Magnetic Field and Work - Department of · PDF filePHY2049: Chapter 28 1 Magnetic Field and Work ÎMagnetic force is alwaysperpendicular to velocity Therefore B field does no work!

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 B⊥rr


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. Let’s calculate radius R

FFv

R

v

B

qF

v


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

=


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

ABBoth masses are equalCannot tell without more info

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


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


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 13

27

K 10 1.6 10 =1.6 10 J

1.67 10 kgm

− −

−

= × ×

= ×

( )1

2 2 / 2v v eBf

T R mv eB mπ π π= = = = 760Hzf =

2900mR =

Frequency is independent of v!


Helical Motion in B FieldVelocity of particle has 2 components

(parallel to B and perp. to B)Only v⊥ = v sinφ contributes to circular motionv|| = v cosφ 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

φ

v⊥

v||


Helical Motion in Earth’s B Field

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


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.


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


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

µ = NiA ⇒ “magnetic dipole moment”Assuming N turnsτ = µB, true for any shape!!

If plane tilted angle θ to B fieldτ = µBsinθθ is angle between normal and B

B

a

b

a

b

( )Fd iBa b iBab iBAτ = = = =Plane normal is ⊥ B(θ = 90°)


Magnetic Dipole Moment


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 µ with B field

30°

( )22 23 3 5 3.14 0.03 0.0339A mi rµ π= = × × × = ⋅

sin30 0.0339 2.5 0.5 0.042 N mBτ µ= = × × = ⋅


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


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


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 w

VF 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


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=τ∆θ

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.

Documents

Magnetic Field and Work - Department of · PDF filePHY2049: Chapter 28 1 Magnetic Field and Work ÎMagnetic force is alwaysperpendicular to velocity Therefore B field does no work!