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!!f " #f
#xi +
#f
#yj +
#f
#zk
$
%&
'
()
Gradient: !
!! •!E = div(
!E) =
"Ex
"x+"Ey
"y+"Ez
"z
Divergence: !
!! "!E = curl(E) =
i j k
#
#x
#
#y
#
#z
Ex Ey Ez
Curl:!
Laplacian:!
!2f "!!i
!!f( ) =
#2f
#x2+#2f
#y2+#2f
#z2
Vector Calculus Operations!
Read Rohlf Appendix C, P576-577!
.
!P"!! id!a =
!"i
!P!!! dv
Divergence Theorem
The flux of a vector over a closed surface = the integral over the enclosed volume of the divergence.!
For example, for the electric field due to a charge distribution:!
(Rohlf, P 576)!
d!a
!P
Gauss's LAW: !Eid!a ="!!
q
"0
=1
"0
#dv!!!
Divergence theorem: !Eid!a =
!$i!!!!Edv"!! =
1
"0
#dv!!!
%!!!!!!$i
!E =
1
"0
#
.
P
!"
id"a#!! =
""iP
!"
!!! dv
Example of Divergence Theorem
(!!"!P)id!a## =
!P•d!l"#
The “flux” of is the circulation of around any closed loop which bounds
the surface. The curl therefore is a measure of the rotation of the vector field. !
!!"!P
!P
Stokes Law
!P
d!l
d!a
(Rohlf, P 577)!
Homework 1 - Vector Calculus!
Due Tuesday, Sept 8. !
2. A sticky fluid is moving past a flat horizontal surface!
such that the velocity is given by m/s.!
Find both magnitude and direction of the curl,.!
!v = 10yi
3. The electric field inside a uniformly charged dielectric!
is . Find the divergence and therefore the charge!
distribution.!
!E = 10xi
1.The gravitational potential is U=Gy J/kg-m. Find the gradient.!
and the gravitational field, which is .!
!g = !
!
"U
Maxwell!s Equations and Electromagnetic Waves!
Read Rohlf !
Chapter 1, Page 8 - 9!
Appendix B, Page Page 572 - 574!
Maxwells Equations
!E !d!a""" =
q
#0
!E !d!l#" = $
%&B
%t
!B !d!a""" = 0
!B !d!l#" = µ
0i + #
0
%&E
%t'()
*+,
!E !d!a""" =
1
#0
-dv""" !E !d!l#" = $
d
dt
!Bid!a""
!B !d!a""" = 0
!B !d!l#" = =µ
0
!J id!a"" + µ
0#
0
%%t
!Eid!a""
Write the right side of each equation as an integral.!
Read Rohlf, Chapter 1, Page 9!
Caution: Rohlf uses 4!k=1/"0!
Convert to differential point notation by using!Stokes Theorem and the Divergence Theorem.!
!E ! d!a""" =
1
#0
$dv""" !E ! d!l#" = %
d
dt
!Bid!a""
Apply the divergence theorem Apply Stokes' theorem
!E ! d!a""" =
!
&i"""!E dv
!E ! d!l#" = (
!
& '!E) ! d
!a""
!
&i"""!E dv =
1
#0
$dv""" (!
& '!E) ! d
!a"" = %
d
dt
!Bid!a""
#0
!
&i
!E = $
!
& '!E +
(!B
dt
= 0
Read Rohlf, Appendix B, P 572-573!
!B ! d!a""" = 0
!B ! d!l#" = µ
0
!J id!a"" + µ
0#
0
d
dt
!Eid!a""
Apply the divergence theorem Apply Stokes' law
!B ! d!a""" =
!
$i"""!B dv
!B ! d!l#" = (
!
$ %!B) ! d
!a""
!
$i
!B = 0
!
$ %!B = µ0
!J + µ0#0
&!E
&t
Read Rohlf, Appendix B, P 572-573!
Maxwell"s Equations!
Integral Form!
!E !d!a""" =
q
#0
!E !d!l#" = $
%&B
%t
!B !d!a""" = 0
!B !d!l#" = µ
0i + #
0
%&E
%t'()
*+,
Differential Form!
!0
!
"i
!E = #
!
" $!E +
%!B
%t= 0
!
"i
!B = 0
!
" $!B = µ
0
!J + µ
0!0
%!E
%t
Read Rohlf, Appendix B-6, Page 573-574!
Homework 2- Electromagnetic waves."
Due Tuesday Sept 8!
Show that Maxwell's equations in vacuo:
1. !0
!
"i
!E = 0 2.
!
" #!E +
$!B
$t= 0
3.!
"i
!B = 0 4.
!
" #!B % µ0!0
$!E
$t= 0
can be combined to produce the differential equation for for electromagnetic waves.
"2!E % µ0!0
$2!E
$t 2= 0. & "2
Ex % µ0!0
$2Ex
$t 2= 0, "2
Ey % µ0!0
$2Ey
$t 2= 0, "2
Ez % µ0!0
$2Ez
$t 2= 0.
Hint:
Combine eqs. 2 and 4 by taking !
" # (!
" #!E)
Use a vector identity: !
" # (!
" #!E) =
!
"!
" •!E( ) % (
!
" •!
")!E
Use eq.1:!
