Lesson 7 Magnetic Fields Magnetic Force on moving charge Motion of charge in Magnetic Field Magnetic Force on current carrying conductor Torque on current carrying loop Hall Effect Magnetic Devices 1

Magnetic Force Some metallic objects cause other metallic objects to accelerate Thus must be producing a force This force is different to Gravitational Force and Electric Force Can plot the directions of lines field By alignment of magnetic dipoles MAGNETIC FORCE 2

Magnetic Field Lines 3

Properties of Magnetic Fields Moving charge (current) is effected Force proportional Q and v FB = 0 if v parallel to Magnetic Field B FB perpendicular to v and B FB on positive charge opposite to that on negative charge FB proportional to Sin of the angle v makes with B Properties of Magnetic Fields 4

Mathematical Expression 5

Vector Product I ( ) ( ) ( ) Vector Product A = A i + A j + A k B = B x y z B = B i + B j + B k x y z i j k A A A A A A A ´ B = y z A A A = i - x z j + x y k x y z B B B B B B y z x z x y B B B x y z ( ) ( ) ( ) = A B - A B i - A B - A B j + A B - A B k y z z y x z z x x y y x 6

Vector Product II A ´ B = - Þ parallel to ( AB ) Sin q if Þ parallel to ( AB ) Sin q if is perpendicular to and 7

Unit vector products 8

Work done by Magnetic Field Work Done by Magnetic Force on charge dW B = F · d s Q v ´ ( ) As is perpendicular to and is parallel to 9

Work done by Electric Field Compare to Electric Force Work done by Electric Field 10

- Implications DK 1 1 = m = vf m vi 2 2 Þ vi = vf i . e . Thus Kinetic Energy of charge is not changed by FB Potential Energy is not changed 1 1 - = DK m 2 2 = vf m vi 2 2 Þ vi = vf i . e . speed is constant under influence of B but v can change direction 11

SI units 12

ò Magnetic Flux [ ] Magnetic Flux F B For constant magnetic field over flat area B with area vector A F = B · A=ABcosq B in general ò F = B · d A B surface [ ] F = Tm 2 = Wb ( Weber ) B 13

Right Hand Rule for Magnetic Force 14

Magnetic Force on Moving Charge B FB v Q k j i F = Q v ´ B = - Q v j ´ B k B y z i j k = - Qv B j ´ k = - Qv B 1 = - Qv B i y z y z y z 1 15

Mathematical expression 16

Uniform Circular Motion of Charge 17

Mathematical analysis v 2 F QvB a = = = Mathematical analysis c r m m mv 2 Þ = QvB r mv r = radius of motion QB v = r w BQ \ w rad = angular speed = angular freqency ( ) s m w BQ f = = frequency ( cps = Hz ) 2 p 2 p m 1 2 p m T = = period of the motion . f BQ 18

Magnetic Bottles magnetic bottles 19

Magnetic Force on Current Force on current carrying conductor in a uniform constant magnetic field : ò b F = I d s ´ = I l ´ B B B a ( ) where l = r - r b a é æ ö ù ds dQ v = = = ( ) = ê dQ dQ s ˆ ç ds ÷ s ˆ I ds s ˆ I d s ú ë è ø û dt dt 25

Want to find total force on charge as it flows from a to b Vector Integral I ds b rb B is constant ra vector integral a Current I Want to find total force on charge as it flows from a to b 22

Vector Integral II ds b rb B rb- ra ra a I 23

Vector Integral III Sum ds 24

Magnetic Force on Current Loop I B k j i 26

Magnetic Force on Current Loop II Total Force on Loop Magnetic Force on Current Loop II 27

Torque on Current Loop I  B I  k z  j i   y 28

Torque on Current Loop II 29

Torque on Current Loop III 30

Magnetic Moment t = IAB k t = M ´ B M = I A , where A Torque on Current Loop in yz plane in a uniform magnetic field in positive y direction t = IAB k In general the torque on a current loop in magnetic field is t = M ´ B M = I A , where A is the area vector of the loop , with the orientation given by the current flow . 31

A=Ai: Area vector with orientation Oriented Area Vector I k A j i 32

I Hall Effect Hall Effect A t vd d FE FB B V - E + 33

Mathematical analysis At Equilibrium Mathematical analysis F = F E B V E = d F = eE = F = v eB E B d E \ v = d B J I Remember v = = d ne neA I E BI BI BI \ = Þ n = = = neA B EAe V Vte dte d 34

can measure vd by measuring the PD V. Measurement can measure vd by measuring the PD V. 35

Lorentz Force 36