1. Magnetic field B (Tesla - College of Engineering · 1. Magnetic field B (Tesla) 2. Force acting...

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Transcript of 1. Magnetic field B (Tesla - College of Engineering · 1. Magnetic field B (Tesla) 2. Force acting...

Page 1: 1. Magnetic field B (Tesla - College of Engineering · 1. Magnetic field B (Tesla) 2. Force acting on a moving charge in B field: F q 3. Cyclotron ... negative plate of a capacitor.

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1. Magnetic field B (Tesla)

2. Force acting on a moving charge in B field: Fq

3. Cyclotron motion Ω

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Historical milestones • 5-th century B.C. Greeks knew about rocks in Magnesia

(Western Turkey) that attract each other. They named rocks “magnetite”, chemical formula

• ~110 A.D. Chinese use compass to navigate • Late 12-th century. Europeans “invent” loadstone=leading

stone • ~1600 William Gilbert. The Earth is a giant magnet • ~1800 Hans Christian Oerstead. Compass is sensitive to

electric current • >1800 Ampere, Faraday, Maxwell. Universal theory of

magnetism

43OFe

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Magnetic field

• We recognize the fact that the magnetic field is established around us by placing a compass needle (test magnet) that would point to magnetic SOUTH

• Postulate: Needle’s North pole shows the direction of magnetic field (towards South Pole)

• Compass cannot say anything about the magnetic field magnitude

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Bar magnet (Permanent magnet)

North pole is marked by RED color South pole is marked by BLUE color

Magnetic field flux lines can be visualized with iron filings acting as little magnets Magnetic field flux lines are continuous loops, they emerge from N pole, arrive to S pole and “continue” inside the magnet Magnetic flux lines do not touch and cross each other One cannot get a magnetic monopole – say S Pole; N and S poles always coexist Magnetic flux lines show the direction of magnetic field at every spot Magnetic flux lines act as rubber bends – try to squeeze themselves

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Earth magnetic field – Permanent magnet

• North Magnetic Pole is located on Canadian territory

• North Magnetic Pole is off the geographic Pole; the separation is ~17 deg

• North Magnetic Pole should be actually called South Magnetic Pole, for doing physics

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North (South) magnetic pole

This figure shows the path of the North Magnetic Pole since its discovery in 1831 to the last observed position in 2001. During the last century the Pole has moved a remarkable 1100 km. What is more, since about 1970 the NMP has accelerated and is now moving at more than 40 km per year. If the NMP maintains its present speed and direction it will reach Siberia in about 50 years. Such an extrapolation is, however, tenuous. It is quite possible that the Pole will veer from its present course, and it is also possible that the pole will slow down sometime in the next half century.

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Properties

• Like poles repel each other

• Unlike poles attract

each other

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Why North magnetic Pole should be South magnetic Pole?

Because unlike poles attract each other, the needle’s North end points towards the South Pole in the Earth magnetic field, but we call this “South Pole” the North Pole, nevertheless

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Force on a moving charge in B field

θsin||];[ qVBFBVqF qq =×=

(Vector product)

This equation can be used as a definition of B field

θ

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Direction of the vector product

Use ALWAYS right hand (right hand rule)

3 2[ ]= ×1V V V

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Direction of the vector product

Use ALWAYS right hand (right hand rule)

3 2[ ]= ×1V V V

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Force on a moving charge in B field

θsin||];[ qVBFBVqF qq =×=

F=0 F=qVB F=qVBsin()

Examples:

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Important points regarding magnetic force Fq

1) POLARITY of magnetic force depends on charge; the force acts in opposite directions for electrons

(“-”charge) and protons (“+” charge) 2) Magnetic force is zero for - non-moving charge - for motion along the B field direction 3) Notation for directions: away, towards you, down to the page out of page B B

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Nikola Tesla Inventor (1856 – 1943)

Units of B field

Unit for B field magnitude is Tesla (T), after Nikola Tesla, Croatian-born US engineer

B=F/ (q v sin ) [1 T ] =N/C/(m/s) or since 1 C/s= 1Amper [1 T ]= N/(Am) Non-SI unit for B is Gauss (G): I G=10-4 T

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Work done by B field Magnetic field does

not do work over a particle and does not change its kinetic energy as the force is always perpendicular to the velocity (scalar product of force and displacement vectors or F and velocity is zero)

/

| | | | cos( / 2) 0

Work second F VFor mag field forceV B π

≡ ⋅

⋅ ⋅ =

V

B

[ ]F q V B= ×

Is q positive here?

