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
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