Physics 2112 Unit 22

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16 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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Physics 2112

Unit 22

Outline


Displacement Current


Maxwell’s Equations (Final Form)


E&M Waves

Electricity & Magnetism Lecture 22, Slide
1

Where we are now




surface
A
d
B
0


ENCL
o
loop
I
l
d
B












A
d
B
dt
d
l
d
E
loop




o
enc
surface
Q
A
d
E






Our equations so far…..


Unit 22, Slide
2

Gauss’ Law

Gauss’ Law for B field

Ampere’s Law

Faraday’s Law

3 Points, all a distance r from axis of a current c
carrying wire connected to capacitor

Displacement Current

Electricity & Magnetism Lecture 22, Slide
3



r



I
1

1

3



2

r
I
B
B
o


2
/
|
|
|
|
3
1


??
0
|
|
2

B
)
(
dis
ENCL
o
loop
I
I
l
d
B







Define

“displacement current”
such that:

Modify Ampere’s Law

Electricity & Magnetism Lecture 22, Slide
4

A
Q
E
0
0





0

Q
EA





0

Q
D
I
dt
d
t
d
dQ



0

Real Current:
Charge
Q

passes through area
A

in time t:



Displacement Current:
Electric flux through area
A

changes in time



Displacement Current

Electricity & Magnetism Lecture 22, Slide
5

dt
d
I
E
D


0

t
d
dQ
I

)
(
dt
d
I
l
d
B
E
o
ENCL
o
loop









A
parrallel

plate capacitor has plates that are 2cm in
diameter and 1mm apart. If the current into the capacitor
is 0.5A, what is the magnetic field between the plates
0.5cm from the axis of the center of the plates?

R

r



I
1

Q
1

d

Conceptual Plan


Use modified Ampere’s Law


Strategic Plan


Find electric flux contained within circle with radius of 0.5cm


Find time rate of change of that flux

Example 22.1

Electricity & Magnetism Lecture 22, Slide
6

1cm

0.5cm

What is the magnetic
field 3cm from the
axis of the center of
the plates?

CheckPoint

1(A)

Electricity & Magnetism Lecture 22, Slide
7

At time t = 0 the switch in the circuit shown below is closed.
Points A and B lie inside the capacitor; A is at the center and
B is
at
the outer
edge.
.

A

After the switch is closed, there will
be a magnetic field at point A which
is proportional to the current in the
circuit.

A.
True

B.
False

CheckPoint

1(B)

Electricity & Magnetism Lecture 22, Slide
8

At time t = 0 the switch in the circuit shown below is closed.
Points A and B lie inside the capacitor; A is at the center and
B is
at
the outer
edge.
.

A

Compare the magnitudes of the
magnetic fields at points A and B
just after the switch is closed:

A.
B
A

< B
B


B.
B
A

= B
B


C.
B
A

> B
B


Switch
S

has been open a long time when at
t


0
, it is closed. Capacitor
C

has circular plates of
radius
R
. At time
t


t
1
, a current
I
1

flows in the
circuit and the capacitor carries charge
Q
1
.


What is the time dependence of the magnetic
field
B

at a radius
r

between the plates of the
capacitor?

0 1
1
2
2
I r
B
R




(A) (B) (C)

Follow
-
Up

Electricity & Magnetism Lecture 22, Slide
9

S

R
a

V

C

A

B

C

Suppose you were able to charge a capacitor
with constant current (does not change in
time).


Does a
B

field exist in between the plates of
the capacitor?


A)
YES
B)
NO

Follow
-
Up 2

Electricity & Magnetism Lecture 22, Slide
10

Final form




surface
A
d
B
0








A
d
E
dt
d
I
l
d
B
o
ENCL
o
loop












A
d
B
dt
d
l
d
E
loop




o
enc
surface
Q
A
d
E






Tada!.....
Maxwell’s Equations


Unit 22, Slide
11

Gauss’ Law

Gauss’ Law for B field

Ampere’s Law

Faraday’s Law

Wave Equation

Electricity & Magnetism Lecture 22, Slide
12

Remember this guy? Not
the spring constant!

Remember
from 2111??

Electricity & Magnetism Lecture 22, Slide
13

+

+

-

-







A
d
B
dt
d
l
d
E
loop




B
h
dx
dt
d
h
E
d
)
*
(
*



dx
B
d
dt
d
dx
E
d




2
2





A
d
E
dt
d
l
d
B
o
o
loop






dt
E
d
dx
B
d
o
o





Some Calculations

see
PHYS
2115


Electricity & Magnetism Lecture 22, Slide
14

2
2
2
2
dx
E
d
dx
E
d
o
o






A wave equation????


o
o
v


1

With a velocity of …?


Some Calculations

Electricity & Magnetism Lecture 22, Slide
15

Electricity & Magnetism Lecture 22, Slide
16

The Sun

Keep us warm

Keep us safe

Electricity & Magnetism Lecture 22, Slide
17

Example 22.2

An electromagnetic plane wave
has a wavelength of
0.100nm.

a)
What is its wave number, k?

b)
What is its frequency?

c)
What portion of the electro
-
magnetic spectrum
does it fall in?

Electricity & Magnetism Lecture 22, Slide
18

Past Confusion

Nothing is moving here.


Arrows only represent strength of field.


Actually a plane wave.


CheckPoint

2(A)

Electricity & Magnetism Lecture 22, Slide
19

E
x



E
0
sin(
kz



w
t
)

E



E
0

sin (
kz



w
t
)
:


E

depends only on
z

coordinate for constant
t
.

z

coordinate is same for
A
,
B
,
C
.

An electromagnetic plane wave is traveling in the +z direction. The
illustration below shows this wave an some instant in time. Points A, B, and
C have the same z coordinate.

Compare the magnitudes of the electric field at points A and B.

A.
E
a

<
E
b


B.
E
a

=
E
b


C.
E
a

>
E
b


CheckPoint

2(B)

Electricity & Magnetism Lecture 22, Slide
20

E
x



E
0
sin(
kz



w
t
)

E



E
0

sin (
kz



w
t
)
:


E

depends only on
z

coordinate for constant
t
.

z

coordinate is same for
A
,
B
,
C
.

An electromagnetic plane wave is traveling in the +z direction. The
illustration below shows this wave an some instant in time. Points A, B, and
C have the same z coordinate.

Compare the magnitudes of the electric field at points A and C.

A.
E
a

<
E
c


B.
E
a

=
E
c


C.
E
a

>
E
c