PowerPoint Presentation

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Earth�s Magnetic Fi= eld

Earth�s Magnetic Fi= eld

Dip Angle of Earth�s Magnetic Field

More About the Eart= h�s Magnetic Poles

Earth�s Magnetic Declination

Source of the Earth= �s Magnetic Field

Reversals of the Ea= rth�s Magnetic Field

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Earth�s Magnetic Fi= eld

The Earth�s geograp= hic north pole corresponds to a magnetic south pole

The Earth�s geograp= hic south pole corresponds to a magnetic north pole

Strictly speaking, = a north pole should be a �north-seeking� pole and a south pole a �south-seeking� pole<= /font>

Earth�s Magnetic Fi= eld

The Earth�s magneti= c field resembles that achieved by burying a huge bar magnet deep in the Earth�s interior

Dip Angle of Earth�s Magnetic Field

If a compass is fre= e to rotate vertically as well as horizontally, it points to the earth�s surface

The angle between t= he horizontal and the direction of the magnetic field is called the dip angl= e

The farther north t= he device is moved, the farther from horizontal the compass needle would be

The compass needle = would be horizontal at the equator and the dip angle would be 0�

The compass needle = would point straight down at the south magnetic pole and the dip angle would be 90�

More About the Eart= h�s Magnetic Poles

The dip angle of 90= � is found at a point just north of Hudson Bay in Canada

This is considered = to be the location of the south magnetic pole

The magnetic and ge= ographic poles are not in the same exact location

The difference betw= een true north, at the geographic north pole, and magnetic north is called the mag= netic declination

The amount of decli= nation varies by location on the earth�s surface

Earth�s Magnetic Declination

Source of the Earth= �s Magnetic Field

There cannot be lar= ge masses of permanently magnetized materials since the high temperatures of the core prevent materials from retaining permanent magnetization

The most likely sou= rce of the Earth�s magnetic field is believed to be electric currents in the liquid = part of the core

Reversals of the Ea= rth�s Magnetic Field

The direction of th= e Earth�s magnetic field reverses every few million years

Evidence of these r= eversals are found in basalts resulting from volcanic activity

The origin of the r= eversals is not understood

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= MAGNETISM

=
CHAPTER� 29� -� DEALS� WITH� MAGNETIC� FORCES<= br>
CHA= PTER� 30� -� DEALS� WITH� MAGNETIC� FIELDS<= br>

= B� -� magnetic� field

=
mov= ing charges� produce� B� -� Chapter� 30
moving charges� respond to B� -� Chapter� 29

I�l= l cover� the� production� of� magnetic� fields� first

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= CHAPTER� 30� SOURCES OF THE MAGNETIC FIELD

=
INSERT SLIDES ON THE BIOT� - SAVART LA= W AND THE
= LONG STRAIGHT WIRE

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in = (2) can be converted to an� integral:

�The

P

r

y

θ= ;

I

r<= /b>

I

d= x

x

c

d

=
a a= nd b in (3) define the limits of the current.
(3) is appropriate for a straight wire of length L.

=
Where L =3D� c � d. Let c =3D -L / 2<= span style=3D'mso-spacerun:yes'>� and d =3D +L / 2
and= P be at x =3D 0 (the midpoint of the wire)

Then= (3) is:

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P

r

y

θ= ;

I

r<= /b>

I

d= x

x

c

d

then= :

and:=

then= (5) is:

HOM= EWORK: Integrate (6) to show:

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Now,= let L

= 734; in (7),� the condition for a lo= ng straight wire

then= y / L

0 an= d (7) is :

The= n� for B a distance r from a lo= ng, straight wire we have:

B=

r

I

I

m= agnitude:

= direction: from the right hand rule applied to dl� X r

whe= re dl is in the direction of I

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= direction: from the right hand rule applied to dl� X r

whe= re dl is in the direction of I

B=

B=

r

r

dl

I

r

I

r

B=

B=

=
RULE ON DIRECTION:� Looking in the direction of I (conventional), B is
�� clockwise around I

r

I

I

B=

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=
RULE ON DIRECTION:� Looking in the direction of I (conventional), B is
�� clockwise around I

r

I

I

B=

=	<= /td>
�� SI�� C= GS (GAUSSIAN)
UNITS�� Tesla, T�� Ga= uss, G
�� 1 T� =3D� 10⁴ G

EXA= MPLE: Find B 1 m from a long straight wire carrying a current of 2 A

<= /td>	<= /td>
compare with earth�s
magnetic field at the
surf= ace :
�� B_E =3D� 0.5 x 10^-4 T

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= The Magnetic Field Due to a Current Loop

A c= ircular loop of radius, R in the x,y plane carries conventional current, I.

