a constant magnetic field cannot be altered by the magnetic field alone

When a charged particle moves with a velocity through a magnetic field, the field can alter the direction of the velocity, but not the speed or the kinetic energy

v

Notation Note The dots indicate the

direction is out of the page

The dots represent the tips of the arrows coming toward you

The crosses indicate the direction is into the page

The crosses represent the feathered tails of the arrows

Charged Particle in a Magnetic Field Consider a particle

moving in an external magnetic field with its velocity perpendicular to the field

The force is always directed toward the center of the circular path

The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle

Force on a Charged Particle Using Newton’s Second Law, you can equate

the magnetic and centripetal forces:

Solving for r:

r is proportional to the linear momentum of the particle and inversely proportional to the magnetic field and the charge

2mvF ma qvB

r

mvr

qB

More About Motion of Charged Particle The angular speed of the particle is

The angular speed, , is also referred to as the cyclotron frequency

The period of the motion is

v qB

r m

2 2 2r mT

v qB

Motion of a Particle, General If a charged particle

moves in a magnetic field at some arbitrary angle with respect to the field, its path is a helix

Same equations apply, with

2 2y zv v v

Bending of an Electron Beam Electrons are

accelerated from rest through a potential difference

Conservation of Energy will give v

Other parameters can be found

Charged Particle Moving in Electric and Magnetic Fields In many applications, the charged particle will

move in the presence of both magnetic and electric fields

In that case, the total force is the sum of the forces due to the individual fields

In general: This force is called the Lorenz force It is the vector sum of the electric force and the

magnetic force

q q F E v B

Velocity Selector Used when all the

particles need to move with the same velocity

A uniform electric field is perpendicular to a uniform magnetic field

Velocity Selector, cont When the force due

to the electric field is equal but opposite to the force due to the magnetic field, the particle moves in a straight line

This occurs for velocities of value v = E / B

Velocity Selector, final Only those particles with the given

speed will pass through the two fields undeflected

The magnetic force exerted on particles moving at speed greater than this is stronger than the electric field and the particles will be deflected upward

Those moving more slowly will be deflected downward

Mass Spectrometer A mass

spectrometer separates ions according to their mass-to-charge ratio

A beam of ions passes through a velocity selector and enters a second magnetic field

Mass Spectrometer, cont After entering the second magnetic field, the

ions move in a semicircle of radius r before striking a detector at P

If the ions are positively charged, they deflect upward

If the ions are negatively charged, they deflect downward

This version is known as the Bainbridge Mass Spectrometer

Mass Spectrometer, final Analyzing the forces on the particles in

the mass spectrometer gives

Typically, ions with the same charge are used and the mass is measured

orB Bm

q E

Thomson’s e/m Experiment Electrons are

accelerated from the cathode

They are deflected by electric and magnetic fields

The beam of electrons strikes a fluorescent screen

Thomson’s e/m Experiment, cont

Thomson’s variation found e/me by measuring the deflection of the beam and the fields

This experiment was crucial in the discovery of the electron

Cyclotron A cyclotron is a device that can

accelerate charged particles to very high speeds

The energetic particles produced are used to bombard atomic nuclei and thereby produce reactions

These reactions can be analyzed by researchers

Cyclotron, 2 D1 and D2 are called

dees because of their shape

A high frequency alternating potential is applied to the dees

A uniform magnetic field is perpendicular to them

Cyclotron, 3 A positive ion is released near the

center and moves in a semicircular path The potential difference is adjusted so

that the polarity of the dees is reversed in the same time interval as the particle travels around one dee

This ensures the kinetic energy of the particle increases each trip

Cyclotron, final The cyclotron’s operation is based on the fact

that T is independent of the speed of the particles and of the radius of their path

When the energy of the ions in a cyclotron exceeds about 20 MeV, relativistic effects come into play

2 2 221

2 2

q B RK mv

m

First Cyclotron Invented by E. O.

