# Magnetic Field and Work - Department of Chapter 28 1 Magnetic Field and Work Magnetic force is alwaysperpendicular to velocity Therefore B field does no work! Why? Because

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• PHY2049: Chapter 28 1

Magnetic Field and WorkMagnetic force is always perpendicular to velocity

Therefore B field does no work!Why? Because

ConsequencesKinetic energy does not changeSpeed does not changeOnly direction changesParticle moves in a circle (if )

( ) 0K F x F v t = = =r rr r

v Brr

• PHY2049: Chapter 28 2

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 xx x x x x x x x x x x x x x

Trajectory in a Constant Magnetic FieldA charge q enters B field with velocity v perpendicular to B. What path will q follow?

Force is always velocity and BPath will be a circle. F is the centripetal force needed to keep the charge in its circular orbit. Lets calculate radius R

FFvR

v

B

qF

v

• PHY2049: Chapter 28 3

x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x

Circular Motion of Positive Particle

BqF

v

2mv qvBR

=mvRqB

=

• PHY2049: Chapter 28 4

Magnetic Force Two particles of the same charge enter a magnetic field with the same speed. Which one has the bigger mass?

x x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xA B

mvRqB

=

Bigger mass means larger inertia, less acceleration, thus bigger radius

• PHY2049: Chapter 28 5

Work and Energy A charged particle enters a uniform magnetic field. What happens to the kinetic energy of the particle?

(1) it increases (2) it decreases(3) it stays the same(4) it depends on the direction of the velocity(5) it depends on the direction of the magnetic field

Magnetic field does no work, so K is constant

• PHY2049: Chapter 28 6

Cosmic Ray ExampleProtons with energy 1 MeV move earth B field of 0.5 gauss or 5 10-5 T. Find radius & frequency of orbit.

212

2KK mv vm

= =

2mv mKReB eB

= =

( )( )6 19 1327

K 10 1.6 10 =1.6 10 J

1.67 10 kgm

=

=

( )1

2 2 / 2v v eBf

T R mv eB m = = = = 760Hzf =

2900mR =

Frequency is independent of v!

• PHY2049: Chapter 28 7

Helical Motion in B FieldVelocity of particle has 2 components

(parallel to B and perp. to B)Only v = v sin contributes to circular motionv|| = v cos is unchanged

So the particle moves in a helical pathv|| is the constant velocity along the B fieldv is the velocity around the circle

v v v= +r r r

mvRqB

=

Bv

v

v||

• PHY2049: Chapter 28 8

Helical Motion in Earths B Field

Particles moving along field lines cause Aurora Borealis and Australis:http://science.nasa.gov/spaceweather/aurora/gallery_01oct03.html

• PHY2049: Chapter 28 9

Mass SpectrometerOriginally developed by physicists, now an important tool in chemistry,

biology, environmental studies, forensics, pharmaceutics, etc.

Sample is vaporized, broken into fragments of molecules, which are positively ionized. Positive ions are first accelerated by a potential difference V, and then their trajectories are bent by B. Varying B (sometimes V) allows ions of different masses to reach the detector.

• PHY2049: Chapter 28 10

Mass Spectrometer (simplified)Sample is vaporized, broken into fragments of molecules,

which are positively ionized. Positive ions are first accelerated by a potential difference V, and then their trajectories are bent by B. Varying B (sometimes V) allows ions of different masses to reach the detector.

Spectrometer determines mass from B (sometimes from V)

D

VBrqm2

)( 2=

2r

detector

• PHY2049: Chapter 28 11

Torque on Current LoopRectangular current loop in uniform magnetic field (lengths a & b)

Forces in left & right branches are 0 Force in top branch is into planeForce in bottom branch is out of plane

Equal forces give net torque!Bottom side up, top side down (RHR)Rotates around horizontal axis

= NiA magnetic dipole momentAssuming N turns = B, true for any shape!!

If plane tilted angle to B field = Bsin is angle between normal and B

B

a

b

a

b

( )Fd iBa b iBab iBA = = = =Plane normal is B( = 90)

• PHY2049: Chapter 28 12

Magnetic Dipole Moment

• PHY2049: Chapter 28 13

Torque ExampleA 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.

Rotation is always in direction to align with B field

30

( )22 23 3 5 3.14 0.03 0.0339A mi r = = = sin30 0.0339 2.5 0.5 0.042 N mB = = =

• PHY2049: Chapter 28 14

Magnetic Force A rectangular current loop is in a uniform magnetic field. What direction is the net force on the loop?

(a) + x (b) + y (c) zero (d) x(e) y

B

x

z

y

Forces cancel onopposite sides of loop

• PHY2049: Chapter 28 15

Electromagnetic Flowmeter

Moving ions in the blood are deflected by magnetic forcePositive ions deflected down, negative ions deflected upThis separation of charge creates an electric field E pointing upE field creates potential difference V = Ed between the electrodesThe velocity of blood flow is measured by v = E/B

E

• PHY2049: Chapter 28 16

Hall Effect: Do + or Charges Carry Current?

+ charges moving counter-clockwise experience upward force

Upper plate at higher potential

charges moving clockwise experience upward force

Upper plate at lower potential

Very quickly, equilibrium between electrostatic & magnetic forces is established and potential difference stops growing:

up driftF qv B= Hdown induced wVF qE q= = H drift "Hall Voltage"V v Bw= =

This type of experiment led to the discovery (E. Hall, 1879) that current in conductors is carried by negative chargesHall effect is used to measure moderate to moderately high B (10-4 T 3 T)It is also used to measure the speed of computer hard drive

• PHY2049: Chapter 28 17

FAQ on Magnetic Field and WorkMagnetic force does no work. But an electric motor (=a current loop in B) does work. Where does this work come from?

Magnetic force does no work on a moving chargeMagnetic torque on a current loop does work: W=

There is no net force, only torque, on a current loop (=magnetic dipole moment) in B. But two bar magnets (=collection of atomic magnetic dipole moments) attract each other. How come?

There is no net force, only torque, on magnetic dipole moment in uniform BWhen B is non-uniform, then there is net force. Can be shown that the direction of this force is such that magnetic dipole moment is attracted to the region of high B.

Magnetic force does no work. But when two bar magnets are attracted to each other, there must be work done by B. Something is not right here.

Magnetic force does no work on a moving chargeWhen a magnetic dipole moment moves as a result of force due to non-uniform B, then this force does work. There is no contradiction.

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