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+ Q free on inner surface. + + + + + + + + + + + +. - - - - - - -. - q bound. Symmetry – fields must be uniform – field lines perpendicular to plates. + q bound. + + + + + +.

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3805816

+Qfreeon inner surface

+ + + + + + + + + + + +

- - - - - - -

-qbound

Symmetry

– fields must be uniform

– field lines perpendicular to plates

+qbound

+ + + + + +

- - - - - - - - - - - -

-Qfreeon inner surface

Interior points electric

field must be zero


3805816

area of plates A

+ + + + + + + + + + + +

+Qfreeon inner surface

plate separationd

-Qfreeon inner surface

- - - - - - - - - - - -


3805816

+

+

+

+

+

conductor

-

-

-

dielectric

Gauss’s Law


3805816

dielectric

Constant

(polar

molecules)

frequency


3805816

Fme

dy

+ + + + + + + + +

+ + + + + + + + +

F

-

-

-

-

-

-

-

-

-


3805816

+ + + + + + + + + + + +

- - - - - - - -

+ + + + + + + +

- - - - - - - - - - - -

Electric field

Electric displacement

Polarization


3805816

only some of the windings are shown

A

Integration paths

B

C


3805816

L

Bz

Bz

dA1

dA2

Br

dA3


3805816

Z

Y

X


3805816

Bz1

Br = 0

Ienclosed = 0

I

A

Bz2

s

x


3805816

Bz1= 0

C

Br = 0

Ienclosed = nsI

I

Bz2

s

Ienclosed = 0

I

Bz1= 0

Bz2

x

x

x

x

x

x


3805816

single turn of wire with current I

B

around integration loop B

dr = 0 and Br = 0

outside loop Bz = 0


3805816

iron core

BFe HFe

gap region

Bgap Hgap

i

coil windings

Bair Hair


3805816

.

.

.

.

.

.

.

.

3

X

X

X

X

X

X

2

Current i

out of page

Circulation loop:

square of length L

1

4

Current i

into page

Cross-section through electromagnet


3805816

thickness t

width L

q = - e

area A

electrons are the charge

carriers in copper


3805816

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -


3805816

+ + + + + + + + +

dy

+ + + + + + + + +

F

- - - - - - - - -

-q

+q


3805816

C

+ + + + + + + + +

V

r

CA

CB

- - - - - - - - -

L-x

C = CA + CB

x


3805816

Induced dipole moment – helium atom

+2e

+2e

-e

-e

-e

-e

Zero electric field – helium atom symmetric  zero dipole moment

A

B

effectively charge +2e at A and -2e at B

dipole moment p = 2ed


3805816

Induced dipole moment – sulfur atom

+16e

+16e

-8e

-8e

-8e

-8e

Zero electric field – helium atom symmetric  zero dipole moment

A

B

effectively charge +16e at A and -16e at B

dipole moment p = 16ed


3805816

Er

E

P

r2  r + (d/2)cos

r1  r – (d/2)cos

r

(d/2)cos

+q

-q


3805816

dA

+f

+ + + + + + + + +

-b

+b

- - - - - - - - -

-f


3805816

-q

+q


3805816

S

+f

-b

O

r

+b

-f


3805816

Width of ring r d

Radius of ring r sin

+

+

+

Area of the shaded ring

between  and  + d

surface S

d

r

-

Pcos

-

-


3805816

+

+

+

electric field at O

due to charge dqe

E0 cos

E0

element of charge dqe

-

-

-


3805816

+Ze

+Ze

d

a

a

d << a


3805816

F

F

+Q

d

F

- Q


3805816

+ p E

U

0

- p E

π

π/2

0


3805816

U = - p E

Lowest energy state

U = 0

U = + p E

highest energy state

-

-

-

 = 90o

+

+

+

 = 0

 = 180o


3805816

r - 1

1/T


3805816

Po

T


3805816

1

slope = 1/3

0

10

pE/kT


3805816

Gaussian surface S

conducting

sphere q

air

a

r

non-conducting

liquid

Symmetry  field lines must be radial


3805816

conducting

sphere q

Eairt

air

Eliquidt

non-conducting

liquid

Symmetry  Eairt = Eliquidt Eair = Eliquid = E


3805816

field lines of E

field lines of D

+


3805816

field lines of D

+

+

+

+

+

+

+

+

+

+

greater concentration of charge

on surface bounded by liquid

field lines of E

+


3805816

-

+

+

-

-

+

+

shift in atoms

due to ionic nature of bond

induced dipoles due

to shift in electron cloud

rotation

orientation of polar molecules


3805816

6

5

4

S

N

HFe

Hair

Circulation loop: square side L

2

1

3


3805816

Cylindrical

Gaussian

surface

Gauss’s Law for magnetism

B-field lines –

form continuous loops


3805816

N pole

im

Bound surface currents im (right hand screw rule)

 


3805816

un-magnetized piece of iron

Bar magnet bought near

un-magnetized piece of iron

N

N

N

 Bar magnet will attract the iron that was initially un-magnetized

north pole attracts

south pole


3805816

Cu ramp

Fe ramp

plastic ramp

N

N

N


3805816

Circulation loop for circulation integration used in applying Ampere’s Law

N

N

Hair

Hiron


3805816

B applying Ampere’s Law

I

I

(0,0)

d

H

B

d


3805816

M applying Ampere’s Lawiron

PERMANENT MAGNET

Hiron

B

B, Hgap

Mgap = 0

B = Bgap = Biron

ELECTROMAGNET

Miron

Hiron

B

Mgap = 0

B, Hgap

B = Bgap = Biron


3805816

Y applying Ampere’s Law

thickness t

area A = wt

width w

X

current in

X direction

Z

magnetic field in Z direction

Schematic diagram of a Hall Probe


3805816

width applying Ampere’s Law

w

Y

+ + + + +

- - - - -

_

VH

+

VH

- - - - -

+ + + + +

X

charge carriers electrons (-)

eg wire, N-type semiconductor

charge carriers positive (+)

eg holes in P-type semiconductor

I

.

