What is magnetism?Magnetism is the properties and interactions of magnetsThe earliest magnets were found naturally in the mineral magnetite which is abundant the rock-type lodestone. These magnets were used by the ancient peoples as compasses to guide sailing vessels.Magnets produce magnetic forces and have magnetic field lines

Magnets have two ends or poles, called north and south poles. At the poles of a magnet, the magnetic field lines are closer together.

Unlike poles of magnets attract each other and like poles of magnets repel.

The earth is like a giant magnet!The nickel iron core of the earth gives the earth a magnetic field much like a bar magnet.

What are magnetic domains?Magnetic substances like iron, cobalt, and nickel are composed of small areas where the groups of atoms are aligned like the poles of a magnet. These regions are called domains. All of the domains of a magnetic substance tend to align themselves in the same direction when placed in a magnetic field. These domains are typically composed of billions of atoms.

Atoms themselves have magnetic properties due to the spin of the atom’s electrons.

These areas of atoms are called “domains”

Groups of atoms join so that their magnetic fields are all going in the same direction

When an unmagnetized substance is placed in a magneticWhen an unmagnetized substance is placed in a magneticfield, the substance can become magnetized.field, the substance can become magnetized.

This happens when the spinning electrons line up in theThis happens when the spinning electrons line up in thesame direction.same direction.

Electricity and Magnetism – how are they related?

When an electric current passes through a wire a magnetic field is formed.

What is an electromagnet?When an electric current is passed through a coil of wire wrapped around a metal core, a very strong magnetic field is produced. This is called an electromagnet.

What is a galvanometer?A galvanometer is an electromagnet that interacts with a permanent magnet. The stronger the electric current passing through the electromagnet, the more is interacts with the permanent magnet.

The greater the current passing through the wires, the stronger the galvanometer interacts with the permanent magnet.

Galvanometers are used as gauges in cars and many other applications.

What are electric motors?An electric motor is a device which changes electrical energy into mechanical energy.

Go to the next slide

How does an electric motor work?

Simple as that!!

We have seen how electricity can produce a magnetic field, but a magnetic field can also produce electricity! How?

What is electromagnetic induction?

Moving a loop of wire through a magnetic field produces an electric current. This is electromagnetic induction.

A generator is used to convert mechanical energy into electrical energy by electromagnetic induction.

Carefully study the next diagrams:

The force on a moving charge is related to the force on a current:

Once again, the direction is given by a right-hand rule.

Force on an Electric Charge Moving in a Magnetic Field

Cont’d

• Negative charge near a magnet.

• A negative charge -Q is placed at rest near a magnet. Will the charge begin to move? Will it feel a force? What if the charge were positive, +Q?

If a charged particle is moving perpendicular to a uniform magnetic field, its path will be a circle.

Cont’d

Cont’d

A helical path.

What is the path of a charged particle in a uniform magnetic field if its velocity is not perpendicular to the magnetic field?

v

v

Magnetism 19

Cont’d

Cont’d

The aurora borealis (northern lights) is caused by charged particles from the solar wind spiraling along the Earth’s magnetic field, and colliding with air molecules.

Cont’d

Velocity selector, or filter: crossed E and B fields.

Some electronic devices and experiments need a beam of charged particles all moving at nearly the same velocity. This can be achieved using both a uniform electric field and a uniform magnetic field, arranged so they are at right angles to each other. Particles of charge q pass through slit S1 and enter the region where E points into the page and B points down from the positive plate toward the negative plate. If the particles enter with different velocities, show how this device “selects” a particular velocity, and determine what this velocity is.

B

EvFF 0BE

Magnetism 22

Hall effect

• Allows the measurement of Magnetic Field if a material is known.

• Allows the determination of the “type” of current carrier in semiconductors if the magnetic field is known.– Electrons

– Holes

Magnetism 23

Hall Geometry (+ Charge)

• Current is moving to the right. (vd)

• Magnetic field will force the charge to the top.

• This leaves a relatively maximum (-) charge on the bottom.

• This creates an electric field and a potential difference.

Magnetism 24

Negative Carriers

• Carrier is negative.• Current still to the

right.• Force pushes

negative charges to the top.

• Positive charge builds up on the bottom.

• Sign of the potential difference is reversed.

Magnetism 25

Hall Math

• Eventually, the field due to the Hall effect will allow the current to travel un-deflected through the conductor.

net

iB

newt

iwBV

wtAneA

iwBBwvV

neA

iv

AinevJ

BwvV

orw

VqqEBqv

balance

Hall

dHall

d

d

dHall

HallHalld

/

:

Cont’d

Electron’s path in a uniform magnetic field.

