B here τ== = = Fd iBa b iBab iBA ()

Torque on Current Loop

?Consider rectangular current loop

Forces in left, right branches = 0

Forces in top/bottom branches cancel No net force! (true for any shape)

b

a

B

b

? But there is a net torque!

Bottom side up, top side down (RHR) Rotates around horizontal axis

= Fd = (iBa)b = iBab = iBA

a Plane normal is B here

? = NiA "magnetic moment" (N turns)

True for any shape!! Direction of given by RHR

PHY2054: Chapter 19

21

General Treatment of Magnetic Moment, Torque

? = NiA is magnetic moment (with N turns)

Direction of given by RHR

?Torque depends on angle between and B

= B sin

PHY2054: Chapter 19

22

Torque Example

?A 3-turn circular loop of radius 3 cm carries 5A current in a B field of 2.5 T. Loop is tilted 30? to B field.

30?

? = 3i r2 = 3? 5? 3.14? (0.03)2 = 0.0339 A m2

? = B sin 30 = 0.0339? 2.5? 0.5 = 0.042 N m ?Rotation always in direction to align with B field

PHY2054: Chapter 19

23

Trajectory in a Constant Magnetic Field

?A charge q enters B field with velocity v perpendicular to B. What path will q follow?

Force is always velocity and B

Path will be a circle. F is the centripetal force needed to keep the charge in its circular orbit. Let's calculate radius R

x x x x x x x x x x x x x x

x x x x x x x x x x x x xBx

x x x x x x x x x x x x x x

x x x x x x x x x x x x vx x

vF

F

R

PHY2054: Chapter 19

v Fq

24

Circular Motion of Positive Particle

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x xB

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

mv2 = qvB R

R = mv qB

PHY2054: Chapter 19

v Fq

25

Cosmic Ray Example

?Protons with energy 1 MeV move earth B field of 0.5 Gauss or B = 5 ? 10-5 T. Find radius & frequency of orbit.

K

=

1 2

mv2

v=

2K m

( )( ) K = 106 1.6?10-19 =1.6?10-13 J

m = 1.67 ?10-27 kg

R = mv = 2mK eB eB

R = 2900 m

f

=1 T

=

v

2 R

=

v

2 (mv / eB)

=

eB

2 m

f = 760 Hz

Frequency is independent of v!

PHY2054: Chapter 19

26

Helical Motion in B Field

?VeloGcityG ofGparticle has 2 components

v = v& + v (parallel to B and perp. to B)

v||

Only v = v sin contributes to circular motion

v|| = v cos is unchanged

v

?So the particle moves in a helical path

v

B

v|| is the constant velocity along the B field

v is the velocity around the circle

R = mv qB

PHY2054: Chapter 19

27

Helical Motion in Earth's B Field

PHY2054: Chapter 19

28

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