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Magnet Forces and Magnetic Fields

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1) Magnets and Magnetic Fields

a) Natural permanent magnets– Like poles repel, unlike attract– come in pairs (no monopoles)– Interact with earth;

define N (or north-seeking) pole as pole attracted to North pole of earth

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b) Magnetic field direction:

- direction of force on N pole

B

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c) Field of dipole

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d) Magnetostatics for poles

(identical to electrostatics for charges)– 2 types: N, S vs +,-– Unlike attract, like repel– Inverse square law– Force along joining line– Magnetic Field:

B

F

qM

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e) Why study magnetism?– No monopoles (yet)– Poles (dipoles) produced by moving charges (no direct

control of pole distribution)– Charges affected by magnetic field

i.e. fundamental unit is still charge; want magnetic field due to charge, and force on charge due to magnetic field

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2) Magnetic field due to current (direction)

• Oersted (1820)

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B I

r

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3) Magnetic force on current

Direction from RHR1: B fingers, I thumb, F palm

F

IB

Force per unit length

defines B

a) Orthogonal case

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Bearth .5 gauss 5 10 5 T

Bfridge magnet .01T

Bsuper conducing 1 10 T

B F

I

N

Amtesla (T)Units:

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b) General case

F

LIBsin

Force per unit length

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4) Force between parallel wires

B I1

d;

F

I2B

F

k

I1I2

d

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Attraction or repulsion?

Does it depend on reference frame?

+

+

FE

FE

FB

FB

v

v

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v

v

+

+

-

-

+

+

-

-

+

+

-

-

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• Define Ampere as the quantity of current that produces a force per unit length of 2 x 10-7 N/m for separation of 1 m

F

k

I1I2

d

k 1

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8.988 109Nm2/C2

k (2 10 7 N/m)(1m)

(1A2)2 10 7N/A2• Then

• This defines C and gives

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• Permeability of free space

0 2 k

F

0

2I1I2

dThen

4 10 7N/A2

k 0

2

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5) Field due to long straight wire (magnitude)

B I

r

B 0

2I

r

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6) Force on a moving charge

• Zero at rest

• Zero parallel to B

• Max perpendicular to B

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F qvBsin(Alternative definition of B)

• Proportional to component of v perp to B

• Perpendicular to B

• Perpendicular to v

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7) Motion of a charge in a magnetic field

a) Constant force

motion is parabolic

electric or gravitational field

not everywhere perp to velocity

not magnetic field

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b) Constant magnitude perpendicular to motion

radial field (circular motion)

mass on a string

magnetic field produces circular motion

(initial vel. perp. to B)

motion is circular

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F Fc mv 2

r

Force due to the field:

F qvB

For circular motion:

So,

mv 2

rqvB

r mv

qBr depends on v, B

v

r

qB

m

angular freq. independent of speed, radius

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Tracks in a bubble chamber

• electron-positron creation

• 1, 3 positive

• 2 negative

• energy: 3 > 2 > 1

• energy decreases by collisions

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Example: Find speed and radius for proton

B = 0.10 T

V = 2100 V

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c) Work done by magnetic field

W Fx cosF

displacement, x

Work by a force F

For a magnetic field,

0

Work = 0

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d) Velocity selector

FE qE

Force due to E (down):

Force due to B (up):

FB qvB

For zero deflection, FE = FB :

qE qvB

v E

B

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e) Mass SpectrometerIon energy:

KE 12 mv 2 qV

v 2qV

m

Radius of motion:

r mv

qB

m

qB

2qV

m

r m

q

2V

B

m

q

r2B2

2V