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

Winter 2004 OSU Sources of Magnetic Fields 1 Chapter 32

Sources Of Magnetic Fields • We learned two ways to calculate Electric Field – Coulomb's Force 1 dq  ˆ dE r Brute Force Calculation  2 4 r 0 • Q    enc E dA High symmetry  0 What are the analogous equations for the Magnetic Field? Winter 2004 OSU Sources of Magnetic Fields 2 Chapter 32

Sources Of Magnetic Fields • Now two ways to calculate Magnetic Field – Biot-Savart Law   ˆ I dl r  0 dB Brute Force Calculation  2 4 r r ˆ • Where unit vector displacement pointing dl from element to the field point P – Ampere’s Law     B dl I High symmetry 0 enc Winter 2004 OSU Sources of Magnetic Fields 3 Chapter 32

Biot-Savart Law • Biot-Savart Law; bits and pieces • Magnetic fields are generated by moving charge. I   ˆ I dl r dl  0 dB r  ^ 2 4 r r P N      7 4 10 0 2 A  0 - permeability of free space Winter 2004 OSU Sources of Magnetic Fields 4 Chapter 32

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   A B I y d l x  C z 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 Winter 2004 OSU Sources of Magnetic Fields 5 Chapter 32

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   A B I y d l x  C z 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 Winter 2004 OSU Sources of Magnetic Fields 6 Chapter 32

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   A B I y d l x  C z 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 Winter 2004 OSU Sources of Magnetic Fields 7 Chapter 32

B-field of  Straight Wire • Calculate field at point P R o dB using Biot-Savart Law : I y r q  ˆ μ I dy r  0 dB d y π r 2 4    2 2 2 dy dl and r R y  μ I dy θ ( )sin     0 B dB π r 2 4  q R R rd q         θ dy R θ dy 2 sin tan csc   2 r y R r Winter 2004 OSU Sources of Magnetic Fields 8 Chapter 32

B-field of  Straight Wire • Calculate field at point P R o dB using Biot-Savart Law : I y  μ I dy θ r q ( )sin     0 B dB π r 2 4 d y  q q 2 rd r d    dy dy   2 R Rr  μ I μ I      q   q θd 0 0 B dB sin cos 0 πR πR 4 4 q  0 μ I  0 B πR 2 Winter 2004 OSU Sources of Magnetic Fields 9 Chapter 32

B-Field of a Circular Loop Circular loop of radius R carries d l • dB q current I . What is B along the r R axis of the loop: q x • Magnitude of dB from element dl : x R   r 0 I 0 I dl 4  dl dB dB = 4  r 2 = x 2 + R 2 x • What is the direction of the field? • Symmetry is B in x -direction. μ I dl  q 0 dB 2 cos  x π x R 2 4 Winter 2004 OSU Sources of Magnetic Fields 10 Chapter 32

B-Field of a Circular Loop μ I d l dl •  q 0 dB 2 cos dB q  x π x R r 2 4 R q R R q   cos x  r 2 2 x z R R μ I R   2   q 0 B ( dl Rd ) x π 3 x 4  0 2 2 2 ( x R ) μ IR 2  0 B x 3  2 2 2 2( x R ) At the center of the loop z = 0 therefore μ I  0 B x 2 R Winter 2004 OSU Sources of Magnetic Fields 11 Chapter 32

B-field of Concentric Semicircles • Homework Problem #9 d l A current I flows in the direction b shown. What is the Magnetic Field a At point P due to the current in P the inner semicircle (at r = a )?  μ I  1) 3) 0 0 B a 2 a μ I μ I   0 0 B 2) B 4) μ I a a  2 a 4 a 0 B 5) a a Winter 2004 OSU Sources of Magnetic Fields 12 Chapter 32

B-field of Concentric Semicircles • Homework Problem #9 d l A current I flows in the direction b shown. What is the Magnetic Field a At point P due to the current in P the outer semicircle (at r = b )?  μ I  1) 3) 0 0 B b 2 b μ I μ I   0 0 B B 2) 4)  μ Ib b b 8 b  4 b 0 B 5) b 4 Winter 2004 OSU Sources of Magnetic Fields 13 Chapter 32

B-field of Concentric Semicircles • Homework Problem #9 d l A current I flows in the direction b shown. What is the Magnetic Field a At point P due to the current in P the outer semicircle (at r = a )? μ I dl  0 dB directed in the screen x π R 2 4 μ I  μ I    q 0  B 0 ( dl Rd ) 0 B a πR 2 x 4 4 R Winter 2004 OSU Sources of Magnetic Fields 14 Chapter 28

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 15 Chapter 32

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 16 Chapter 32

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 B  0 M C T • If the material is lowered below the Curie temperature, it will maintain its magnetic alignment. Winter 2004 OSU Sources of Magnetic Fields 17 Chapter 32

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 Winter 2004 OSU Sources of Magnetic Fields 18 Chapter 32