static magnetic fields

48

description

static magnetic fields. Static magnetic fields. Charge in motion yields a current I I = j  area j is a vector -- current density -- amperes/meter 2 Ampere’s circuital law  B  dl =  o I enc. 1 Tesla = 10 4 Gauss B at equator  1 Gauss. current ==> magnetic field. B. a. - PowerPoint PPT Presentation

Transcript of static magnetic fields

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Static magnetic fields• Charge in motion yields a current I• I = j area• j is a vector -- current density --

amperes/meter2

• Ampere’s circuital law B dl = o Ienc

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•1 Tesla = 104 Gauss

•B at equator 1 Gauss

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current ==> magnetic field

Henries104 7o

jB o

dsjdsB o

dsjdlB o

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r2IB enco

Iar

r2B 2

2o

r a a

IIarI 2

2

enc

a r

B

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r2IB enco

Ir2

B o

r a a

IIIenc

a r

B

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B dl = o Ienc

• due to one wire • Ienc = I• B dl = B [2 r]• B = o {1 / 2 r} I

• due to other wire• B = o {1 / 2 r} I

• superposition

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•Ienc = N I

B dl = B [2 L ]

•B = o N I / 2L

•B = o N I / L -- in center / top and bottom

L

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we know that • B = 0vector potential A

we know that • [ x vector] = 0we can now specify the vectorlet vector be A such that B = x A

William Thomson shows that Neumann's

electromagnetic potential A is in fact the

vector potential from which may be

obtained via B = x A.

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vector potential A

we also know x B = µo jB = x A

x x A = • A) - A

- A A = - µo j is similar to Poisson’s

equation but we have to solve three PDE’s

A and j are in the same direction!!

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'dv

''

4O

rrrjrA

z'dz'zI'dv' urj

22 'zr

z22

22O

LrL

LrLln4

I urA

R' rrRz’

r

dz’2 L

I

A

z

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Rz’

r

dz’2 L

I

B

z

rArB

urBr

A z

urB

22o

rL

Lr2I

urBr2I

Llim o

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Biot-Savart integral

'dv'

'4 2

'O

rrurjrB rr

Rz’

r

dz’2 L

I

B

z

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'dv'

'4 2

'O

rrurjrB rr

'dzr'z

r'zI4 2/322

rzz

O uuu

u

22O

rL

Lr2I

Rz’

r

dz’2 L

I

B

z

ur2I

Llim o

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'adza

zaI4 2/322

zrO uuu

2/322

2O

zza

aI2

B

'dv'

'4 2

'O

rrurjrB rr

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large hadron collider

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Earth’s magnetic field protects us

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Inductance L

currentflux magneticL

current

)area(xdensityflux magnetic

I

)area(xLNIo

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Coaxial cable

abcurrentflux magneticL

I

(length) x drr2Ib

ao

(length) x abln

2o

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Inductance of a microstrip

z

d

w

I

I

B

currentflux magneticL

d0

0z0 dy

wIdz

I1

zw

d0

x

y

z

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An induced electric current flows in a direction such that the current opposes the change that induced it. This law was deduced in 1834 by the Russian physicist Heinrich Friedrich Emil Lenz (1804-65).

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Faraday’s law dt

tdtV m

ΔsBtm

either B or s individually change in time or they both

change in time together

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dt

tdILtV

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zudxdyds

T

BeatVT

t2

zuB Tt

eB

dt

dtV m

dt

BeadtV

Tt2

x y

z

dsBm

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dsBm

zudxdydszuB B

dt

dtV m

dt

tsinABdtV

x y

z tsinAAs

tcosABtV

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x y

z dsBm

dt

dtV m

cosBLWm

dtd

ddtV m

tsinBLWtV R

tVtI

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t

dsBdlE

dsEdlE x

Faraday’s law

apply Stoke’s theorem

t

dsB

tx

BE

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t

0x

BE

0 B

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wire carrying current I

I

2w

L u

V

Luwx2

IV o

22o

wxILwuV

Luwx2

Io

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Bewley’s book• trick questions• not every motion

generates a voltage

• uniform B & v• substitution of

circuit• Vgen = 0!

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XB

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XB

1 2

cu

V12= 0

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XB

1 2

cu

V12= Bcu

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XB

1 2

cu

V12= Bcu

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XB

V12= Bcu1 2

cu

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XB

1 2

cu

V12= -Bcu

V12= Bcu

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XB

1 2

cu

V12= -Bcu

V12= Bcu

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I1 dl1

1 B1

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I2 dl2

1B2

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I1 dl1 I2 dl2

1B2 B1

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BvdF dQ

dvv BvdF

BIdldF

I1 dl1 I2 dl2

1B2 B1

dsdlBjdF

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