An electron travels at 2.0 x 107 m/s in a plane perpendicular to a uniform 0.010-T magnetic field. Describe its path quantitatively.

Can a magnetic field be used to stop a single charged particle, as an electric field can?

Magnetism 27

The Magnetic Force is Different From the Electric Force.

Whereas the electric force acts in the same direction as the field:

The magnetic force acts in a direction orthogonal to the field:

And --- the charge must be moving !!

FqBvFqE(Use “Right-Hand” Rule to determine direction of F)

Magnetism 28

So…

• A moving charge can create a magnetic field.

• A moving charge is acted upon by a magnetic field.

• In Magnetism, things move.• In the Electric Field, forces and the

field can be created by stationary charges.

Magnetism 29

Magnetic force on current Wires

• A wire with a current contains moving charges.

• A magnetic field will apply a force to those moving charges.

• This results in a force on the wire itself.– The electron’s sort of

PUSH on the side of the wire.

F

Remember: Electrons go the “other way”.

Magnetism 31

The Wire in More Detail

B out of plane of the paper

Assume all electrons are moving with the same velocity vd.

(i). charge POSITIVE of

motion theofdirection in the

:

L

BLF

i

vector

iLBBvv

LiBqvF

v

Litiq

dd

d

d

Magnetism 32

Magnetic Levitation

Current = img

Magnetic Force

Where does B point???? Into the paper.

iL

mgB

mgiLB

Cont’d

34

Torque on a Current Carrying Loop

• Consider a small rectangular current carrying loop in a region permeated by a magnetic field.

• Assuming a uniform magnetic field, the force on the upper wire is:

• The force on the lower wire is: x

y

I

B Fm1

Fm2

L

W

)̂(1kILBFm)̂(2kILBFmN.B: we considered that the plane of the loop is parallel with B field

35

Cont’d

• The forces acting on the loop have a tendency to cause the loop to rotate about the x-axis.

• The quantitative measure of the tendency of a force to cause or change rotational motion is torque.

Cont’d

• The torque acting on a body with respect to a reference axis is given by

Fr

distance vector from the reference axis

Cont’d• Hence, the torque acting on the loop is;

• The above eqn. can be rewritten as; where and

)ˆ()ˆ(

)ˆ(2

)ˆ(2

)ˆ()ˆ(2

)ˆ(ˆ2 21

iIABiIwLB

iILBw

iILBw

kFjw

kFjw

mm

BjBk mm

ˆˆ

ANIm

Magnetic dipole moment of the current loopMagnetic dipole moment of the current loop

N represents the number of turns in the loop

Cont’d

• The torque acting on the loop tries to align the magnetic

dipole moment of the loop with the external B field.

Bm

holds in general regardless of loop shape

General derivation of the Torque

• Motors are most common apps of magnetic force on

current caring wire.

• The force on the top and bottom segments are vertical

and produce no torque.

N.B: Here the plane of the Current loop is neither parallel nor perpendicular to B field.

Cont’d• Below is the top view of the loop in the left side.

Cont’d

• So, net torque for the orientation of the loop in part (a) is:

BN

or

jBkiBiiAB

iwilBiwF

kFjw

kFjw

B

BB

ˆˆ)ˆ(sin)ˆ(sin

)ˆ(sin)ˆ(sin

ˆ)ˆ(sin2

)ˆ(ˆsin2

B

B

z

Y

X

A Video showing Torque on current loop

Example

• Torque on a coil.

• A circular coil of wire has a diameter of 20.0 cm and contains 10 loops. The current in each loop is 3.00 A, and the coil is placed in a 2.00-T external magnetic field. Determine the maximum and minimum torque exerted on the coil by the field.

Magnetic force on a current carrying wire: Biot-Savart Law

• 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 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

dBr

dsr

Biot-Savart Law – Observations

• The vector is perpendicular to both 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

dBr

r̂dB

rdsr

dsr

dsr

Biot-Savart Law – Observations, cont’d

• The magnitude of is proportional to the current and to the magnitude ds of the length element

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

dsr

r̂dsr

dBr

dBr

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

• The magnetic field described by the law is the field due to the current-carrying conductor– Don’t confuse this field with a field external to the

conductor

Biot-Savart Law – Equation

24oμ d

dπ r

s rB

rr ˆI

Permeability of Free Space

• The constant o is called the permeability of free space

o = 4 x 10-7 T. m / A

Total Magnetic Field

• is the field created by the current in the length segment ds

• To find the total field, sum up the contributions from all the current elements I

– The integral is over the entire current distribution

dBr

24oμ d

π r

s r

Brr ˆI

dsr

Biot-Savart Law – Final Notes

• The law is also valid for a current consisting of charges flowing through space.