Lawrence and M. S. Livingston

Invented in 1934

Force on a Current-Carrying Conductor A current carrying conductor

experiences a force when placed in an external magnetic field

The current represents a collection of many charged particles in motion

The resultant magnetic force on the wire is due to the sum of the magnetic forces on the charged particles

Force on a Wire In this case, there is

no current, so there is no force

Therefore, the wire remains vertical

Force on a Wire,cont The magnetic field

is into the page The current is

upward, along the page

The force is to the left

Force on a Wire, final The field is into the

page The current is

downward along the page

The force is to the right

Force on a Wire, equation The magnetic force is

exerted on each moving charge in the wire

The total force is the

product of the force on one charge and the number of charges

B dq F v B

B dq nA F v B

Force on a Wire, cont In terms of the current, this becomes

l is a vector that points in the direction of the current

Its magnitude is the length of the segment This applies only to a straight segment of wire

in a uniform external magnetic field

B I F B

Force on a Wire, Arbitrary Shape Consider a small

segment of the wire,

The force exerted on

this segment is

The total force is

b

B I d aF s B

ds

Bd I d F s B

Torque on a Current Loop The rectangular loop

carries a current I in a uniform magnetic field

No magnetic force acts on sides & The wires are

parallel to the field and cross product is zero

Torque on a Current Loop, 2 There is a force on sides &

These sides are perpendicular to the field The magnitude of the magnetic force on

these sides will be: F2 = F4 = I a B

The direction of F2 is out of the page

The direction of F4 is into the page

Torque on a Current Loop, 3 The forces are equal

and in opposite directions, but not along the same line of action

The forces produce a torque around point O

Torque on a Current Loop, Equation The maximum torque is found by:

The area enclosed by the loop is ab, so max = I A B This maximum value occurs only when the

field is parallel to the plane of the loop

max 2 4 ( ) ( )2 2 2 2

b b b bF F IaB IaB

IabB

Torque on a Current Loop, General Assume the

magnetic field makes an angle of <90o with a line perpendicular to the plane of the loop

The net torque about point O will be = I A B sin

Torque on a Current Loop, Summary The torque has a maximum value when the

field is perpendicular to the normal to the plane of the loop

The torque is zero when the field is parallel to the normal to the plane of the loop

where A is perpendicular to the plane of the loop and has a magnitude equal to the area of the loop

I A B

Direction of A The right-hand rule

can be used to determine the direction of

Curl your fingers in the direction of the current in the loop

Your thumb points in the direction of

A

A

Magnetic Dipole Moment The product I is defined as the

magnetic dipole moment, of the loop Often called the magnetic moment

SI units: A m2

Torque in terms of magnetic moment:

A

B

Biot-Savart Law – Introduction Biot and Savart conducted experiments

on the force exerted by an electric current on a nearby magnet

They arrived at a mathematical expression that gives an expression for the magnetic field at some point in space due to a current

Biot-Savart Law – Set-Up The magnetic field is

at some point P The length element

is The wire is carrying

a steady current of I

dB

ds

Biot-Savart Law – Observations The vector is perpendicular to both

ds and to the unit vector directed from

toward P The magnitude of is inversely

proportional to r2, where r is the distance from to P

r̂dB

dBds

ds

Biot-Savart Law – Observations, cont The magnitude of is proportional to

the current and to the magnitude ds of the length element ds

The magnitude of is proportional to sin where is the angle between the vectors and r̂

dB

dB

ds

The observations are summarized in the mathematical equation called Biot-Savart Law:

The Biot-Savart law gives the magnetic field only for a small length of the conductor

Biot-Savart Law, Equation

2

ˆm

I dd k

r

s rB

Permeability of Free Space

The constant o is called the permeability of free space

o = 4 x 10-7 T. m / A The Biot-Savart Law can be written as

7104

om

T mk

A

2

ˆ

4o I d

dr

s rB

Total Magnetic Field To find the total field, you need to sum

up the contributions from all the current elements You need to evaluate the field by

integrating over the entire current distribution

The magnitude of the field will be

2oIBr

B Compared to E Distance

The magnitude of the magnetic field varies as the inverse square of the distance from the source

The electric field due to a point charge also varies as the inverse square of the distance from the charge

