!!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!