Chapter 29: Electromagnetic Waves Thursday November 10th
• Transformers - demo • Maxwell’s equations • Electromagnetic waves
• Wave equations • Speed of light • Relations between quantities • Energy flux and intensity
Reading: up to page 515 in the text book (Ch. 28/29)
Mini-exam 5 next Thursday (AC circuits and EM waves) 55 unregistered iClickers – any takers?
Transformers
Soft iron
• Flux the same on both sides, but number of turns, N, is different • Total flux through primary and secondary coils depends on N1 and N2
V1= N
1!; V
2= N
2!; "
V2
V1
=N
2
N1
The basic equations of electromagnetism so far.....
Gauss’ law for B (no magnetic equivalent of charge):
!B =!B "d
!A"# = 0
!E =
!E "d
!A"# =
qenc
$o
Gauss’ law:
!B !d
!l = µ
o!! Ienc
Ampère’s law:
Faraday’s law:
! =
!E !d!l ="" #
d$B
dt
The basic equations of electromagnetism so far.....
In vacuum:
} Symmetry
Is it possible that a time-varying electric field could produce a magnetic field, thereby restoring symmetry?
!E =!E !d
!A!! = 0
!B =!B "d
!A"# = 0
} No Symm- etry
!B !d
!l = 0!"
!E !d
!l =!! "
d#B
dt
I I
I I
Maxwell’s displacement current
!B !d
!l = µ
oI
enc"" = µo#
J
= µo
!J !d
!A"
Stokes’ theorem: The choice of surface should not matter
I I
I I
Maxwell’s displacement current
!B !d!s = µ
oI + I
d( )""
I
d= !
o
d!E
dt
!B !d
!l = µ
oI + µ
o"" !o
d#E
dt
Stokes’ theorem: The choice of surface should not matter
Maxwell’s equations
The main thing to note here is the symmetry in the last two equations: a time varying magnetic field produces an electric field; similarly, a time varying electric field produces a magnetic field.
Maxwell’s equations in vacuum
!E =!E !d
!A!! = 0
!B =!B "d
!A"# = 0
!B !d
!l = µ
o!
o
d"E
dt"#
!E !d!l ="" #
d$B
dtThe main thing to note here is the symmetry in the last two equations: a time varying magnetic field produces an electric field; similarly, a time varying electric field produces a magnetic field.
Electromagnetic waves
Propagation Electromagnetic perturbation breaks completely free from the charge/current
Dipole radiation
Field does not appear instantaneously
Maxwell’s equations guarantee that electric and magnetic fields are perpendicular to each other and perpendicular to the direction of propagation; they are polarized.
Spherical waves
Plane waves
!E!
!B
Electromagnetic waves
Maxwell’s equations in vacuum
!E =!E !d
!A!! = 0
!B =!B "d
!A"# = 0
!B !d
!l = µ
o!
o
d"E
dt"#
!E !d!l ="" #
d$B
dt
!"!B = µ
o!o
d!Edt
#
$%%%
&
'(((
!"
!E =#
d!Bdt
$
%&&&
'
()))
Stokes’ Theorem: Gives differential form of Maxwell’s equ’ns
Let there be light!!
!"
!B = µ
o!
o
d!Edt
!"
!E =#
d!Bdt
Stokes’ Theorem: Gives differential form of Maxwell’s equ’ns
Maxwell’s equations in vacuum can be solved simultaneously to give identical differential equations for E and B:
!2!E = µ
o!
o
"2!E"t 2
and !2!B = µ
o!
o
"2!B"t 2
!2 ="2
"x 2+!2
!y2+!2
!z 2The Laplacian differential operator
The Electromagnetic Wave Equations
Let there be light!!
!"
!B = µ
o!
o
d!Edt
!"
!E =#
d!Bdt
Stokes’ Theorem: Gives differential form of Maxwell’s equ’ns
Maxwell’s equations in vacuum can be solved simultaneously to give identical differential equations for E and B:
!2!E!x 2= µ
o!o
!2!E!t 2
and!2!B!x 2= µ
o!o
!2!B!t 2
The Electromagnetic Wave Equations
Waves in one-dimension (traveling along x)
Review of waves (PHY2048)
2k πλ
= k is the angular wavenumber.
Tran
sver
se w
ave
2Tπω = w is the angular frequency.
12
fT
ωπ
= =
v = !!k= ! "T
= ! f "
frequency
velocity
my(x,t) = Asin(kx ±! t +")
Amplitude Displacement Phase }
Phase shift
angular wavenumber angular frequency
Electromagnetic waves
x
Direction of motion
!E x,t( )= Ep sin kx !!t( ) j!B x,t( )= Bp sin kx !!t( )k
• The E and B fields are still related via Ampère’s and Faraday’s laws.
• For a plane wave traveling in the x direction (see text):
Electromagnetic waves • The E and B fields are still related via Ampère’s and Faraday’s laws.
• For a plane wave traveling in the x direction (see text):
!Ez
!x=!B
y
!t,
!Ey
!x=!"B
z
!t, E
x= B
x= 0
x
Direction of motion
Electromagnetic waves • Plugging these wave solutions into the wave equation:
!2Ey=!k 2E
y= µ
o!o
!2Ey
!t 2=!!2µ
o!oEy
!!2
k 2= c2 =
1µ
o!
o
, or c =1µ
o!
o
• Plugging these wave solutions into Faraday’s law: !E
y
!x= kE
pcos kx !!t( )=!"Bz
!t= !B
pcos kx !!t( )
!Ep
Bp
=!k= c