Transmission line demo to illustrate whyvoltage along transmission lines is high

Connect to step down transformer120V to 12V to lightbulb

Lights up brightly

12 V6.5 A

Connect it to long fat wires

Lights up brightly

Connect it tolong thin wires

Lights up dimly

P=I2R loss is high along wires

Connect tostep up

transformer12V to 120 V6.5 to .65 A

Connect tostep down transformer

Lights brightly

P=I2R loss is lowalong wires

\

Lecture 8 Magnetic Fields Ch. 29• Cartoon Magnesia, Bar Magnet with N/S Poles, Right Hand Rule• Topics

– Permanent magnets– Magnetic field lines,– Force on a moving charge,– Right hand rule,– Force on a current carrying wire in a magnetic field,– Torque on a current loop

• Demos– Compass, declinometer, globe, magnet– Iron fillings and bar magnets– Compass needle array– Pair of gray magnets– CRT illustrating electron beam bent bent by a bar magnet - Lorentz law– Gimbal mounted bar magnet– Wire jumping out of a horseshoe magnet.– Coil in a magnet

• Elmo

• Polling

Magnetic Fields

• Magnetism has been around as long as there has been an Earth withan iron magnetic core.

• Thousands of years ago the Chinese built compasses for navigation inthe shape of a spoon with rounded bottoms on which they balanced(Rather curious shape for people who eat with chopsticks).

• Certain natural rocks are ferromagnetic – having been magnetized bycooling of the Earth’s core.

• Show a sample of natural magnetic rock. Put it next to manycompasses.

Magnetism’s Sociabilities

• Magnetism has always has something of a mystic aura about it. It isusually spoken of in a favorable light.

• Animal magnetism, magnetic personality, and now you can wearmagnetic collars, bracelets, magnetic beds all designed to make youhealthier – even grow hair.

• We do not have the same feeling about electricity. If you live nearelectric power lines, the first thing you want to do is to sue the electriccompany.

Compass and Declinometer• In 1600 William Gilbert used a compass needle to show how it oriented

itself in the direction of the north geographic pole of the Earth, whichhappens to be the south magnetic pole of the Earth’s permanentmagnetic field.

• Show compass and declinometer. Each has a slightly magnetizedneedle that is free to rotate. The compass lines up with the componentof the magnetic field line parallel to the surface of the Earth. Thedeclinometer lines up with the actual magnetic field line itself. It saysthat the angle between the field lines and the surface is 71 degrees asmeasured from the south.

• Earth’s magnetic field

• Basically there are two types of magnets: permanent magnets andelectromagnets

• Show field lines for a bar magnet. Show bar magnet surroundedby compass needle array.

Permanent Magnets• Bar magnet is a model of a ferromagnetic material that can be

permanently magnetized. Other ferromagnetic materials arecobalt and nickel.

• The origin of magnetism in materials is due mostly to the spinningmotion of the charged electron on its own axis. There is a smallcontribution from the orbital motion of the electron.

+

a

v

Electronorbitingnucleus

Magnetic dipole

Magneticdipole

sElectronspinning onits axis

e

-

Atomic origin of magnetic field

Permanent Magnets (continued)

• In ferromagnetic materials there are whole sections of the ironcalled domains where the magnetism does add up fromindividual electrons. Then there are other sections or domainswhere contributions from different domains can cancel.However, by putting the iron in a weak magnetic field you canalign the domains more or less permanently and produce apermanent bar magnet as you see here.

• In nonmagnetic materials the contributions from all The electrons cancel out. Domains are not even formed.

Magnetic field lines donot stop at surface.

They are continuous.

They make completeloops.

Field lines for a barmagnet are the same asfor a current loop

Magnetic field linesSimilarities to electric lines• A line drawn tangent to a field line is the direction of the field at that

point.• The density of field lines still represent the strength of the field.

Differences• The magnetic field lines do not terminate on anything. They form

complete loops. There is no magnetic charge on as there was electriccharge in the electric case. This means if you cut a bar magnet in halfyou get two smaller bar magnets ad infinitum all the way down to theatomic level – Magnetic atoms have an atomic dipole – not a monopoleas is the case for electric charge.

• They are not necessarily perpendicular to the surface of theferromagnetic material.