• represents the length of a small segment of space in which the charges flow– For example, this could apply to the electron

beam in a TV set

dsr

Compared to

• 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

Br

Er

Compared to , 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

r

Br

Er

Compared to , 3

• 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

Br

Er

Magnetism 55

Magnetic Field of a Straight Wire

• We intimated via magnets that the Magnetic field associated with a straight wire seemed to vary with 1/d.

• We can now PROVE this!

Magnetism 56

From the Past

Using Magnets

Magnetism 57

Right-hand rule: Grasp the element in your right hand with your extended thumb pointing in the direction of the current. Your fingers will then naturally curl around in the direction of the magnetic field lines due to that element.

Magnetism 58

Let’s Calculate the FIELD

Note:

For ALL current elements

ds X r

is into the page

Magnetism 59

The Details

02

0

20

)sin(

2B

it. DOUBLE and to0 from integrate

wesoamount equalan scontribute

wire theofportion Negative

)sin(

4

r

dsi

r

idsdB

Magnetism 60

Moving right along

R

i

Rs

rdsiB

SoRs

R

Rsr

22

)sin(sin

0

02/322

0

22

22

1/d

Magnetism 61

A bit more complicatedA finite wire

Magnetism 62

P1

)sin()sin(: NOTE

r

ds

2/122

20 )sin(

4

)sin(

)sin(

Rsr

r

dsidB

r

R

rdsrds

Magnetism 63

More P1

R

i

whenRL

L

R

iB

and

Rs

dsiB

L

L

2B

,L 42

4

0

22

0

2/

2/2/322

0

Magnetism 64

P2

22

0

0

2/322

0

4

4

Rs

L

R

iB

or

Rs

dsiRB

L

Magnetism 65

HOME TASK:Find the magnetic field B at point P in for i = 10 A and a = 8.0 cm.

Magnetism 66

Circular Arc of Wire

Magnetism 67

More arc…

Cpoint at 4

44

44

0

0

0

02

0

20

20

R

iB

dR

i

R

iRddBB

R

iRd

R

idsdB

Rdds ds

Magnetism 68

How do you do that??

0rsdNo Field at C

Magnetism 69

Force Between Two Current Carrying Straight Parallel

ConductorsWire “a” createsa field at wire “b”

Current in wire “b” sees aforce because it is movingin the magnetic field of “a”.

Magnetism 70

The Calculation

d

iLi

iFd

iB

ba

b

a

2F

angles...right at are and Since

2

:calculatedjust what weis a"" wire

todue b"" at wire FIELD The

0

b""on

0b""at

BL

BL

Magnetism 71

Ampere’s Law

enclosedid 0 sB

USE THE RIGHT HAND RULE IN THESE CALCULATIONS

Magnetism 72

COMPARE

enclosedid 0 sB

0enclosedq

d AE

Line Integral

Surface Integral

Magnetism 73

Simple Example

Magnetism 74

Field Around a Long Straight Wire

enclosedid 0 sB

r

iB

irB

2

2

0

0

Magnetism 75

Field INSIDE a WireCarrying UNIFORM Current

Magnetism 76

The Calculation

rR

iB

andR

rii

irBdsBd

enclosed

enclosed

20

2

2

0

2

2

sB

Magnetism 77

R r

B

R

i

2

0

for a Circular Loop of Wire

• Consider the previous result of the half loop, with a full circle� = 2

• This is the field at the center of the loop

24 4 2

o o oμ μ μBθ π

πa πa a I I I

Br

for a Circular Current Loop

• The loop has a radius of R and carries a steady current of I

• Find the field at point P

Br

for a Circular Current Loop

• The loop has a radius of R and carries a steady current of I

• Find the field at point P

• Due to symmetry the field along y-axis vanishes.

• The net field is along x only and becomes;

2

32 2 22

ox

μ aB

a x

I

Br

Comparison of Loops

• Consider the field at the center of the current loop

• At this special point, x = 0 • Then,

– This is exactly the same result as from the curved wire

2

32 2 2 22

o ox

μ a μB

aa x

I I