Magnetic Fields

Certain iron containing materials have been known to attract or repel each other.Compasses align to the magnetic field of earth.Analogous to positive and negative charges, every magnet has a north and a south pole (but always as a pair).Magnetic fields can be created by electrical currents.

Magnetic Field and Magnetic Force

vF rr,qB ∝

0// =⇒ BFBvrrr

vBF rrr,0 ⊥⇒≠ Bθ

θsin∝BFr

θ

B

v

( )vF rr,0<qB ( )vF rr

,0>qB

oppositedirection

FB

+q

Magnetic ForceBvFrrr

×= qBθsinvBqFB =

Magnetic Field, B: Tesla (T)=N/(C.m/s)=N/(A.m)

Right Hand Rule Magnetic Field Representation

Electric and Magnetic Forces

Electric forces act in the direction of the electric field, magnetic forces are perpendicular to the magnetic field.A magnetic force exists only for charges in motion.The magnetic force of a steady magnetic field does no work when displacing a charged particle.The magnetic field can alter the direction of a moving charged particle but not its speed or its kinetic energy.

BvFrrr

×= qB EFrr

qE =

Motion of Charged Particles in a Uniform Magnetic Field

Consider a positive charge moving perpendicular to a magnetic field with an initial velocity, v.The force FB

is always at right angles to v and its magnitude is,

qvBFB =

So, as q moves, it will rotate about a circle and FB

and v

will always be perpendicular. The magnitude of v

will always be

the same, only its direction will change.

Cyclotron FrequencyTo find the radius and frequency of the rotation:

∑ = rmaF

rvmqvBFB

2

==

qBmvr =

mqB

rv==ω

qBm

vrT π

ωππ 222===

The radial force

The angular speed

The period

Cyclotron frequency

Example 29.3Electron beam in a magnetic field

ΔV = 350 Vr = 7.5 cmq = e = 1.6 x 10-19 Cm = me = 9.11 x 10-31 kg

B = ?ω = ?

v

r Vevm

UK

e Δ=

=Δ

2

21

smm

Veve

/1011.12 7×=Δ

=

qBmvr = T

ervmB e 4104.8 −×==

sradmeB

mqB

e

/105.1 8×===ω

FBΔV

B

Guiding Charged Particles

Cosmic Rays in the Van Allen Belt

Helical Motion – vx , vy and vz

Complex Fields –Magnetic Bottles

Lorenz Force and ApplicationsEssentially it is the total electric and magnetic force acting upon a charge.

BvEFrrrr

×+= qqVelocity Selector

BEv

FF Be

=

=

Mass Spectrometer

EBrB

qm 0=

Magnetic Force on a Current Carrying Conductor

Magnetic force acts upon charges moving in a conductor.The total force on the current is the integral sum of the force on each charge in the current.In turn, the charges transfer the force on to the wire when they collide with the atoms of the wire.

Magnetic Force on a Current Carrying Conductor

Magnetic Force on a Current Carrying Conductor

( )BvFrrr

×= dqB q,

( )nALq dB BvFrrr

×=

qnAvI d=

BLFrrr

×= IB

For a straight wire in a uniform field

BsFrrr

×= Idd B

∫ ×=b

aB dI BsF

rrr For an arbitrary wire in an arbitrary field

A Special Case – arbitrary wire in a uniform field

∫ ×=b

aB dI BsF

rrr

BsFrrr

×⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫

b

aB dI

BLFrrr

×′= IB

The net magnetic force acting on a curved wire in a uniform magnetic field is the same as that of a straight wire between the same end points.

Closed Loop in a Magnetic Field

( ) BsFrrr

×= ∫ dIB

0=∫ srd

0=BFr

Net magnetic force acting on any closed current loop in a uniform magnetic field is zero.

Forces on a Semicircular Conductor

IRBILBF 21 ==

On the straight wire

dsIBdIdF θsin2 =×= Bsrr

On the curved wire

θθRdds

Rs==

θθdIRBdF sin2 =

∫∫ ==ππ

θθθθ00

2 sinsin dIRBdIRBF

IRBF 22 =

Directed out of the board

Directed in to the board

021 =+= FFFrrr

Torque on a Current Loop in a Uniform Magnetic Field

0

0

=

=×

BF

BLr

rr

For 1 and 3,

IaBFF == 42

( ) ( )

( )bIaB

bIaBbIaBbFbF

=

+=+=

max

42max 2222τ

τ

If the field is parallel to the plane of the loop

IAB=maxτ

For 2 and 4,

Torque on a Current Loop in a Uniform Magnetic Field

If the field makes an angle with a line perpendicular to the plane of the loop:

θτ

θθθτ

θθτ

sin

sinsin2

sin2

sin2

sin2 42

IAB

IabBbIaBbIaB

aFaF

=

=+=

+=

BAτrrr

×= I

Concept QuestionA rectangular loop is placed in a uniform magnetic field with the plane of the loop perpendicular to the direction of the field.If a current is made to flow through the loop in the sense shown by the arrows, the field exerts on the loop:

1. a net force.2. a net torque.3. a net force and a net torque.4. neither a net force nor a net torque.

Magnetic Dipole Moment

Aμrr I=

Bμτrrr

×=

(Amperes.m2)

If the wire makes N loops around A,

BμBμτrrrrr

×=×= coilloopN

Bμrr⋅−=U

The potential energy of the loop is:

Hall Effect

BvEqEBqv

dH

Hd

==

BdvV dH =Δ

nqAIvd =

tIBR

nqtIBV H

H ==Δ

SummaryThe right hand ruleMagnetic forces on charged particlesCharged particle motionMagnetic forces on current carrying wiresTorque on loops and magnetic dipole moments

For Next ClassReading Assignment

Chapter 30 – Sources of the Magnetic Field

WebAssign: Assignment 7