B Compared to E, 2 Direction

The electric field created by a point charge is radial in direction

The magnetic field created by a current element is perpendicular to both the length element and the unit vector r̂ds

Source An electric field is established by an

isolated electric charge The current element that produces a

magnetic field must be part of an extended current distribution

Therefore you must integrate over the entire current distribution

B Compared to E, 3

B for a Long, Straight Conductor, Direction The magnetic field lines

are circles concentric with the wire

The field lines lie in planes perpendicular to to wire

The magnitude of the field is constant on any circle of radius a

The right hand rule for determining the direction of the field is shown

B for a Circular Current Loop The loop has a

radius of R and carries a steady current of I

Find at point P

2

32 2 22

ox

IRB

x R

B

Field at the Center of a Loop Consider the field at the center of the

current loop At this special point, x = 0 Then,

2

32 2 2 22

o ox

IR IB

Rx R

Magnetic Field Lines for a Loop

Figure a shows the magnetic field lines surrounding a current loop

Figure b shows the field lines in the iron filings Figure c compares the field lines to that of a bar

magnet

Magnetic Force Between Two Parallel Conductors Two parallel wires

each carry a steady current

The field due to the current in wire 2 exerts a force on wire 1 of F1 = I1l B2

2B

Magnetic Force Between Two Parallel Conductors, cont

Substituting the equation for B2 gives

Parallel conductors carrying currents in the same direction attract each other

Parallel conductors carrying current in opposite directions repel each other

1 21 2

oI IF

a

Magnetic Force Between Two Parallel Conductors, final The result is often expressed as the

magnetic force between the two wires, FB

This can also be given as the force per unit length, FB/l

a2IIF 21oB

Definition of the Ampere The force between two parallel wires

can be used to define the ampere When the magnitude of the force per

unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A

Definition of the Coulomb The SI unit of charge, the coulomb, is

defined in terms of the ampere When a conductor carries a steady

current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 s is 1 C

Magnetic Field of a Wire A compass can be used

to detect the magnetic field

When there is no current in the wire, there is no field due to the current

The compass needles all point toward the earth’s north pole

Due to the earth’s magnetic field

Magnetic Field of a Wire, 2 The wire carries a

strong current The compass needles

deflect in a direction tangent to the circle

This shows the direction of the magnetic field produced by the wire

Magnetic Field of a Wire, 3 The circular

magnetic field around the wire is shown by the iron filings

André-Marie Ampère 1775 –1836 Credited with the

discovery of electromagnetism The relationship

between electric currents and magnetic fields

Died of pneumonia

Ampere’s Law The product of can be evaluated for

small length elements on the circular path defined by the compass needles for the long straight wire

Ampere’s Law states that the line integral of

around any closed path equals oI where I is the total steady current passing through any surface bounded by the closed path

od I B s

dB s

ds

dB s

Ampere’s Law, cont Ampere’s Law describes the creation of

magnetic fields by all continuous current configurations Most useful for this course if the current

configuration has a high degree of symmetry Put the thumb of your right hand in the

direction of the current through the amperian loop and your figures curl in the direction you should integrate around the loop

Amperian Loops Each portion of the path satisfies one or

more of the following conditions: The value of the magnetic field can be

argued by symmetry to be constant over the portion of the path

The dot product can be expressed as a simple algebraic product B ds

The vectors are parallel

Amperian Loops, cont Conditions:

The dot product is zero The vectors are perpendicular

The magnetic field can be argued to be zero at all points on the portion of the path

Field Due to a Long Straight Wire – From Ampere’s Law Want to calculate

the magnetic field at a distance r from the center of a wire carrying a steady current I

The current is uniformly distributed through the cross section of the wire

Field Due to a Long Straight Wire – Results From Ampere’s Law

Outside of the wire, r > R

Inside the wire, we need I’, the current inside the amperian circle

(2 )

2

o

o

d B r I

IB

r

B s

2

2

2

(2 ) ' '

2

o

o

rd B r I I I

RI

B rR

B s

Field Due to a Long Straight Wire – Results Summary

The field is proportional to r inside the wire

The field varies as 1/r outside the wire

Both equations are equal at r = R

Magnetic Field of a Toroid Find the field at a

point at distance r from the center of the toroid

The toroid has N turns of wire

(2 )

