CAPA #14 Friday
Reading: 34.7
pretest and Tutorial hw for tomorrow.
_________________________
Last: AC circuits and Faraday's law
Today: Maxwell's equations + FCQ's (if time, otherwise Wed)
Next: More of Maxwell and EM radiation
1
Faraday's Law:
"=#
d$mag
dt
And thus Faraday’s Law can be written as:
!
v
v
v
d
v
# E " dl = $ dt ## B " dA
Loop
surface
!
2
A changing B-field creates a new kind of E-field.
B
If B is decreasing
E-field is CCW
E
If B is increasing  E-field is Clockwise
B
E
3
1
The (non-Coulomb) E-field is very different from the Coulomb
E-field created by single electric charges.
E
B
q
v v
#V = " $ E ! dl = 0
$=
E
v v
E
# " dl ! 0
Loop
4
CT 33.12
A solenoid has an increasing current causing an
increasing B-field inside. An end-on view of the solenoid is
shown. What is the direction of the force on an electron
(shown below the solenoid)?
I
A
B
D
C
E: 0
B(in)
I is Increasing with time
B(out)=0
Electron (neg charge)
5
What have we seen so far…
v
v
! E " dA =
Qenc
#0
Gauss’ Law for Electric Fields
What about for Magnetic Fields?
v
v
! B " dA = ?
6
2
Magnetic Field lines always close on themselves.
The Magnetic Flux through any closed surface must be
zero, since the number of lines entering = number of lines
leaving.
v
v
v
! B " dA = 0
v
! B " dA = 0
v
v
could only
! B # dA " 0 This
be from a
?
B
Magnetic
Monopole,
which we have
never seen.
7
Electromagnetism so far:
r
v
"" E • dA = Q
r
v
"" B • dA = 0
enclosed
/# 0
!
r r
d%
" E • d l = $ dtB
r r
" B • d l = µ0ithrough
r
r
r r
F = qE + qv " B
Gauss’ Law
No magnetic monopoles
Faraday’s Law
Ampere’s Law
!
8
Electromagnetism in Full: Maxwell’s equations
r
v
"" E • dA = Q
r
v
"" B • dA = 0
enclosed
r
r
r
r
" E• dl
" B• dl
=$
/# 0
d%B
dt
= µ0ithrough + µ0#0
d%E
dt
!
9
3
Maxwell then found an amazing solution to these equations.
1. Changing E-field creates a B-field
2. Changing B-field creates an E-field
Thus, there is a solution of a traveling electromagnetic wave !
10
Suppose you have a charge q and you shake it up and down.
B
E
E B
E
B E
B
q
As it is moving, the Electric Field nearby is changing.
This changing Electric Field creates a Magnetic Field.
This Magnetic Field is changing and it creates an Electric Field.
etc…………
11
This simple shaking charge, creates an electromagnetic wave
that travels outward.
Wave = self propagating disturbance
Maxwell computed the speed of this new E/M wave and found
1
speed v =
= (4# "10 !7 N / A2 " 8.85 "10 !12 F / m) !1/ 2
µ 0$ 0
8
speed v = 3 "10 m /s
=c, Speed of Light !
!
12
4
CT 33.14
A long solenoid (radius R) contains a uniform B field that
increases steadily, B(t) = c t.
Consider the dashed “Amperian loop”.
What is " E • dL around the loop? (going CCW)
R
A: zero
B: 2πrE
C: 2πRE
!
D: π r^2 E
E: π R^2 E
B
r
13
CT 33.14b
What is
"" B • dA
through the the dashed curve?
A: zero
!
B: 2πrB
C: 2πRB
D: π r^2 B
E: π R^2 B
R
B
r
14
CT 33.14c
Let’s put this together. Recall B = ct.
d#
d
Faraday says $ E • dl = " = " $$ B • dA
dt
dt
Recall the left side is 2πrE
2
Recall ! ## B • dA = " r B
So… how does E depend on r?
A: E=0 B: E ~ 1/r
C: E ~ r D: E~ r^2
E: Other
!
R
B
r
15
5