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Comparing effects of E field and B field

• Positive charge is shifted along E field towards negatively charged plate. The trajectory is a parabola

• Positive charge is

shifted upwards. The trajectory is a circle

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Velocity separator and making a straight trajectory for a charged particle

E/BVBVqEq =⇒⋅⋅=⋅

Particle detector/ collector

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Circular motion in B field

1) Centripetal acceleration, centripetal force 2) Radius of the orbit 3) Radial frequency

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Note on centripetal (“center seeking”) acceleration

Even though (circular motion with constant speed), the velocity changes its direction so that

0| ( ) | | ( ) |V t V t=

0( ) ( ) 0V t V t V− = ∆ ≠

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Centripetal acceleration

car

VtV

rtV

VV

==∆∆

∆=

2

For circular motion, the resultant force should always be directed toward the center

Direction of ac: Toward the center 0( ) ( )V t V t V− = ∆

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Derivation of expressions for the radius of gyration and angular (cyclotron) frequency

2

; ;c qV mVma F m qVB rr qB

= ⇒ = ⇒ =

mqB

=Ωsec

r VSo much of a distancecovered per

⋅Ω =

( / )V V qBr mV qB m

Ω = = =

22 ( ) [ ]( ) sec

. . sec( 2 s in )

radianf angular frequencyT period

ang freq so many radians perthere are radian one circle

ππ

π

= = = Ω

=

1 1( ) [ ]( )sec

f frequencyT period s

so many turns per

=

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Cyclotron frequency of gyration

mVrqB

=

Radius of the orbit

mqB

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Aurora Borealis

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Aurora borealis is a global phenomenon

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Plasma (electrons & protons) is ejected from the Sun toward the Earth

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Particles from the Sun are diverted by the Earth magnetic field, but some of them precipitate into the upper atmosphere, causing aurora borealis

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Example of a problem: aurora Aurora borealis occurs because energetic

electrons and protons bombard the atmospheric gases at the height of ~ 100-150 km. Assume that the precipitating protons and electrons have energy of 5 eV and move almost perpendicular the Earth’s magnetic field (0.5 G). (a) Find the radius of proton (electron) gyration and the radial frequency of their motion. (b) If you are looking along the Earth’s magnetic field, what is the direction of the proton (electron) gyration, clockwise or counterclockwise?

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Solution: Electron motion

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Solution: Proton motion

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Applications: The mass spectrometer

• Ions are accelerated in electric field

• Ions travel in B field along trajectories of different radius, depending on mass

• Detector selects particles of proper mass, certainly you have to calibrate the instrument

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Example of a problem: A proton is released from rest

at point A. The proton is then accelerated toward the negative plate of a capacitor. It leaves the capacitor at B through a small hole in the plate. The electric potential of the positive plate with respect to the negative plate is 2.1 kV. Outside the capacitor, there is constant magnetic field of 0.1 T, directed perpendicular to the plates. Find: (a) the speed of the proton at point B and (b) the largest separation of the proton from B.

Point B

A

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Solution: Mass spectrometer

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Comments of particle motion in electric field

chargeForce Electric

q= =

electrFE

1) Electric field is directed from “+” charge towards ” – “ charge:

2) Positive charge (free of other forces) moves in the direction of E field

3) Negative charge (free of other forces) moves against the direction of E field

+ -

E

E E

- +

Positive q Negative q

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Conceptual analogy between E & B fields

Electric field Magnetic field

In both cases, the field characterizes force acting on a particle

(positive)

(shown is force on a negative particle)

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Summary of equations

θsin||];[ qVBFBVqF qq =×=

rVac

2

=

qBmVr =

mqB

( & )EVel Crossed E BB

=