2

o

o

d B r NI

NIB

r

B s

Magnetic Field of a Solenoid A solenoid is a long wire wound in the form

of a helix A reasonably uniform magnetic field can be

produced in the space surrounded by the turns of the wire

Each of the turns can be modeled as a circular loop The net magnetic field is the vector sum of all the

fields due to all the turns

Magnetic Field of a Solenoid, Description The field lines in the interior are

Approximately parallel to each other Uniformly distributed Close together

This indicates the field is strong and almost uniform

Magnetic Field of a Tightly Wound Solenoid The field distribution

is similar to that of a bar magnet

As the length of the solenoid increases The interior field

becomes more uniform

The exterior field becomes weaker

Ideal Solenoid – Characteristics An ideal solenoid is

approached when The turns are closely

spaced The length is much

greater than the radius of the turns

For an ideal solenoid, the field outside of solenoid is negligible

The field inside is uniform

Ampere’s Law Applied to a Solenoid Ampere’s Law can be used to find the

interior magnetic field of the solenoid Consider a rectangle with side l parallel

to the interior field and side w perpendicular to the field

The side of length l inside the solenoid contributes to the field This is path 1 in the diagram

Ampere’s Law Applied to a Solenoid, cont Applying Ampere’s Law gives

The total current through the rectangular path equals the current through each turn multiplied by the number of turns

1 1path path

d d B ds B B s B s

od B NI B s

Magnetic Field of a Solenoid, final Solving Ampere’s Law for the magnetic

field is

n = N / l is the number of turns per unit length

This is valid only at points near the center of a very long solenoid

o o

NB I nI

Magnetic Moment – Bohr Atom

The electrons move in circular orbits

The orbiting electron constitutes a tiny current loop

The magnetic moment of the electron is associated with this orbital motion

The angular momentum and magnetic moment are in opposite directions due to the electron’s negative charge

Magnetic Moments of Multiple Electrons In most substances, the magnetic

moment of one electron is canceled by that of another electron orbiting in the opposite direction

The net result is that the magnetic effect produced by the orbital motion of the electrons is either zero or very small

Electron Spin Electrons (and other particles) have an

intrinsic property called spin that also contributes to its magnetic moment The electron is not physically spinning It has an intrinsic angular momentum as if

it were spinning Spin angular momentum is actually a

relativistic effect

Electron Magnetic Moment, final In atoms with multiple

electrons, many electrons are paired up with their spins in opposite directions

The spin magnetic moments cancel

Those with an “odd” electron will have a net moment

Some moments are given in the table

Ferromagnetic Materials Some examples of ferromagnetic materials

are Iron Cobalt Nickel Gadolinium Dysprosium

They contain permanent atomic magnetic moments that tend to align parallel to each other even in a weak external magnetic field

Domains All ferromagnetic materials are made up

of microscopic regions called domains The domain is an area within which all

magnetic moments are aligned The boundaries between various

domains having different orientations are called domain walls

Domains, Unmagnetized Material The magnetic

moments in the domains are randomly aligned

The net magnetic moment is zero

Domains, External Field Applied A sample is placed

in an external magnetic field

The size of the domains with magnetic moments aligned with the field grows

The sample is magnetized

Domains, External Field Applied, cont The material is

placed in a stronger field

The domains not aligned with the field become very small

When the external field is removed, the material may retain most of its magnetism

Magnetic Levitation The Electromagnetic System (EMS) is

one design model for magnetic levitation

The magnets supporting the vehicle are located below the track because the attractive force between these magnets and those in the track lift the vehicle

EMS, cont

The proximity detector uses magnetic induction to measure the magnet-rail separation

The power supply is adjusted to maintain proper separation

EMS, final Disadvantages

Instability caused by the variation of magnetic force with distance

Compensated for by the proximity detector Relatively small separation between the magnets

and the tracks Usually about 10 mm Track needs high maintenance

Advantage Independent of speed, so wheels are not needed

Wheels are in place for “emergency landing” system

German Transrapid – EMS Example