Z direction

out of page


3805816

e applying Ampere’s Law-

-

e

length L

area

A

resistance R

resistivity 

conductivity 

number densityn

+

I

_

_

v

electron

V


3805816

e applying Ampere’s Law-

-

e


3805816

image applying Ampere’s Law

object

Z

A

electron beam

Y

X


3805816

+Z applying Ampere’s Law

Electron at A moving parallel to +Y-axis

vy

Electron acted upon by the radial component of the magnetic field  force on electron in +X direction  +X-component to the velocity

Fx

due to Bz

By

axis for the motion of the electron beam

+Y

Bz

radial component

of magnetic field

+X


3805816

+Z applying Ampere’s Law

Electron at B has a velocity component in the +X direction

Fy

due to Bz

Electron acted upon by the axial component of the magnetic field By force on electron in -Z direction i.e. towards to axis  focusing action

Fz

due to By

vx

By

axis for the motion of the electron beam

+Y

Bz

radial component

of magnetic field

+X


3805816

i applying Ampere’s Lawfree

.

.

.

.

.

.

.

.


3805816

external magnetic field applying Ampere’s Law


3805816

Electrostatic applying Ampere’s Law

capacitor

Electrolytic

capacitor


3805816

Electrostatic capacitor applying Ampere’s Law

Electrolytic capacitor


3805816

Electrochemical double layer capacitor applying Ampere’s Law

d

conductive

electrode

conductive

electrode

separator

activated carbon


3805816

+ applying Ampere’s Law

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-


3805816

+ applying Ampere’s Law

+

-

Electric Field

-

-


3805816

+ applying Ampere’s Law

Zero applied stress

-

Compressive stress

Induces a voltage

Applied voltage produces

An expansion


3805816

- applying Ampere’s Law

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Ferroelectric material

Antiferroelectric material


3805816

Interior points electric applying Ampere’s Law

field must be zero

+0.2Q on outer surface

+

+

+ + + + + + + + + +

+Q on inner surface

- - - - - - - - - -

-Q on inner surface

Interior points electric

field must be zero

Symmetry

– electric field must be uniform

– electric field lines perpendicular to conductive plates


3805816

Symmetry applying Ampere’s Law

– fields must be uniform

– field lines perpendicular to plates

Interior points electric

field must be zero

+ + + + + + + + + +

+Q on inner surface

- - - - - - - - - -

-Q on outer surface

+ + + + + + + + + +

+Q on outer surface

-Q on inner surface

- - - - - - - - - -

Interior points electric

field must be zero


3805816

Electric field between applying Ampere’s Law

Adjacent plates

+q

-q

-q

+q

+q

-q

+V


3805816

... applying Ampere’s Law

...

...

V

Capacitors in series (charge on each plate)

Capacitors in parallel (voltage across each capacitor is the same)

Series

branch


3805816

Capacitors in parallel applying Ampere’s Law

-Q1

+Q1

C1

Q =Q1+Q2

+Q2

-Q2

Ceq = C1+C2

C2

V

V

Capacitors in series

Q

+Q

-Q

1/Ceq = 1/C1+1/C2

-Q

+Q

C1

C2

V

V


3805816

air applying Ampere’s Law

fuel

fuel

w

l

h


3805816

+ applying Ampere’s LawQ

- +

r

Induced dipole


3805816

a applying Ampere’s Law

Slab 1

a

a

5a

Slab 2

a

a


3805816

S applying Ampere’s Law1

S3

S2

S4


3805816

+ applying Ampere’s LawQ

+ Q

- Qb1

C1

- Q

+ Qb1

- Qb2

+ Qb2

+ Q

- Q

C2

- Q

Capacitors in series


3805816

E applying Ampere’s Law = 0


3805816

+ applying Ampere’s Law

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

E = 0

-

-

-

-

-

-

-

-


3805816

+Q applying Ampere’s Law/2

+Q/2

C1 = Q / 2V1

C1

C1

V1

Q = 2 C1V1

- Q/2

- Q/2

qA = C2V2 = r C1 V2

qB = C1 V2

Q = qA + qB

= C1 V2 (r + 1)

= 2 C1 V1

V2 = 2 V1 / (r + 1)

qA = 2 C1 V1 r / (r + 1)

qB = 2 C1 V1 / (r + 1)

+qA

+qB

V2

C1

C2

- qA

- qB


3805816

+ applying Ampere’s LawQf

-Qf


3805816

+q applying Ampere’s Law

d

t

r

- q


3805816

Homogenous dielectric – uniformly polarized applying Ampere’s Law

Dielectric is neutral

+

+

+

+

+

+

+

-

-

-

-

-

-

-

The electrical field is reduced in the dielectric material


3805816

+ applying Ampere’s Law

+

+

+

+

+

+

+

+

+

+

Thin long rod L = 0

Zero polarization

Sphere L = 1/3

Concentration of charges

At surface given by

-

-

-

-

-

-

-

-

-

-

-

Flat plate L = 1

Max polarization


3805816

+ applying Ampere’s LawQ

- Q


3805816

+ applying Ampere’s Law

Q

-

Q

b

1

E1

+

Q

b

1

-

Q

b

2

E2

+

Q

b

2

-

Q


3805816

R applying Ampere’s Law1

R2


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