!!"==#

"==#

AdE flux Electric

AdB flux Magnetic

E

B!!

!!

Definition of magnetic Field

• definition of a magnetic field

• The units of B are or in SI units(MKS).

This is called a Tesla (T). One Tesla is a very strong field.

• A commonly used smaller unit is the Gauss. 1 T = 104 G(Have to convert Gauss to Tesla in formulas in MKS)

• In general the force depends on angle . This iscalled the Lorentz Force

qv

FB =

)( .s

mC

N

).( mA

N

BvqF!!!

!=

In analogy with the electric force on a point charge, the correspondingequation for a force on a moving point charge in a magnetic field is:

Magnitude of

– Direction of F is given by the right hand rule (see next slide).

• If θ = 90, then he force = and the particle moves in a circle.

v B sin(0o) = 0 F = 0

FB

v

BvqFm

!!!

!= EqFe

!!

=

!sinqvBFm =

qvB

If the angle between v and B is θ = 0, then the force = 0.

• Consider a uniform B field for simplicity.

B v

!!

Use right hand rule to find the direction of F

+Positive Charge

Rotate v into B through the smaller angle φ and the force F will be in the directiona right handed screw will move.

BvqFm

!!!

!=

!Fm= q!v !!B

x

y

z

i

jk

!v = vxi + vy j!B = Bxi + By j

!F =

i j k

vx vy 0

Bx By 0

!

"

###

$

%

&&&=

vy 0

By 0

!

"#$

%&i +

0 vx

0 Bx

!"#

$%&j +

vx vy

Bx By

!

"#$

%&k

!F =

i j k

vx vy 0

Bx By 0

!

"

###

$

%

&&&= (vxBy ' Bxvy )k

Note

!F ! xy plane

Motion of a point positive charge “ ” in a magnetic field.

For a “+” charge, the particle rotates counter clockwise.For a “-” charge, the particle rotates counter clockwise.•Since F ⊥ v, the magnetic force does no work on the particle.

W = F • d = 0 ; F ⊥ d•This means kinetic energy remains constant.•The magnitude of velocity doesn’t change.•Then the particle will move in a circle forever.•The B field provides the centripetal force needed for circular motion.

= qvBsin90o

Direction is given by the RHR (righthand rule)

Magnitude of F = qvB

B is directed into the paper

r

x

x x

x

x

x

F

v

F

v

Fv +

BvF

!!!

!!

BvqFm

!!!

!=

Find the radius r and period of motion for a + chargemoving in the magnetic field B. Use Newtons 2nd Law.

What is the period of revolution of the motion?

qvBr

mvmaF ===

2

qB

mvr =

Radius of the orbitImportant formula inPhysics

v

ar

r

va

2

=

�

T =2!r

v=2!m

qB= period = T

Note the period is independent of the radius, amplitude, and velocity. Example ofsimple harmonic motion in 2D.

T is also the cyclotron period.

It is important in the design of the cyclotron accelerator. Of course, this is importantbecause today it is used to make medical isotopes for radiation therapy.

m

qBf

tf

!2

1

=

=

Cyclotron frequency

�

v = qBr /m

x

x x

x

Example: If a proton moves in a circle of radius 21 cm perpendicular to aB field of 0.4 T, what is the speed of the proton and the frequency ofmotion?

1

m

qBf

!2=

f =1.6 !10

"19C (0.4T )

(2# ) 1.67 !10"27kg

f =1.6 (0.4)

(6.28) 1.67!10

8Hz = 6.1!10

6Hz

Hzf 6101.6 !=

2

m

qBrv =

v =1.6 !10

"19C (0.4T ) 0.21m

1.67 !10"27kg

v =1.6 (0.4) 0.21

1.67!10

8 m

s= 8.1!10

6 m

s

s

m

v6

101.8 !=

v

r

x x

x x

x x

x

x

Use right hand rule to find the direction of F

+

Negative Charge

Rotate v into B through the smaller angle φ and the force F will be in the opposite Direction a right handed screw will move.

BvqFm

!!!

