04/19/23 Lecture XII 1

Physics 122

Magnetism

04/19/23 Lecture XII 2

Magnetic force on moving charge• Magnetic force F is

perpendicular to the velocity v of a particle with charge q

F=qvB sin

• Magnetic field does not change the magnitude of particle’s velocity, only its direction. Charged particles move in circles in constant magnetic field

vF

BvqF

04/19/23 Lecture XII 3

Magnetic force on moving charge• Circular motion • Magnetic force =

Centripetal forceF=qvB

• Centripetal acceleration a=v2/R

• Newton’s second lawF=maqvB=mv2/R

qB

mvR

In experiments B is known,q=e most of the time, Measure R- measure mvFor a given v measure m – magnetic spectrometer

04/19/23 Lecture XII 4

Cyclotron frequency

qB

mvR

vTR 2 :circle full

Bm

q

Tf

qB

mT

vTqB

mv

2

1

2

2

Frequency does not depend on v!For a given particle type (q,m=const) it depends only on B

04/19/23 Lecture XII 5

Particle selector• Crossed E and B fields:

• For one given velocity electric force exactly compensates magnetic force

• Only particles with this value of velocity emerge from particle selector

BvqEqF

+ + + +

- - - -

+

BvqEqF

B

Ev

BvqF

EqF

v

04/19/23 Lecture XII 6

Crossed fields

• Electric field changes kinetic energy of charged particle

qV=mv2/2

• Magnetic field does not change KE, only direction

• Lorentz force:

BvqEqF

04/19/23 Lecture XII 7

Torque

sinFrFr

sinrFrF

04/19/23 Lecture XII 8

Current loop in magnetic field

IaBF 1

IaBF 2

211

bF

222

bF

IABIabB

bIaB

bIaB

22

21

IAB

• A=ab – area of the loop• Magnetic field exerts a torque

on a loop parallel to the magnetic field:

04/19/23 Lecture XII 9

Current loop in magnetic field

– angle between magnetic field and a perpendicular to the loop !!!

• Magnetic field orients a loop current perpendicular to B.

• N loops:211

bF

222

bF

IAB

2sin11

bF

2sin22

bF

sinIAB

sinNIAB

04/19/23 Lecture XII 10

Magnetic moment• Current loop (area A, current I,

number of turns N) in constant magnetic field B experiences torque

• Define magnetic moment perpendicular to the plane of the loop

• Torque

NIAB

ANI

B

NIAB

04/19/23 Lecture XII 11

Magnetic moment• Equilibrium position for current loop

is when B is perpendicular to the face of the loop, or when magnetic moment is parallel to B

• Inside magnetic field magnetic moments orient parallel to the field

• Galvanometer: B is know – measure I

BBU

cos

04/19/23 Lecture XII 12

Magnetic field created by currents• Electric currents – moving electric charges - create magnetic field

• Stationary electric charges do not create magnetic filed

• First right hand rule– Thumb along the current

– Wrap your fingers around the wire

– fingers show the direction of the magnetic field

04/19/23 Lecture XII 13

Magnetic field created by current

• Magnetic field B created by current I at a distance r from the conductor is

Tm/Amagneticpermeability of free space

r

IB

2

0

04/19/23 Lecture XII 14

Ampere’s Law

• In a way similar to Gauss’s law for E and Q

• Not flux, integral over closed path

• Only component of B parallel to dl contributes to integral

enclIldB 0�

04/19/23 Lecture XII 15

Application of Ampere’s Law• Thick wire of radius R carries current I. What

is B(r ) in and outside of the wire?• Current through unit area:

• Current through circle radius r<R

• Path integral over radius r

• Ampere’s law

• For r>R – same as thin wire

20

2

2

00

2)(

2

R

IrrB

R

rIIrB encl

rB 2

2

2

R

rI

2R

I

20

2

2

00

2)(

2

R

IrrB

R

rIIrB encl

r

IrB

2

)( 0

04/19/23 Lecture XII 16

Practical application: coaxial cable

• Cylindrical braid carries return current magnetic field outside is zero.