Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

John David Jackson

Dai-Sik Kim

Nano Optics Lab.

School of Physics and Astronomy

Seoul National University

Classical Electrodynamics I

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

The Laplace equation in rectangular coordinates is

02

02

2

2

2

2

22

zyx

By separation of variables

)()()(),,( zZyYxXzyx

0111

2

2

2

2

2

22

dz

Zd

Zdy

Yd

Ydx

Xd

X

Day 72. Boundary-Value Problems in Electrostatics: I2.9 Separation of Variables; Laplace Equation in Rectangular Coordinates

,1 2

2

2

dx

Xd

X,

1 2

2

2

dy

Yd

Y

2

2

21

dz

Zd

Z

222 where

zyixi eee22

Fourier and heat dissipation

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

]sinhsinsin[1,

zyb

mx

a

nA mn

mn

mn

02

There remains only the boundary condition ),( yxV at cz

]sinhsinsin[),(1,

cyb

mx

a

nAyxV mn

mn

mn

yb

mx

a

nyxVdydx

cabA

ba

mn

mn

sinsin),(

)(sinh

4

00

Day 72. Boundary-Value Problems in Electrostatics: I

As an example, consider a rectangular box,

zyixi eee22

0,0,00 zyxfor

byaxat ,0

Boundary

conditions:

2

2

2

2

b

n

a

mmn where

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

,),( VyxV

yb

m

a

nyxVdydx

cabA

ba

mn

mn

sinsin),(

)(sinh

4

00

If

m & n: odd abmncab

V

mn

22

)(sinh

4

oddmn mn

mn

cmn

zyb

mx

a

n

V,1,

2 )(sinh

sinhsinsin

16

)(sinh

162 cmn

V

mn

Day 72. Boundary-Value Problems in Electrostatics: I

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

]sin[,1

ya

n

n

n exa

nAyx

By the boundary condition,

Vxa

nAyx

n

n

1

sin)0,(

0

14sin)0,(

2

0 n

Vx

a

nxdx

aA

a

n

for n odd

for n even

oddn

ya

n

xa

ne

n

Vyx

sin14

),(

oddn

iyxa

ni

en

V ))((1Im

4

Day 72. Boundary-Value Problems in Electrostatics: I2.10 A Two-Dimensional Potential Problem; Summation of a Fourier Series

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

oddn

n

n

ZVyx Im

4),(

432

4

1

3

1

2

1)1ln( ZZZZZ

Z

Z

n

Z

oddn

n

1

1ln

2

1

432

4

1

3

1

2

1)1ln( ZZZZZ

)(tan)]arg(Im[ln]Im[ln 122

a

bbiaibabia

2

2

2

*

|1|

Im2||1

|1|

11

1

1

Z

ZiZ

Z

ZZ

Z

Z

))(( iyxa

i

eZ

Day 72. Boundary-Value Problems in Electrostatics: I

With the definition,

We can recall the expansion,

Evidently,

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

2

1

||1

)(Im2tan

2]

1

1Im[ln

2Im

4),(

Z

ZV

Z

ZV

n

ZVyx

oddn

n

ysmallforV

a

xeV

a

ya

xV a

y

;ylargeforsin4

sinh

sintan

2 1

Day 72. Boundary-Value Problems in Electrostatics: I

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

From the Laplace equation in 2-D

By using separation of variables

)()(),( R

This leads to

01

2

2

d

dR

d

d

R

2

d

dR

d

d

R

2

2

21

011

2

2

2

2

2

2

z

Day 72. Boundary-Value Problems in Electrostatics: I2.11 Fields and Charge Densities in Two-Dimensional Corners and Along Edges

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

The general solution is

ln)( 00 baR

00)( BA

0

baR )(

)(sin)(cos)( BA

0

: integer

1

00 ))(sin)(cos()(lnn

nn

n

n

n

n nBnAbaBA n: integer

Day 72. Boundary-Value Problems in Electrostatics: I

If there is no restriction on , for single value of at 0 and ,2

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

1

/ )/(sin),(m

m

m maV

v

vv

v

v

v

v vBvAbaBA ))(sin)(cos()(ln 00

0&00 vbAgivesV )or0,0(

VBAv 0&0V )0,( gives

V ),( gives ...,2,1, mm

Day 72. Boundary-Value Problems in Electrostatics: I

0

0V

0 .

for all

when and

Boundary conditions

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

1

/ )/(sin),(n

n

n naV

near 0 the potential is approximately

)/(sin),( /

1 aV

Electric field components are

)/(sin),( 1)/(1

aE

)/(cos1

),( 1)/(1

aE

Surface charge densities at the boundary are

1)/(100 )0,(),()0,(

aE

Day 72. Boundary-Value Problems in Electrostatics: I

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

Day 72. Boundary-Value Problems in Electrostatics: I

1)/(10)0,(

a

The charge density near 0 all vary with distance as .1/

For , the field quantities become independent of .

, For The 2-dimensional corner becomes an edge.

.2/1,2 For the singularity is as

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

Day 7

End of Chapter 2;

close to first exam!

Seoul National University Classical Electrodynamics I

Department of Physics & Astronomy Dai-Sik Kim

1. Let us assume that the electron has a radius r0 with uniform charge distribution. Show that the electrostatic energy is given by:

00

22

20

3

r

emc

2. Estimate the classical radius of electron such that

00

2

20

3

r

eE

•You should know some of the constants by heart or at least should be able toestimate them. Please don’t disappoint me by asking.