CHAPTER FOUR: Magnetic Field and Its SourcesCHAPTER FOUR: Magnetic Field and Its Sources4.1. Magnetic Field

4.2. Sources of Magnetic Fields

CHAPTER FOUR: Magnetic Field and Its SourcesCHAPTER FOUR: Magnetic Field and Its Sources

1

•Calculate the force exerted by magnetic fields•Calculate the force exerted by magnetic fields4 1 Calculate the force exerted by magnetic fields•Calculate kinematical quantities in problems

involving motion in magnetic fields

Calculate the force exerted by magnetic fields•Calculate kinematical quantities in problems

involving motion in magnetic fields4.1

•Calculate the magnetic field due to different •Calculate the magnetic field due to different current and moving charge configurations

•Define Ampere’s Law current and moving charge configurations

•Define Ampere’s Law 4.2•Define Gauss’ Law for Magnetism•Define Gauss’ Law for Magnetism

2

Outline:Outline:1. The Magnetic

Phenomenon2. The Force of Magnetic

Fields3. The Gauss and the

Tesla4 Magnetic Field Lines4. Magnetic Field Lines5. Motion in Magnetic

FieldsFields6. Applications of the

Velocity Selector

3

Ancient Greeks (around 2000 years ago) were aware that y g )magnetite (Fe3O4) attracts pieces of iron.

h fThere are written references to the use of magnets for navigation during the 12th

centurycentury.

In 1269, Pierre de Maricourtdiscovered using simplediscovered using simple observations, the existence of magnetic “poles”.

In 1600, William Gilbert discovered that the earth itself is a natural magnet.

4

Magnetic PolesMagnetic Poles◦ Are the basic elements of

the magnetic phenomenon

◦ It comes with two varieties that exist as pairs: the “ ( )” d h“NORTH (N)” and the “SOUTH (S)” poles

◦ No matter what the shape nor the size, all magnets have two poles which can’thave two poles which can t be isolated like electric charges! Magnetic Fields emerges

from the N pole and enters

5

the S pole

Magnetic Interaction:Magnetic Interaction:

The Fundamental Law of Magnetostatics:The Fundamental Law of Magnetostatics: “Like Poles Repel, Unlike Poles Attract”

6

The magnetic effect and ginteraction can be better studied by analyzing the force exerted by aforce exerted by a magnetic field on a test specimenp

The Magnetic Field only i i h hinteracts with the following tests:

1 Moving Charges1. Moving Charges2. Current Carrying Wires

7

The north-pole end of a bar magnet is heldThe north-pole end of a bar magnet is held near a positively charged piece of plastic. Is the plasticIs the plastic (a) attracted, (b) ll d(b) repelled, or (c) unaffected by the magnet?

Ans: (c)

8

9

When a test specimen isWhen a test specimen is in a magnetic field, it experiences a magnetic p gforce.

The Magnetic Force on a moving test charge

Th M i FThe Magnetic Force on a current carrying wire

10

Since the force is a vector, we can determine it’s magnit de and directionit’s magnitude and direction.

For the magnitude:For the magnitude:

|F | B i θ |F | ILB i θIf the given are in •|FB| = qvB sinθ or |FB| = ILB sinθgNEWSUD config

If the given are in Unit Vector configUnit Vector config

If the given are in the

11•|FB| = qvB sinθ or |FB| = ILB sinθIf the given are in the

simple 2-D config

There is ZERO FORCEThere is ZERO FORCE if:

v(or L) and B are parallel (θ 0) or areparallel (θ=0) or are anti parallel (θ=180)

Proof: It’s simple: take the sine of thetake the sine of the angles!

12

For the direction:For the direction:

• Use the Right Hand Rule for 3If the given are in • Use the Right Hand Rule for 3 Dimensions

If the given are in NEWSUD config

If the given are in Unit Vector Notation

• Use the Right Hand Rule for 2 If the given are in

13

gDimensions

gsimple 2-D config

Steps:Steps: 1. Point your 4 fingers to thefingers to the direction of v (or L).2 Curl your 4 fingers2. Curl your 4 fingers to the direction of B.3 Release your3. Release your thumb and this will be the direction ofbe the direction of the force!

