1. Sources of the Magnetic Fieldbb
Lesson 8
Biot-Savart Law
Force between parallel conductors
Ampere’s Law
Use of Ampere’s Law 1

2. E Field
Interaction between stationary charges is mediated by an Electric Field
E Field 2

3. Interaction between currents is mediated by a Magnetic Field
B Field
Interaction between currents is mediated by a Magnetic Field
B Field 3

4. Electric Dipoles align along an E Field- +
E Dipole 4

5. Magnetic Dipoles align along a B Field
B Dipole 5

6. Biot-Savart Law
Biot-Savart Law 9

7. E and B Fields due sources
For a point source charge d
q the
E and B Fields due sources
electric field produced at a position r
from the source is 1 dq 1 dq d E = ˆ r = r 4 pe r 2 4 pe r 3
For a length of wire d s
with current I (
this is the same as charge per unit length d
q with velocity v )
the magnetic field produced at a position r
from
the source d s is m I m I d B = d s ˆ r = d s r 4 p r 2 4 p r 3 7

8. B Field due to source m = 4 p 10 - 7 Tm A
Permeability 8

9. B Field due to loop 10

10. Right Hand Rules
Right Hand Rule I
Need B and v then get FB 11

11. Right Hand Rule II
Need I get B 12

12. Magnetic Fields from conductors of different shapes
B Field from wire
Magnitude of Magnetic Field about
thin conductor k m I ( ) ( ) B r = B r = 2 p r é ò æ ò ù ö ê ç m I ÷ ú ( ê B r ) = d B = ç
sin q ds ÷ k ú  ds p r 2 R ê ç 2 ÷ ú è ø ê - ú ê r ú ê
sin q =
sin ( p - q ) = ; r = s 2 + R 2 ú r s 2 + r 2 ê ú ê ò ú ê æ m I ö r ú B ( r ) ê = ç ÷ ds k ú è 2 p ø ( ) 3 ê 2 + ú s r 2 2 ê - ú ë ê û ú 13

13. B Field from closed wire segment X B = m I 4 p r q 14

14. B Field from closed loop15

15. Picture B I 17

16. Two parallel conductorsk
Parallel wires I
Two parallel conductors j i B2 I1 a B1 I2 L 18

17. Force on wire 1 due to field of wire 2 ´ = B F = ´ B m B = - i 2 p a m
L j B 1 2 2 m I2 B = - i 2 2 p a m I1 I2 m I1 I2 L L F = - j i = + k 1 2 p a 2 p a
Force on wire 2
due to field of wire 1 F = I2 L B = - I2
L j B 2 1 1 m I2 B = - i 1 2 p a m I1 I2 m I1 I2 L L F = j i = - k 2 2 p a 2 p a 19

18. Two parallel conductorsk
Parallel wires II
Two parallel conductors j i B2 F1 a B1 F2 20

19. Parallel Currents Attract Anti Parallel Currents Repel
Parallel wire rules
Parallel Currents Attract
Anti Parallel Currents Repel 21

20. Definition of Amp a = 1 m = = = 1 A m N F = = 2 ´ 10 p m 2= 2 10 - 7 p m 2
This defines the Amp . as :
The current flowing when a force of 2 10 - 7 N m
measured between wires one meter apart
This in turn defines the Coulomb as :
The quantity of charge that flows through any
cross section of a conductor in one second when
a steady current of one amp is flowing . 23

21. ò ò Gauss's Law for E Recall Gauss ' s Law Q = E · d A = EA e = as E
Gaussian surface = as E
constant and E || d A
everywhere on surface
For magnetism ò B d A =
as magnetic field has ALWAYS been observed to
be produced by magnetic dipoles 24

22. In electrostatics, potential difference is defined byò b
Gauss's Law for B V = E d s ab a
where integral is evaluated along ANY path
starting at a
and finishing at b ,
thus ò E d s = as E
is a conservative force per unit charge
For magnetism, in general ò B d s
and ò ò B d s = B ds
Amperian Loop
Amperian Loop : B =
constant and B d s 25

23. ò ò B R Amperes Law I ( ) ( ) B · d s = B R ds for steady current æ m
Amperian Loop
Amperian Loop
for steady current B æ m ö ( ) = ç ÷ p = m I 2 R I R è p ø 2 R
true for ANY closed path
Amperes
Law 26

24. sum of all currents threading
Amperes Law II ò B s = m d I
Total = I
sum of all currents threading
loop
Total 27

25. Gauss’s Law is Equivalent to Coulombs Law
Equivalence of Laws
Gauss’s Law is Equivalent to Coulombs Law
Biot-Savart Law is Equivalent to Amperes Law. 28

26. Toroids and Solenoids I29

27. Toroids and Solenoids IIò ò ò B B d s = B d s = B ds = BL
loop
path 1
path 1 ò w 2 B d s = m NI = BL
loop 1 N B = m I = m nI 3 l L 4
Amperian Loop 30

28. Picture 31

29. Toroids and Solenoids III
One can perform the same Amperian
Toroids and Solenoids III
calculation for the Toroid
and get the same result B = m nI
where n
is the number of turns per length N = n =
but
now , R
Radius of Toroid 2 p R
instead of N n = ,
for Solenoid L 32

30. Galvanometer
Galvanometer 36