Winter 2004 OSU Sources of Magnetic Fields 1 Chapter 32 Sources - - PowerPoint PPT Presentation

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 1

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 2

Sources Of Magnetic Fields

• We learned two ways to calculate Electric Field

– Coulomb's Force Brute Force Calculation

• High symmetry

What are the analogous equations for the Magnetic Field?

2

1 ˆ 4 dq dE r r  

enc

Q E dA   



Winter 2004 OSU Sources of Magnetic Fields Chapter 32 3

Sources Of Magnetic Fields

• Now two ways to calculate Magnetic Field

– Biot-Savart Law Brute Force Calculation

• Where unit vector displacement pointing

from element to the field point P

– Ampere’s Law High symmetry

ˆ r

dl

0 enc

B dl I   



2

ˆ 4 I dl r dB r    

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 4

Biot-Savart Law

• Biot-Savart Law; bits and pieces
• Magnetic fields are generated by moving

charge.

0 - permeability of free space

I r ^ r

dl

P

2

ˆ 4 I dl r dB r    

2 7

A N 10 4



   

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 5

Biot-Savart Law

• Concept Checks

– A wire lying in x-y plane carries a current I as shown. Points A,B, & C lie in the x-y plane as well. Let z-axis point out of the screen I

I) What is the direction of the magnetic field contribution at point A due to the segment dl .

1) +x 2) -x 3) +y 4) -y 5) +z 6) -z

A B C    dl

x z y

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 6

Biot-Savart Law

• Concept Checks

– A wire lying in x-y plane carries a current I as shown. Points A,B, & C lie in the x-y plane as well. Let z-axis point out of the screen I

I) What is the direction of the magnetic field contribution at point B due to the segment dl .

1) +x 2) -x 3) +y 4) -y 5) +z 6) -z

A B C    dl

x z y

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 7

Biot-Savart Law

• Concept Checks

– A wire lying in x-y plane carries a current I as shown. Points A,B, & C lie in the x-y plane as well. Let z-axis point out of the screen I

I) What is the direction of the magnetic field contribution at point C due to the segment dl .

1) +x 2) -x 3) +y 4) -y 5) +z 6) -z

A B C    dl

x z y

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 8

B-field of  Straight Wire

• Calculate field at point P

R o using Biot-Savart Law: I y

r q dy

2

ˆ 4 μ I dy r dB π r  

2 2 2

dy dl and r R y   

2

( )sin 4 μ I dy θ B dB π r

 

 

 

dB

 

2 2

sin tan csc R R rd θ dy R θ dy r y R r q q        

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 9

B-field of  Straight Wire

• Calculate field at point P

R o using Biot-Savart Law: I y

r q dy

2

( )sin 4 μ I dy θ B dB π r

 

 

 

dB  

2 2

sin cos 4 4 2 rd r d dy dy R Rr μ I μ I B dB θd πR πR μ I B πR

  q

q q q q



       

 

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 10

B-Field of a Circular Loop

Circular loop of radius R carries dl current I. What is B along the axis of the loop:

x

• x

R R r x dB q q r dB

• Magnitude of dB from element dl:

dB = 

0 I

4  dl r 2 = 

0 I

4  dl x

2 +R 2

• What is the direction of the field?
• Symmetry is B in x-direction.

2 2 cos

4

x

μ I dl dB π x R q  

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 11

B-Field of a Circular Loop

dl

At the center of the loop z = 0 therefore

x

• x

R R r x dB q q

2 2 cos

4

x

μ I dl dB π x R q  

2 2 2 3 2 2 2

cos ( ) 4 ( )

x

R R r z R μ I R B dl Rd π x R



q q      



2 3 2 2 2

2( )

x

μ IR B x R  

2

x

μ I B R 

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 12

B-field of Concentric Semicircles

• Homework Problem #9

A current I flows in the direction b

• shown. What is the Magnetic Field a

At point P due to the current in the inner semicircle (at r = a)?

1) 3) 0 2) 4) 5)

dl P

2

a

μ I B a  

2

a

μ I B a 

4

a

μ I B a 

a

μ I B a 

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 13

B-field of Concentric Semicircles

• Homework Problem #9

A current I flows in the direction b

• shown. What is the Magnetic Field a

At point P due to the current in the outer semicircle (at r = b)?

1) 3) 0 2) 4) 5)

dl P

2

b

μ I B b  

8

b

μ I B b 

4

b

μ I B b 

4

b

μ Ib B  

Winter 2004 OSU Sources of Magnetic Fields Chapter 28 14

B-field of Concentric Semicircles

• Homework Problem #9

A current I flows in the direction b

• shown. What is the Magnetic Field a

At point P due to the current in the outer semicircle (at r = a)?

dl P

2

4

x

μ I dl dB directed in the screen π R 

2 0 (

) 4

a

μ I B dl Rd πR



q  



4

x

μ I B R 

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 15

Magnetism in Matter

• Sources L

– Orbital magnetic moment of an electron. – Spin magnetic moment of an electron e – 

• Types of Magnetism

– Ferromagnetic – Paramagnetic spin – Diamagnetic

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 16

Magnetism in Matter Ferromagnetism

• Consists of small regions

(called domains) where the magnetic moments are aligned.

• With placing the material

(domains) an external magnetic field you can align the domains.

• At critical temperature called Curie

Temperature the substance loses its magnetization and becomes Paramagnetic.

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 17

Magnetism in Matter Paramagnetism

• Magnetic moments within these materials interact

weakly with one another.

• Magnetic moments will align in the presence of an

external magnetic field.

• Curie’s Law
• If the material is lowered below the Curie

temperature, it will maintain its magnetic alignment.

B M C T 

Winter 2004 OSU Sources of Magnetic Fields Chapter 32 18

Magnetism in Matter

• Diamagnetism
• Has no permanent magnetic moment.
• The presence of an external magnetic field causes a

weak opposing magnetic moment in the material Hence the total field will be less than the external field

• Diamagnetism is present in all materials