14

15

First assume that theFirst, assume that the charge is positive and apply the Right Hand R l b i hRule to obtain the direction of the force,

Then, the direction of a force acting on a gnegative charge is opposite to that of the direction attained fordirection attained for the positive charge

16

Some Conventions:Some Conventions:

y-axis z-axis or “page” directions

x-axisX

x axisin or –z axis out or +z axis

17

F X vv

Fv FB X+

B

FB=0+ B

FB

+ BB +

+v

v B +v

B X

BFB + FB X

FBFB

18

The Gauss and the Tesla serves as unit forThe Gauss and the Tesla serves as unit for Magnetic Field!

The Gauss-Tesla Relation:

The Tesla can be represented by:

19

A proton is moving with aA proton is moving with a velocity of 10Mm/s.

It i tiIt experiences a magnetic field of 0.6G which is directed downward and

h d knorthward, making an angle of 70o with the horizontal. o o ta

Find the magnetic force on the protonon the proton.

20

A wire segment 3mm longA wire segment 3mm long carries a current of 3A in the x-direction. It lies in a magnetic field of 0.02T that is in the xy-yplane and makes an angle of 30o with the x-axis, as h h fshown in the figure.

What is the magnetic f t d th iforce exerted on the wire segment.

21

Magnetic Field Lines are very similar to Electric FieldMagnetic Field Lines are very similar to Electric Field lines in the following aspects◦ The direction of the field is the direction of the field lines

h d f h f ld h d f h l◦ The magnitude of the field is the density of the lines◦ Field Lines never cross

Magnetic Field Lines are very different to Electric Field lines in the following aspects◦ Electric field lines are in the direction of the electric force while

magnetic field lines are in a perpendicular direction of the magnetic forceg

◦ Electric field lines have beginnings and ends, while magnetic field lines form closed loops 22

For the motion of charges in magnetic fieldFor the motion of charges in magnetic field, we consider two situations:

1. The Charges are moving in a pure Magnetic FieldField

2. The Charges are moving in a E-B crossed fieldsfields

23

v is alwaysv is always perpendicular to FB.

From this we haveFrom this we have general ideas:

FB l h th1. FB only changes the direction of the velocity but not the

it dmagnitude

2. FB does no work on the charge

3 FB therefore does not SPECIAL CASE: When v is 3. FB therefore does not change the kinetic energy of the charge! 24

perpendicular to B, the charge undergoes UCM

In this special case, the p ,magnetic force provides the centripetal force necessary for the centripetal acceleration in i l ticircular motion.

We use Newton’s Second Law l h i ito relate the quantities.

T and f are known as the l i d d

Cyclotron Radius

cyclotron period and frequency respectively.

h l d dThe cyclotron period and frequency depend on the charge-to-mass ratio q/m but are independent of r and v of

Cyclotron Period Cyclotron Frequency

are independent of r and v of the particle!

25

There are two interesting motion paths andThere are two interesting motion paths and behaviors involved with moving charges in magnetic fieldsmagnetic fields

THE HELIX:THE HELIX:Happens when charges enter B with a non-

THE BOTTLE:perpendicular velocity

Happens when the magnetic field is not uniform (being strong

d k h

26

at ends, weak at the center

The magnetic forceThe magnetic force can be balanced by an electric field by choosing a correctchoosing a correct configuration, such as the figure to the

hright

Once there is aOnce there is a balance of the force, we have a region of

d fi ld !

“Lorentz Force”

crossed -fields!

To achieve the “Velocity Selector”To achieve the balance, the velocity must be chosen! 27

With the velocity selectorWith the velocity selector, the charged particle will traverse the crossed fields

d fl d!+

undeflected!

If it enters the field with a v greater than the selector Deflect to FBv greater than the selector

If it enters the field with a v lesser than the selector Deflect to FE

28

The velocity selector for crossed-fields haveThe velocity selector for crossed fields have very important applications that were discovered during the late 19th and early 20thg ycentury.

In this course, two applications will be discussed:

1 Th M S1. The Mass Spectrometer2. The Cyclotron

29

The mass spectrometer, first designed by Francis William g yAston in 1919, was developed as a means of measuring the masses of isotopes.

Such measurements are an important way of determining b th th f i tboth the presence of isotopes and their abundance in nature.

F l lFor example, natural magnesium has been found to have mass ratios of 24:25:26.

The mass ratios are computed from the radius of curvature!

