Physics 2961

Intro. to Modern Physics!

Textbook: Rohlf, Modern Physics, Wiley!

Imaginary Numbers and Complex Notation!

x= Re(z)

y=

Im(z) !

z z = x + iy = r(cos! + i sin! )

ei!= cos! + i sin!

z = r cos! + i sin!( ) = rei!

z* = r cos! " i sin!( ) = re" i!

cos! =

ei!+ e

"i!

2; sin! =

ei!

" e"i!

2i

z

2

= zz* = (x+ iy)(x! iy)= x2+ y2

= r 2 or z2

= rei"re!i"= r 2

Magnitude!

Solve above two equations for ei! !Euler"s identity.!

Representation on a 2-dimensional complex plane.!

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#( ), Real part of f : # = cos!1 f (0,0)

A

'()

*+,

Intensity: I - ff*= Aei(kx!"t+# )A*e!i(kx!"t+# ) = AA* = A2

k!!k , x!

!r , kx! kx x + ky y + kzz =

!k i!r

f (x, y,z,t) = Aei(kxx+kyy+kzz"#t+$ )

= Aei(!k i!r"#t+$ )

Three Dimensions!

Vector Wave!

!E =!E0ei(!k i!r!"t+# )

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 . !

!

!F "#F

#s1

u1+#F

#s2

u2+#F

#s3

u3

!

!F "#F

#xi +

#F

#yj +

#F

#zk

Cartesian coordinates.!

General coordinates.!

F(x, y, z)

The del or gradient operator in Cartesian and spherical coordinate systems.!Rohlf, Appendix E, P 587.!

Cartesian coordinates

Spherical Coordinates:

f = f (r,!,")

!#f = u

r

$f

$r+ u!

1

r

$f

$!+ u"

1

r sin!

$f

$"

!!f "

#f

#xi +

#f

#yj +

#f

#zk

$

%&'

()

!

!

Divergence and Curl

Since is a vector one can “dot” it with another vector. !

This is known as the divergence of a vector field.!!!

!! •!E = div(

!E) =

"

"xE

x+

"

"yE

y+

"

"zE

z

Cylindrical coordinates: !

!! •!E =

1

r

"

"rrE

r( ) +

1

r

"

"#E#( ) +

"

"zE

z( )

Spherical coordinates:

!! •!E =

1

r2

"

"rr

2E

r( ) +1

r sin#

"

"#E#

sin#( ) +1

r sin#

"E$

"$

Cartesian coordinates.!

Cross product or curl !

!! "!E = curl(E) =

i j k

#

#x

#

#y

#

#z

Ex

Ey

Ez

!A!!E =

i j k

Ax

Ay

Az

Ex

Ey

Ez

Cross product or curl !

!! "!E = curl(E) =

i j k

#

#x

#

#y

#

#z

Ex

Ey

Ez

!! "!E =

1

r

ur

ru#

uz

$

$r

$

$#

$

$z

Er

rE#

Ez

!! "!E =

1

r2

sin#

ur

ru#

r sin#u$

%

%r

%

%#

%

%$

Er

rE#

r sin#E$

Cartesian coordinates:!

Circular cylindrical coordinates: !

Spherical coordinates:

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.

!!

!!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

.

!Aid!a"!! =

!"i

!A!!! dv

Example of Divergence Theorem

!Eid!a =

1

!0

""" #dv =q

!0

"""

!$i"""!Edv =

q

!0

#dv =q

!0

""" !0# =

!$i

!E (Gauss's Law)

Gauss" Law!

(!!"!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)!

Class Exercise - Vector Calculus !

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