=
See� Ex. 30.3 p 841� but� NOTE: The text finds B at an arbitrary point on the
= loop axis , and the text uses �a� for the loop radius.

z

B= _z

y

R

r<= /b>

I

dl

x

=
The magnetic field at the center of the loop in the x,y plane is B_z.
Fro= m the Biot-Savart Law:

<= /td>
where dl is an element of length vector tangent to
the = loop

(9)<= /font>

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r<= /b>

φ= ;

I

d= l

=
= dl X r� =3D� dl� r� sin θ;

r�� =3D�� 1�� and� sin θ� =3D� 1

and = (9) gives:

(10)=

the= refore,�� dl X �r�� =3D� dl

=	<= /td>
whe= re: dl� =3D� R dφ
and φ� is� the azimuthal angle
with� r� =3D� R,� the loop radius

(9)<= /font>

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r<= /b>

φ= ;

I

(10)=

Equ= ation (10) gives:

or:<= /font>

<= /td>
NOTE: This is the last expression in
Ex. = 30.3� p 842

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(11= ) gives B_z at the center of the loop

B=

I

R

M= agnitude:

<= /td>	=
Direction:� from the right hand rule applied
to <= b>dl X r� -
IF THE LOOP IS VIEWED FROM A
PERSPECTIVE SUCH THAT� I� IS
CLOCKWISE, THEN B IS� AWAY

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For= N closely packed turns :

B=

Thi= s is an approximation

=
HOMEWORK: Show that B_z at a point on the axis of the loop, a distance
z f= rom the loop center is:

B=

z

R

y

x

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=
EXAMPLE � Find B at the center of a circular, conducting loop of radius,=
�� R =3D 10 cm if= the loop carries a current, I =3D 2 A.

T

com= pare with B for the straight wire example

B� 1.26 x 10^-5 T

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=	<= /td>
The magnetic dipole moment, m, due to a
closed , circular current loop in the x,y
pla= ne carrying current I is given by:

z

m� =3D�� m�� =3D� I A� =3D� I π R²

=
where R is the radius of the loop
and= A is the loop area

m=

y

<= /td>	<= /td>
NOTE: The text uses � for the dipole
mome= nt
which could be confusing

x

<= /td>	<= /td>
NOTE : A dipole exists for any closed
curr= ent path
not just circular loops

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INS= ERT� SLIDES� ON� MAGNETIC� DIPOLE� MOMENTS

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=
Recall:� B_Z� for points on the loop axis a distance, z from a circular
�loop of radius, R carrying current I= is given by:

=
where m =3D I π r^2��= �is the magnitude of the magnetic
dip= ole moment of the loop

Bz

z

m

I

y

I

x

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N )<= /font>

= B due to m� off axis

z

Bz

r

θ= ;

B=

N

r

θ= ;

m

m

I

I

y

S

I

Bz

<= /td>
In general, a distance r from the dipole there, are two components of
B= :

whi= ch shows the 1 / r³ dipole drop-off

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charset="windows-1252" PowerPoint Presentation

= Earth�s� Dipole� Field (See Sec. 30.7)

Ng

Sm

B=

B=

B=

Nm

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text/html; charset="windows-1252" PowerPoint Presentation

Earth�s Magnetic Field

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�	The Earth�s geographic north pole

	corresponds to a magnetic south pole

�	The Earth�s geographic south pole

	corresponds to a magnetic north pole

	�	Strictly speaking, a north pole should be a

	�north-seeking� pole and a south pole a �south-

	seeking� pole

------=_NextPart_01CA5271.87F409E0 Content-Location: file:///C:/C6F2A690/PHYSICS2212K-WEB-CH30_files/v3_slide0023.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" PowerPoint Presentation

Earth�s Magnetic Field

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�	The Earth�s magnetic

	field resembles that

	achieved by burying a

	huge bar magnet deep

	in the Earth�s interior

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nP8A7CiigA/4XH/1Af8Ayc/+wo/4XH/1Af8Ayc/+woooA//Z ------=_NextPart_01CA5271.87F409E0 Content-Location: file:///C:/C6F2A690/PHYSICS2212K-WEB-CH30_files/v3_slide0024.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" PowerPoint Presentation

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Dip Angle of Earth�s Magnetic

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If a compass is free to rotate vertically as well as

horizontally, it points to the earth�s surface

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The angle between the horizontal and the direction

of the magnetic field is called the dip angle

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The farther north the device is moved, the farther from

horizontal the compass needle would be

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The compass needle would be horizontal at the equator and the

dip angle would be 0�

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The compass needle would point straight down at the south

magnetic pole and the dip angle would be 90�

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More About the Earth�s Magnetic

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The dip angle of 90� is found at a point just north

of Hudson Bay in Canada

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This is considered to be the location of the south

magnetic pole

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The magnetic and geographic poles are not in the

same exact location

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The difference between true north, at the geographic

north pole, and magnetic north is called the magnetic

declination

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The amount of declination varies by location on the earth�s

surface

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Earth�s Magnetic Declination

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=	<= /td>
Source of the Earth�s Magnetic

	<= /td>		=	=	<= /td>	<= /td>			<= /td>
�	There cannot be large masses of

	permanently magnetized materials since the

	high temperatures of the core prevent

	materials from retaining permanent

	magnetization

�	The most likely source of the Earth�s

	magnetic field is believed to be electric

	currents in the liquid part of the core

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<= /td>	=
Reversals of the Earth�s

	Magnetic Field

<= /td>			<= /td>		=	<= /td>	<= /td>	<= /td>
�	The direction of the Earth�s magnetic field

	reverses every few million years

	�	Evidence of these reversals are found in basalts

	resulting from volcanic activity

	�	The origin of the reversals is not understood

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=
Nuclear magnetic� dipole moments expose= d to a large external magnetic
= field, B₀ precess at a resonant frequency angular frequency ω

z

m=

<= /td>	<= /td>	<= /td>	<= /td>
Nuclear Magnetic Resonance
is based on the precession
phenomenon. Magnetic resonance
also occurs for environments other
than= the nucleus.