!=

Suppose we have an electron . Which picture is correct?

x

Noyes B

v

F F

x

x

x

x

x

x

x

v

Example of the force on a fast moving proton due to the earth’smagnetic field. (Already we know we can neglect gravity, but canwe neglect magnetism?) Magnetic field of earth is about 0.5 gauss.Convert to Tesla. 1 gauss=10-4 Tesla

Let v = 107 m/s moving North.What is the direction and magnitude of F?Take B = 0.5x10-4 T and v⊥ B to get maximum effect.

T105.010C106.1qvBF 4

s

m719

m

!!"##"==

!

N108F17

m

!"=

!(a very fast-moving proton)

meter

volts19 100C106.1qEFe

!"==#

!

!Fe= 1.6 !10

"17N

B

N

F

vV x B is into thepaper (west).Check with globe

Earth

Force on a current-carrying wire

When a wire carries current in a magnetic field, there is a force on the wire that is the sum of the forces moving charges that carry the current.

vd is the driftvelocity of theelectrons.

Cross sectionalarea A

ivd

F

L .

. .

.

.

.

. . .

n = density of mobile charges

Number of charges = nAL

or L is a vector in the directionof the current i withmagnitude equal to the lengthof the wire.Also

v ⊥ B

B (Out of the paper)

!F = (q

!v !!B)(nAL)

F = nqvALB Current,i = nqvA

iLBF =

BLidFd!!!

!=

BLiF

!!!

!=

Show force on a wire in a magnetic field

BLiF

!!!

!=

Currentup

Currentdown

Drift velocity of electrons

Magnetic bottle.The charge istrapped inside andspirals back andforth

Torques on current loopsElectric motors operate by connecting a coil in a magnetic field to a current

supply, which produces a torque on the coil causing it to rotate.

P ia

b

Above is a rectangular loop of wire of sides a and b carrying current i.B is in the plane of the loop and ⊥ to a.

Equal and opposite forces are exerted on the sides a.No forces exerted on b since

Since net force is zero, we can evaluate T (torque) at any point. Evaluate it at P.T tends to rotate loop until plane is ⊥ to B.

Bn

F = iaB

Bi

iABiaBbFbT ===

!sinNiABT =

Bθ

i

!B

!B

!F

!F

Torque on a current loop

B

B

NiA

NiAB

!=

=

=

=

µ"

#µ"

µ

#"

sin

sin

n

θ

Galvanometer

Magnetic dipole moment µ

B

B

NiA

NiAB

!=

=

=

=

µ"

#µ"

µ

#"

sin

sin

BU !"= µ

Ep-U

Ep!!

!!!

!=

"=#

Recall that for Electric dipole moment p

Demo: show torque on current loop (galvanometer)

Can you predict direction of rotation?

Example

A square loop has N = 100 turns. The area of the loop is 4 cm2 and itcarries a current I = 10 A. It makes an angle of 30o with a B fieldequal to 0.8 T. Find he magnetic moment of the loop and the torque.

Demo: Show world’s simplest electric motor

(scratch off all insulation on one end)

Scratch off half on the other end

Momentum will carry it ½ turn

(no opportunity for current to reverse coil direction)

mNTmABT

mAmANiA

.16.05.08.0.4.030sin

.4.0104101002

224

=!!=°=

=!!!== "

µ

µ

Cathode Ray Tube

Chapter 28 Problem 18 An alpha particle (q = +2e, m = 4.00 u) travels in a circular path ofradius 5.00 cm in a magnetic field with B = 1.60 T. Calculate thefollowing values.(a) the speed of the particle(b) its period of revolution(c) its kinetic energy(d) the potential difference through which it would have to beaccelerated to achieve this energy

Chapter 28 Problem 37

A 2.3 kg copper rod rests on two horizontal rails2.4 m apart and carries a current of 60 A from onerail to the other. The coefficient of static frictionbetween rod and rails is 0.51. What is the smallestmagnetic field (not necessarily vertical) thatwould cause the rod to slide?

(a)magnitude (b)direction counterclockwise from the horizontal

Chapter 28 Problem 47

A circular coil of 130 turns has a radius of 1.50 cm.

(a) Calculate the current that results in a magneticdipole moment of 2.30 A·m2.

(b) Find the maximum torque that the coil, carryingthis current, can experience in a uniform 20.0 mTmagnetic field.