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

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

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

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

Magnetic Dipoles align along a B Field B Dipole 5

Biot-Savart Law Biot-Savart Law 9

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

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

B Field due to loop 10

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

Right Hand Rule II Need I get B 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

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

B Field from closed loop 15

Picture B I 17

Two parallel conductors k Parallel wires I Two parallel conductors j i B2 I1 a B1 I2 L 18

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

Two parallel conductors k Parallel wires II Two parallel conductors j i B2 F1 a B1 F2 20

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

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

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

In electrostatics, potential difference is defined by ò 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 ¹ and ò ò B · d s = B ds Amperian Loop Amperian Loop : B = constant and B d s 25

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

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

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

Toroids and Solenoids I 29

Toroids and Solenoids II ò 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

Picture 31

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

Galvanometer Galvanometer 36