Each element, by the way has a unique q/m ratio! 30

A 58Ni ion of charge +e and mass 9 62 x 10-26A 58Ni ion of charge +e and mass 9.62 x 10 26

kg is accelerated through a potential difference of 3kV and deflected in a magneticdifference of 3kV and deflected in a magnetic field of 0.12T.

(a) Find the radius of curvature of the orbit of(a) Find the radius of curvature of the orbit of the ion.

(b) Find the difference in the radii of curvature(b) Find the difference in the radii of curvature of 58Ni ions and 60Ni ions. (Assume that the mass ratio is 58/60 )mass ratio is 58/60.)

31

The cyclotron wasThe cyclotron was invented by E.O. Lawrence and M SLawrence and M.S. Livingston in 1934 to accelerate particlesaccelerate particles such as protons or deuterons to highdeuterons to high kinetic energies*.

32

A cyclotron for accelerating protons has aA cyclotron for accelerating protons has a magnetic field of 1.5T and a maximum radius of 0 5mof 0.5m

(a) What is the cyclotron frequency?(b) What is the kinetic energy of the protons(b) What is the kinetic energy of the protons

when they emerge?

33

1. Due to Moving ChCharges2. Due to Currents: Biot Savart LawBiot Savart Law◦ 2.1 Current Loops◦ 2.2 Current of Solenoids◦ 2 3 Current of Straight◦ 2.3 Current of Straight

Wires◦ 2.4 Current of Toroids3 Gauss’ Law for3. Gauss Law for Magnetism4. Ampere’s Law 5. Magnetism in Matter

34

Permanent Magnets were the earliest known sources of magnetism.magnetism.

Oersted announced his discovery that a compass needle is deflected pby an electric current.

Jean Baptiste Biot and Felix Savartd th lt f th iannounced the results of their

measurements of the force on a magnet near a long current-carrying wire and analyzed results y g yin terms of the magnetic field.

Andre-Marie Ampere extended th i t d h dthese experiments and showed that current elements also experience a force in the presence of a magnetic field and that two gcurrents exert forces on each other.

35

There are twoThere are two possible sources of magnetic fieldsmagnetic fields

1 Moving Charges1. Moving Charges2. Currents in Wires

Our Quest is to find the ti fi ld imagnetic field in a

certain field point P!

36

You will almost always encounter theYou will almost always encounter the permeability of free space during your computations in this chapter so might ascomputations in this chapter, so might as well introduce it here:

In your calculators, press π first before multiplying it to 4 x 10-7!

37

When a point charge q If the given are in UV Notation

moves with a velocity v, it produces a magnetic f ld h f ldfield B at the field point P given by: If the given are in basic geo

notation

38

A point charge ofA point charge of magnitude q = 4.5 nC is moving with speed v = 3 6 107 / ll l3.6 x 107 m/s parallel to the x-axis along the line y = 3my = 3m. Find the magnetic fields produced by this chargeg(x = -4m, y = 3m) at(1) the origin(2) h i (0 3 )(2) the point (0,3m)(3) the point (0, 6m)

Ans: (1) 3.89 x 10-10 T in the paper. (2) 0 (Why?). (3) 3.89 x 10-10 T out the paper 39

Since currents areSince currents are basically moving charges only in wires, we can y ,extend our earlier formula into a more definite law:To compute for the

f ld d bmagnetic field caused by currents on a certain field point P we implementpoint P, we implement BIOT-SAVART LAW!

40

Throughout this chapter we shall encounterThroughout this chapter, we shall encounter four basic sources of magnetic fields that produces the field by carrying currents!produces the field by carrying currents!

They are:They are:1. Current Loops

S l idTo find the direction of2. Solenoids

3. Straight Wiresdirection of

the field, apply RHD

4. Toroids

41

Steps:Steps:1. Grab the wire in such a way that yoursuch a way that your thumb is in the same direction and the current.2. Your 4-fingers determines the direction of the

ti fi ld itmagnetic field as it wraps around the wire

42

RI

Figure shows a currentFigure shows a current loop. x

P

To calculate the magnetic fieldTo calculate the magnetic field caused by the loop at a certain point P along the central axis of h l l h f lthe loop, we apply the formula:

Special Case: If P is the center of the loop, x =0, then the magnetic field at the center ofmagnetic field at the center of the loop is

43

A circular loop of radius 5.0cmA circular loop of radius 5.0cm has 12 turns and lies in the yz-plane, where it is centered at the origin. It carries a gcurrent of 4A. The current is counterclockwise from thecounterclockwise from the perspective of the x-axis looking to the yz-plane.

