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Physics IClass 23

Magnetic Forceon Moving Charges

Rev. 07-Apr-04 GB

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Hendrick Antoon Lorentz

H.A. Lorentz1853-1928

Hendrick A. Lorentz was a Dutchphysicist who refined certain aspects ofelectromagnetic theory. He, along withIrish mathematical physicist George F.FitzGerald, deduced fundamentalproperties of the electromagneticbehavior of moving bodies that formedthe basis of Einstein’s Special Theoryof Relativity.

The force of a magnetic field on amoving charge is sometimes called theLorentz Force.

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Important Facts About Velocity and Net Force Vectors (Review)

v F

v F

Same direction: speeding up.

Opposite directions: slowing down.

Right angles: changing direction, same speed.v

F

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Vector Cross Product (Review)

bac ; )sin(|b||a||c|

The direction comes from theright-hand rule. It is at a rightangle to the plane formed by a

and b. In other words, the cross

product is at right angles to both

a and b

. (3D thinking required!)

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Drawing 3D Vectors in 2D

X

Z

Y

+Z-Z

+Y-Y

-X

+Z is out of page

+X

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Magnetic Force on aMoving Charge

BvqF

q: charge of the particle (C; + or –)v:velocity of the particle (m/s)B:magnetic field (T) Force is at a right angle to velocity. Force is at a right angle to magnetic field.

Important: If q is negative, that reverses the direction of force.

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An ExampleAn Electron in a Magnetic Field

X

Z

Y

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Analysis of the Magnetic Force

X

Z

Y

F

BvqF

We will evaluate this expression beforethe electron starts turning.

First, evaluate Bv. In this case, they are 90° apart, so all we

need is the direction. v is +X, B

is –Z, so Bv

is +Y.

Next, we need to account for q. This is an electron, so q isnegative. Therefore, the magnitude of the force is (e v B) and thedirection is –Y.

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Uniform Circular Motion

Speed stays constant because acceleration isalways perpendicular to velocity.

The electron travels in a circle at a constant speed.

As the electron turns, so does the force vector.

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The Radius of the Circle

v

r

F

A l t h o u g h t h e d i r e c t i o n s o f t h e v e c t o r s a r ec h a n g i n g , t h e m a g n i t u d e s s t a y t h e s a m e .

rv

mamF2

BvqF

rv

mBvq2

Bqvm

Bvqv

mr2

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The Period and Frequency

v

r

FT h e c i r c u m f e r e n c e o f t h e c i r c l e i s 2 r .

Tr2

TimeDistance

v

Bqm2

vBqvm

2

vr2

T

m2Bq

T1

f

mBq

f2

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Bubble Chamber

The red and green lines in the figureto the left are tracks of chargedparticles in a bubble chamber. Eachcharged particle makes a trail oftiny bubbles as it moves in thechamber. There is a magnetic fieldof 1.0 T directed into the page.

What are the signs of the charges of the particles?

Why do they spiral inward?

What are they?

What created them at the points where the tracks start?

Bqvm

r

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The Aurora

There is no acceleration in the direction of themagnetic field line. (Why?)

The component of velocity in the direction of thefield line remains constant. (Why?)

The component of velocity at a right angle to thefield line continually changes direction. (Why?)

The result is that the charged particle (electron)travels in a spiral path along the magnetic field line,giving off light when it hits the atmosphere.

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The AuroraAs Seen from the Space Shuttle

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The Effect of the Solar Windon the Magnetic Field of Earth

Energetic charged particles travel alongmagnetic field lines on the sun.Some escape into interplanetary space.These are called the solar wind.The solar wind interacts with the magneticfield lines of Earth and distorts them. Thecomplex interaction of flowing chargedparticles with the electromagnetic field iscalled Magneto-Hydrodynamics or MHD.

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Class #23Take-Away Concepts

1. The magnetic (Lorentz) force on a moving, charged particle:

BvqF

2. The magnetic force cannot change a particle’s speed, only thedirection of its velocity.

3. Radius and angular frequency of a charged particle in uniformcircular motion in a magnetic field:

Bqvm

r

mBq

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Class #23Problems of the Day

___1. A charged, non-magnetic particle is moving in a uniformmagnetic field. Which of the following conditions (if any)would cause the particle to speed up?

A) The velocity of the particle is at a right angle to the magneticfield.B) The velocity of the particle is in the same direction as themagnetic field.C) The velocity of the particle is in the opposite direction as themagnetic field.D) Any of the above (A-C) would cause the particle to speed up.E) None of the above; the magnetic force cannot cause theparticle to speed up.

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Class #23Problems of the Day

2 . A n e l e c t r o n i s t r a v e l i n g i n a v a c u u m t u b e a t 1 . 4 x 1 0 7 m / s i n ah o r i z o n t a l d i r e c t i o n t o w a r d t h e s o u t h . T h e r e i s a c o n s t a n tm a g n e t i c f i e l d i n t h e t u b e w i t h a m a g n i t u d e o f 0 . 5 g a u s s . T h ed i r e c t i o n o f t h e m a g n e t i c f i e l d i s t o w a r d t h e n o r t h a n d 3 0 º d o w n( t o w a r d t h e g r o u n d ) . W h a t a r e t h e m a g n i t u d e a n d d i r e c t i o n o f t h em a g n e t i c ( L o r e n t z ) f o r c e o n t h e e l e c t r o n ? ( 1 T = 1 0 , 0 0 0 g a u s s . )

D o w n

- ev

U p

B

S N3 0 °

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Activity #23Magnetic Field and Force

Objective of the Activity:

1. Consider the implications of the magnetic force on speed anddirection of a charged particle.

2. Determine the direction and magnitude of the magnetic field atyour table in the classroom using a compass, a coil of wire, apower supply, and a current meter.