" •!E = 0, and the definition of the Laplacian: (
!
" •!
")!E ' "2
!E
(
)
***
+
,
---
Ex = E0xei(!k i!r%.t+/ ) , Ey = E0ye
i(!k i!r%.t+/ )
, etc. are solutions.
Repeat the above, but for the magnetic field.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !0
!"i
!E = 0
!" #!E +
$!B
dt= 0
!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!"i
!B = 0
!" #!B % µ
0!0
$!E
$t= 0
!!!!!!!!!!!!!!!!!!!!!"2!E % µ
0!0
$2 !E
$t2= 0!!!!!!!!!!!!!!!!!
!E =!!!Ex i
"!!+Ey j"!!+Ezk
"
"2Ex % µ
0!0
$2Ex
$t2= 0!!!!!!!!"2
Ey % µ0!0
$2Ey
$t2= 0!!!!!!"2
Ez % µ0!0
$2Ez
$t2= 0
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"2=
$2
$x2+
$2
$y2+
$2
$z2
Electromagnetic waves.
Maxwell’s equations in a vacuum.
One dimensional wave is traveling in the x direction!
and polarized in the y direction!
!!2=
"2
"x2+
"2
"y2+
"2
"z2!#
"2
"x2!!!!!!!!!!!!!
!E# Ey
!!2!E $ µ
0%0
"2 !E
"t2= 0!!!!!#
"2
"x2Ey $ µ
0%0
"2Ey
"t2= 0!!!
Exercise: Show!!!!!!Ey = E0 yei(kx$&t )
!!is a solution.
Verify that the conditions on k and & for the equation to be satisfied?
v = f' =&
k=
1
%0µ0
= 2.998 (108 m/s = c
Homework 3. Due Friday Sept 11.!
Show that this is a solution provided the speed of the wave is !
v = f! ="
k=
1
#0µ
0
= 2.998 $108 m/s = c
Show that each component of is a solution "to the electromagnetic the wave equation for"A wave traveling in an arbitrary direction "
E
!"
Write the expression for in components: !
!E
Ex = E0 xei(kxx+kyy+kzz!"t+# )
And similarly for! Ey !!and!!!Ez
!A ! Axi + Ay j + Az k ! A1n1 + A2 n2 + A3n3 ! ni
1
3
" Ai
Matrix representation!A !
Ax
Ay
Az
#
$
%%
&
'
((
C11
C12
C13
C21
C22
C23
C31
C32
C33
!
"
##
$
%
&&
Representations of a vector by a column matrix
3X3 Matrix, etc
Matrices!
Vector and Matrix Multiplication
!Ai
!B ! (Axi + Ay j + Az k)i(Bxi + By j + Bz k)
i ii = ji j = kik = 1
i i j = 0, " " " " etc.!Ai
!B ! AxBx + AyBy + AzBz
(Ax Ay Az )
Bx
B y
Bz
#
$
%%
&
'
((= Ai
1
3
) Bi
Dot product or scalar product.!
Matrix representation.!
=scalar!
C11
C12
C13
C21
C22
C23
C31
C32
C33
!
"
##
$
%
&&
B1
B2
B3
!
"
##
$
%
&&=
C11B1+C
12B2+C
13B3
C21B1+C
22B2+C
23B3
C31B1+C
32B2+C
33B3
!
"
##
$
%
&&=
D1
D2
D3
!
"
##
$
%
&&
Matrix Multiplication
D
i= C
ijj
! Bj
" CijB
j
Inner Product - Contraction!
F
ijk= D
ijB
k
Outer Product - Tensor Product!
where C(i, j) = i • j ' = cos!ij…
x '
y '
z '
!
"
##
$
%
&&=
C(i, i ') C( j, i ') C(k, i ')
C(i, j ') C( j, j ') C(k, j ')
C(i, k ') C( j, k ') C(k, k ')
!
"
##
$
%
&&
x
y
z
!
"
##
$
%
&&
Shorthand representation of a coordinate transformations: r’=Rr
Two consecutive rotation: r’’=R2r’ then r’’=R
2R
1r
Matrices are rotation operators.
!r = xi + yj + zkPosition vector:!
Unit vectors: !i , j, k
Rotate coordinates by angle !.!
!!r = !x !i + !y ˆ!j + !z ˆ!kPosition vector :!
Unit vectors: !!i , ˆ!j , ˆ!k
The components of a vector in the rotated coordinates can be obtained!
from the vector in the original coordinate by a matrix operation as follows!
y
C(i, i ') = cos!; C( j, i ') = cos(90 +!) = sin!; C(k, i ') = 0
C(i, j ') = sin!; C( j, j ') = cos!; C(k, j ') = 0
C(i, k ') = 0; C( j, k ') = 0; C(k, k ') = 1
so that !r ' =
cos! " sin! 0
s in! cos! 0
0 0 1
#
$
%%%
&
'
(((
!r = T
!r
CLAS Exercise:
Suppose
!r = 5i + 5 j and !rot = 30
". Find the
Vector !"r in the primed coordinate system.
Example: given a vector in 2-dimensions !
Express the vector in a coordinate sysytem rotated clockwize by an angle !. !
!r = xi + yj
x
x’
y’
#$
j!
i
!j
!i
r!