Your Comments
Llamas, and why it is so difficult to get someone to buy me one as a pet. All I ask is that it
breathes fire. I have simple needs people. Seriously. p.s. this lecture was hard.
did not understand any of this
wTF. so over my head
I have no idea what I am doing :(
Wow! That was cool. There is so much that you learn in high school that you just take as factual
and move on. But when you can finally see it derived and shown that it is actually true...WOW!
Pretty awesome.
This was undoubtedly my favorite prelecture of the semester, its so cool!
I was doing fine until we combined ampere's modified law and faraday's
law. Also go over the magnetic and electric fields in empty space. Are
they generated from the empty space itself, or do they come from some
ambient source like stars?
its mindblowing that E0 and mu0 are related to the speed of light. Can
you calculate the speed of light more accurately through measuring E0
and mu0, or by measuring it directly (with lasers, etc)?
I suspect that people's reactions to electric and magnetic waves that
are out of phase will be quite polarized.
Electricity & Magnetism Lecture 22, Slide 1
Physics 212
Lecture 22
Electricity & Magnetism Lecture 22, Slide 2
What We Knew Before Prelecture 22
Electricity & Magnetism Lecture 22, Slide 3
After Prelecture 21: 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
Free space
Electricity & Magnetism Lecture 22, Slide 5
Calculation
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.
At time t1, what is the magnetic field B1 at a radius r (point d) in
between the plates of the capacitor?
●d
I1
S
C
V
Ra
r
R
Q1
Conceptual and Strategic Analysis
Charge Q1 creates electric field between the plates of C
Charge Q1 changing in time gives rise to a changing electric flux between the plates
Changing electric flux gives rise to a displacement current ID in between the plates
Apply (modified) Ampere’s law using ID to determine B
Electricity & Magnetism Lecture 22, Slide 6
Calculation
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.
S
C
V
Ra
c●
●d
r
I1
r
R
Q1
Compare the magnitudes of the B fields at points c and d.
A) Bc < Bd
B) Bc  Bd
What is the difference?
Apply (modified) Ampere’s Law
point c:
I(enclosed)  I1
r
X
C) Bc > Bd
r
R
point d:
ID(enclosed) < I1
Electricity & Magnetism Lecture 22, Slide 7
Calculation
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.
S
C
V
Ra
●d
I1
E
r
R
E
Q1

0
What is the magnitude of the electric field between the plates?
Q
A) E  21
R  0
B) E 
Q1
R
Q1R 2
C) E 
0
E

0
Q1
D) E  Q1
r
0

Q1
Q
 12
A R
E
Q1
 0R 2
Electricity & Magnetism Lecture 22, Slide 8
Calculation
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.
●d
I1
E
r
S
C
V
Q1
E
 R 2 0
Ra
R
Q1
What is the electric flux through a circle of radius r in between the plates?
A)  E 
Q1
0
r
r
B)  E 
2
Q1
0
R
2
2
Q
r
C)  E  1 2
0R
Q1 r 2
D)  E 
 0R2
R
 
E  E  A
Q1
2
E 

r
 0 R 2
Q1 r 2
E 
 0 R2
Electricity & Magnetism Lecture 22, Slide 9
Calculation
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.
●d
I1
E
r
R
Q1
S
C
V
Q1 r 2
E 
 0R2
ID  0
Ra
d E
dt
What is the displacement current enclosed by circle of radius r ?
2
R
A) I D  I1 2
r
r
B) I D  I1 r
R
R
2
r
C) I D  I1 2
R
D) I D  I1 R
r
d  E dQ1 r 2
r2
ID  0

 I1 2
2
dt
dt R
R
r2
I D  I1 2
R
Electricity & Magnetism Lecture 22, Slide 10
Calculation
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.
●d
I1
E
r
R
Q1
S
C
V
Ra
r2
I D  I1 2
R
 
 B  d    o I  I D 
What is the magnitude of the B field at radius r ?
A) B 
0 I1
2 R
B) B 
0 I1
2 r
I R
C) B  0 1 2
2 r
D) B 
 
Ampere’s Law:  B  d   o I  I D 
r
R
0 I1 r
2 R 2

r2 
B  2 r  0  0  I1 2 
R 

I r
B 0 1 2
2 R
Electricity & Magnetism Lecture 22, Slide 11
CheckPoint 1B
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 toward 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
From the
calculation we
just did:
B
0 I1 r
2 R 2
Electricity & Magnetism Lecture 22, Slide 12
CheckPoint 1a
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 toward 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
0 I1 r
B
2 R 2
B is proportional to I
but
At A, B  0 !!
Electricity & Magnetism Lecture 22, Slide 13
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.
S
C
V
Ra
What is the time dependence of the magnetic field B at a radius
r between the plates of the capacitor?
B1 
(A)
(B)
0 I1 r
2 R
2
(C)
B at fixed r is proportional to the current I
Close switch: VC  0  I  V/Ra (maximum)
I exponentially decays with time constant t  RaC
Electricity & Magnetism Lecture 22, Slide 14
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
Constant current  Q increases linearly with time
Therefore E increases linearly with time (E  Q/(A0)
dE/dt is not zero  Displacement current is not zero
 B is not zero !
Electricity & Magnetism Lecture 22, Slide 15
We learned about waves in Physics 211
Electricity & Magnetism Lecture 22, Slide 16
Electromagnetic Spectrum
Electricity & Magnetism Lecture 22, Slide 17
“How can light move at the
same velocity in any inertial
frame of reference? That's
really trippy. ”
see PHYS 225
Electricity & Magnetism Lecture 22, Slide 18
Electricity & Magnetism Lecture 22, Slide 19
CheckPoint 2a
An electromagnetic plane-wave is traveling in the +z direction. The illustration below shows
this wave at some instant in time. Points A, B and C have the same z coordinate.
Ex  E0sin(kz - wt)
Compare the magnitudes of the electric fields 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 20
CheckPoint 2b
An electromagnetic plane-wave is traveling in the +z direction. The illustration below shows
this wave at some instant in time. Points A, B and C have the same z coordinate.
Ex  E0sin(kz - wt)
Compare the magnitudes of the electric fields 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 21
Clicker Question
Ex  E0sin(kz - wt)
Consider a point (x,y,z) at time t when Ex is
negative and has its maximum value.
At (x,y,z) at time t, what is By?
A)
B)
C)
D)
By is positive and has its maximum value
By is negative and has its maximum value
By is zero
We do not have enough information
Electricity & Magnetism Lecture 22, Slide 22