B= ₀

y

x

`= 9;� =3D� γ B₀

whe= re γ is the gyro magnetic ratio for the nucleus

=
This behavior results from the fact that the nuclear dipole moment is=
pro= portional to the nuclear spin angular momentum, S

= m_z =3D� γ S_z� =3D� γ �� h�� where� h� is� Plank�s Constant

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<= /td>	=	<= /td>	<= /td>
SECTION� 30.3� AMPERE�S LAW &� 30.4= � B OF A SOLENOI= D-

�� A closed path of length, L� bounds a surface of area, A. A curr= ent density
J is directed perpendicular to the surface. The total current through the=
surface is I. A magnetic field B on the path has a component B_{l tangent to}
the = path.

dl

A

B=

=	<= /td>
From Maxwell�s IV Equation
it can be shown that if B_l is
= uniform along the path:

B= _l

J<= /b>

Thi= s is Ampere�s Law in its useful form

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= Finding B due to a long, straight wire using Ampere�s Law

A l= ong, straight� wire carries current,= I in the +z direction

z

<= /td>
B_φ � azimuthal component of B, a distance
r fr= om I

I

her= e� B_l =3D B_{φ and is
uniform over L}

L� =3D� 2π r

r

y

φ= ;

=
Ampere�s Law says:
BφL� =3D� μ₀ I�� or:

x

Bφ

Bφ(2π r)� =3D �μ₀ I

Bφ� =3D� �(μ₀ I) / (2π r)=

=
B� of� a long, straight wire, originally
Der= ived from the Biot-Savart Law

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= Finding B inside a solenoid -

=
A solenoid consists of N turns, each of radius, r wound on a core of leng= th, L.
The= current in each turn is I.

L

1

r

B<= /b>

4

2

3

=
For the solenoid geometry, we assume� r << L and the only contribution
To = the product� B L� occurs along path segment 1 insi= de the coil

<= /td>
For this situation it can be shown that the magnetic, B field inside=
The = coil is:

B� =3D� μ₀ (N/L) I

<= /td>
And defining n to be the number of turns per unit length:
n = =3D N/L we have:

B = =3D μ₀ n I

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<= /td>	=
EXAM= PLE :

�� A Tesla Coil is 1.5 m long and ha= s 2000 turns. If a current of 2 A is carried,
what is the magnitude of B inside the coil.

<= /td>	<= /td>
B� =3D� μ₀ n I

B� =3D� 4π x 10^-7 N / A² (2000 / 1.5 m) 2 A

B =3D 3.35x10^-3 T

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<= /td>	=
MAGNETISM IN MATTER� (See Sec. 30.6 for discussion of individual
dipo= le moments)
(Analogous to Dielectric Response in Electricity)

<= /td>	=
An external current I in an electromagnet produces a magnetic field Streng= th,
H. In a material with magneitc susceptibility, χ a magnetization vector, M
i= s induced.

I

M=

H=

H=

m= _i

mi � i^th atomic magnetic dipole moment

=		<= /td>
NOTE: H and M have
uni= ts of� A m² / m³ or<= /font>
A / m which is appropriate
for dipole moment / volume

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= B� =3D� μ₀ ( H + M)� or

= B� =3D� μ₀ (1 + χ) H� =3D� μ H

whe= re μ� =3D μ₀ (1 + χ)=

=
non magnetic materials (free space):

= 67; =3D 0� then,� μ� =3D� μ₀� and B� =3D μ₀ H

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MAGNETISM

CHAPTER 29 - DEALS WITH MAGNETIC FORCES

CHAPTER 30 - DEALS WITH MAGNETIC FIELDS

B - magnetic field

moving charges&nb= sp; produce B - Chapter 30

moving charges&nb= sp; respond to B - Chapter 29

IR= 17;ll cover the production of magnetic fields first

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CHAPTER 30 SOURCES OF THE MAGNETIC FIELD

INSERT SLIDES ON THE BIOT - SAVART LAW AND THE

LONG STRAIGHT WIRE

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=

The<= /span>

in (2) can be converted to an integral:

=
I

I

r

r

x

dx

c

d

a and b in (3) define the limits of the current.&#= 13;

(3) is appropriate for a straight wire of length L= .

Where L =3D = c – d. Let c =3D -L / 2 = and d =3D +L / 2

and P be at x =3D 0 (the midpoint of the wire)

Then (3) is:

θ

y

P