Find the magnetic field at(a) center of the loop (x = 0)(b) 15(b) x = 15 cm(c) x = 3 cm.

44

A solenoid is a wire tightly wound f

g yinto a helix of closely spaced turns as illustrated.

It is used to produce a strongIt is used to produce a strong, uniform magnetic field in a region surrounded by its loops.

Its role in magnetism is analogous toIts role in magnetism is analogous to that of the parallel-plate capacitor, which produces a strong, uniform electric field between its plates.

The magnetic field of a solenoid is essentially that of a set of N identical current loops placed side by side.

We have two types of solenoids:1. Finite Solenoids2 I fi it S l id2. Infinite Solenoids

45

For PHYSICS 13 weFor PHYSICS 13, we shall use the long solenoidsolenoid approximation.n (turn density) = N/Ln (turn density) = N/LWe can calculate the magnetic field causedmagnetic field caused by solenoids at two locations:

At the center

locations:1. At its center2 At its ends

At the ends2. At its ends

46

Find the magnetic field at the center of aFind the magnetic field at the center of a solenoid of length 20 cm, radius 1.4 cm, and 600 turns that carries a current of 4A600 turns that carries a current of 4A.

47

As with solenoidsAs with solenoids, we there are two kinds of straight gwires1. Finite Wires2. Infinite Wires

We wish to obtain the magnetic field

Field caused by finite wires. R is the distance of P from the wires

at a point P perpendicular f th li

Field caused by infinite wires, R is the distancefrom the line

48

wires, R is the distance of P from the wires!

Find the magnetic fieldFind the magnetic field at the center of a square current loop of q pside L = 50cm carrying a current of 1.5A.

Picture the Problem:The magnetic field at the center is the sum f h b fof the contributions of

each side!

49

A toroid consists ofA toroid consists of loops of wire wound around a doughnut-h d fshaped form.

he magnetic field at ahe magnetic field at a distance r from the center of the toroid are given as:

If r<a (inner radius) or if r>b (outer radius)

If a<r<b

50

We know that magnetic field lines differ fromfield lines differ from electric field lines.Magnetic field lines formMagnetic field lines form closed loops. The magnetic equivalent f h l hof the electric charge is

called a magnetic pole.Gauss’s Law forGauss s Law for Magnetism is stated as:

That is, no magnetic monopoles! 51

Ampere’s Law is very analogous to Gauss’s Lawanalogous to Gauss s Law for Electricity.

It relates the magnetic field to the current enclosed by an imaginaryenclosed by an imaginary loop (called Amperian Loop).

Ampere’s Law works for configurations that have a ghigh degree of symmetry.

52

Check

Ampere’s Law will only work

point: IAmpere s Law will only work if and only if the following statements hold:

n whichAm

1. The configuration has a very high level of symmetry

h of them

pere’s 2. The current is continuous everywhere in space.

ese fourLaw

ho

Therefore, there are only three cases where Ampere’s

r configold?

pLaw can be used:◦ 1. Long straight lines◦ 2. Long, tightly wound

gurationg, g ysolenoids

◦ 3. Toroids 53

n does

In our discussion of the magnetism in matter, we

t t th t ireturn to the atomic model.

El bi d hElectrons orbit around the nucleus, and since we can consider it as a current, it produces a magneticproduces a magnetic field!

If we look back at theIf we look back at the electric dipole (and the corresponding electric dipole moment), we candipole moment), we can also make the same analogy to create a magnetic dipole (and the

di icorresponding magnetic dipole moment

54

We can classify tt i t th All materials have matter into three

based on their reaction to an

randomly oriented magnetic moments

reaction to an external magnetic field! Bext

Diamagnetsg

Paramagnets

Magnetic Moments align to oppose the external magnetic field, thus diagmagnets are slightly repelled by magnets Bext

Paramagnets

Ferromagnets

Magnetic Moments align slightly with the external magnetic field, thus paramagnets interact weakly with magnets

BextFerromagnets55

Magnetic Moments align strongly with the external magnetic field, causing permanent magnetization, thus ferromagnets interact strongly with magnets