Physics 2961
Intro. to Modern Physics!
Textbook: Rohlf, Modern Physics, Wiley!
Imaginary Numbers and Complex Notation!
Representation on a 2-dimensional complex plane.!
z = x + iy = r(cos ! + i sin ! )
i!
e +e
cos ! =
2
"i!
i!
e "e
; sin ! =
2i
y=
Im(z)
"i!
x= Re(z)
Solve above two equations for ei! !
Euler"s identity.!
ei! = cos ! + i sin !
) = rei!
"i!
z* = r ( cos ! " i sin ! ) = re
z = r cos ! + i sin !
Magnitude!
z 2 = zz* = (x + iy)(x ! iy) = x 2 + y 2 = r 2
or
z 2 = rei" re!i"
= r2
Plane Wave in Complex Notation
Plane wave:
f (x,t) = A cos(kx ! " t + # )
Wave number: k = 2$ / %
Angular Frequency: " = 2$& = 2$ / T
Complex notation: f (x, t) = Aei(kx!" t+# ) = A ( cos(kx ! " t + # ) + i sin(kx ! " t + # ))
Phase angle: f (0, 0) = A ( cos # + i sin # ) ,
' f (0, 0) *
Real part of f : # = cos )
( A ,+
Intensity: I - ff * = Aei(kx!" t +# ) A*e!i(kx!" t +# ) = AA* = A
!1
Three Dimensions!
! !
!
!
k ! k, x ! r, kx ! kx x + ky y + kz z = kir
i(kx x+ky y+kz z"# t+$ )
f (x, y, z,t) = Ae
! !
i( k i r "# t+$ )
= Ae
Vector Wave!
! ! i( k! ir! !" t+# )
E = E0 e
Vector Calculus!
Read Rohlf, P576 to 577!
Gradient Operator
The gradient operator gives the direction and magnitude of the steepest!
rate of increase of a scalar function
. !
Cartesian coordinates.!
!
#F #F #F
!F "
i+
j+
k
#x
#y
#z
General coordinates.!
!
#F
#F
#F
!F "
u1 +
u2 +
u3
#s1
#s2
#s3
F(x, y, z)
The del or gradient operator in Cartesian and spherical coordinate systems.!
Rohlf, Appendix E, P 587.!
Cartesian coordinates
!
$ #f
#f #f
!f " &
i+
j+
#y
#z
% #x
'
k)
(
Spherical Coordinates:
f = f (r,! , " )
!
$f
1 $f
1 $f
#f = ur
+ u!
+ u"
$r
r $!
r sin ! $"
Divergence and Curl
!
! is a vector one can dot it with another vector.
Since
This is known as the divergence of a vector field.!
Cartesian coordinates.!
! !
!
"
"
"
! E = div( E) =
Ex + E y + Ez
"x
"y
"z
Cylindrical coordinates:
! ! 1 "
1 "
"
! E =
rEr +
E# +
E
r "r
r "#
"z z
( )
( )
( )
Spherical coordinates:
! ! 1 " 2
1
"
1 "E$
! E = 2
r Er +
E# sin# +
r sin# "#
r sin# "$
r "r
Cross product or curl !
! !
A! E =
Ax
Ay
Az
Ex
Ey
Ez
i
! !
#
! " E = curl(E) =
#x
Ex
j
#
#y
Ey
k
#
#z
Ez
Cross product or curl !
i
! !
#
! " E = curl(E) =
#x
Ex
Cartesian coordinates:!
ur
Circular cylindrical coordinates:
Spherical coordinates:
! ! 1 $
!" E=
r $r
Er
! !
!" E=
j
#
#y
Ey
ru#
uz
$
$#
rE#
$
$z
Ez
k
#
#z
Ez
ur
ru#
r sin # u$
%
r sin # %r
Er
%
%#
rE#
%
%$
r sin # E$
Some Vector Calculus Identities:
!
! ! ! !
!
! " u = "! u + u !"
!
! ! !
!
!
! # " u = "! # u + !" # u
! ! ! ! ! ! ! ! !
!u # v = v !#u $u!# v
!
! !
! !! ! !! ! ! ! ! ! !
! # u # v = v !u $ u !v + u ! v $ v ! u
) (
! ! !
! !! ! !! ! ! ! ! ! !
! u v = u !v + v !u + u " ! " v + v " ! " u
!
!
! " !# = 0
! ! !
! !"u = 0
!
! !
! ! ! ! !!
! " ! " u = ! ! u $ ! !u
! !
!
! !#1 " !# 2 = 0
( )
( )
( ) (
(
)
In the above, assume that ! operates on all terms to its right
that are not separated from it by intervening parentheses.
Vector Calculus Operations!
Read Rohlf Appendix C, P576-577!
Gradient: !
!
$ #f
#f #f '
!f " &
i+
j+
k)
#x
#y
#z
%
(
Divergence:
! !
! "Ex "E y "Ez
! E = div( E) =
+
+
"x
"y
"z
Curl:!
Laplacian:!
j k
i
! !
# # #
! " E = curl(E) =
#x #y #z
Ex E y Ez
! !
#2 f #2 f #2 f
! f " !i !f = 2 + 2 + 2
#x
#y
#z
2
( )
Divergence Theorem
(Rohlf, P 576)!
! !
! !
"
!! Pida = !!! "i Pdv
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:!
!
da
!
P
Example of Divergence Theorem
! !
! !
"
!! Aida = !!! "i Adv
Gauss" Law!
.
! ! 1
q
Eid
a
=
#
dv
=
"
""
! 0 """
!0
! !
q
$i
E
dv
=
"""
!0
q
#
dv
=
"""
!0
! !
! 0 # = $i E (Gauss's Law)
Stokes Law
(Rohlf, P 577)!
! ! !
! !
## (! " P)ida = "# P dl
! !
!
The flux of ! " P is the circulation of P around any closed loop which bounds
the surface. The curl therefore is a measure of the rotation of the vector field. !
!
da
!
P
!
dl
Class Exercise - Vector Calculus !
1.The gravitational potential is U=Gy J/kg-m. Find the gradient.!
!
!
and the gravitational field, which is g = !"U .!
2. A sticky fluid is moving past a flat horizontal surface!
such that the velocity is given by
!
v = 10yi m/s.!
Find both magnitude and direction of the curl,.!
3. The electric field inside a uniformly charged dielectric!
!
is E = 10xi. Find the divergence and therefore the charge!
distribution.!
