### Lecture 5

```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
  Qenc
E  dA 

o
surface
 
d  
loopE  dl   dt  B  dA
Gauss’ Law
 
 B  dA  0
surface
 
 B  dl  o I ENCL
loop
Gauss’ Law for B field
Ampere’s Law
Our equations so far…..
Unit 22, Slide 2
Displacement Current
3 Points, all a distance r from axis of a current c
carrying wire connected to capacitor
1
2
●
●
3
●
r
I1
| B1 || B3 | I o / 2r
| B2 | 0 ??
Define
“displacement current”
such that:
 
 B  dl  o ( I ENCL  I dis )
loop
Electricity & Magnetism Lecture 22, Slide 3
Modify Ampere’s Law

Q
E 
0 0 A
  EA 
Q
0
Q   0
dQ
d
 0
 ID
dt
dt
Electricity & Magnetism Lecture 22, Slide 4
Displacement Current
Real Current:
Charge Q passes through area A in time t:
I
dQ
dt
Displacement Current: Electric flux through area A changes in time
ID  0
d E
dt
 
d E
loopB  dl  o ( I ENCL   o dt )
Electricity & Magnetism Lecture 22, Slide 5
Example 22.1
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?
●d
I1
r
1cm
0.5cm
R
Q1
Conceptual Plan
What is the magnetic
field 3cm from the
axis of the center of
the plates?
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
Electricity & Magnetism Lecture 22, Slide 6
CheckPoint 1(A)
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..
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
A
Electricity & Magnetism Lecture 22, Slide 7
CheckPoint 1(B)
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..
Compare the magnitudes of the
magnetic fields at points A and B
just after the switch is closed:
A. BA < BB
B. BA = BB
C. BA > BB
A
Electricity & Magnetism Lecture 22, Slide 8
Follow-Up
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  t1, a current I1 flows in the
circuit and the capacitor carries charge Q1.
What is the time dependence of the magnetic
field B at a radius r between the plates of the
capacitor?
(A)
A
(B)
B
S
C
V
Ra
B1 
0 I1 r
2 R 2
(C)
C
Electricity & Magnetism Lecture 22, Slide 9
Follow-Up 2
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
Electricity & Magnetism Lecture 22, Slide 10
Final form
  Qenc
E  dA 

o
surface
Gauss’ Law
 
 B  dA  0
surface
Gauss’ Law for B field
 
d  
loopE  dl   dt  B  dA
 
d  
loopB  dl  o I ENCL   o dt  E  dA
Ampere’s Law
Unit 22, Slide 11
Wave Equation
Remember from 2111??
Remember this guy? Not
the spring constant!
Electricity & Magnetism Lecture 22, Slide 12
Some Calculations
+
+
-
 
d  
E  dl    B  dA

dt
loop

d
dE * h   (dx * h) B
dt


d E
d dB

2
dx
dt dx
2
 
d  
loopB  dl  o  o dt  E  dA


dB
dE
  o o
dx
dt
Electricity & Magnetism Lecture 22, Slide 13
Some Calculations


2
2
d E
d E
   o o 2
2
dx
dx
A wave equation????
v
1
 o o
With a velocity of …?
see PHYS 2115
Electricity & Magnetism Lecture 22, Slide 14
Electricity & Magnetism Lecture 22, Slide 15
Keep us warm
Keep us safe
The Sun
Electricity & Magnetism Lecture 22, Slide 16
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 17
Past Confusion
Nothing is moving here.
Arrows only represent strength of field.
Actually a plane wave.
Electricity & Magnetism Lecture 22, Slide 18
CheckPoint 2(A)
Ex  E0sin(kz  wt)
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. Ea < Eb
B. Ea = Eb
C. Ea > Eb
E  E0 sin (kz  wt):
E depends only on z coordinate for constant t.
z coordinate is same for A, B, C.
Electricity & Magnetism Lecture 22, Slide 19
CheckPoint 2(B)
Ex  E0sin(kz  wt)
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. Ea < Ec
B. Ea = Ec
C. Ea > Ec
E  E0 sin (kz  wt):
E depends only on z coordinate for constant t.
z coordinate is same for A, B, C.
Electricity & Magnetism Lecture 22, Slide 20
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