Classical Electrodynamics

Physics II

Reference Books

1. Introduction to Electrodynamics by David. J. Griffiths

2. Classical Electrodynamics by John David Jackson

3. Electricity and Magnetism by Edward M. Purcell

%

45

10

45

Final exam

quizzes

Mid term exam.

� Electrostatics

�Special techniques

�Concepts of Dipole

�Electric Field in Materials

� Magnetostatics

�Magnetic Field in Materials

�Electrodynamics

�Maxwell’s Equation

� Review of Mathematical Tools 4 lectures

1 lectures

1 lectures

1 lectures

1 lectures

2 lectures

2 lectures

3 lectures

2 lectures

(x,y,z)

(x′,y ′,z ′)

θcosABAr)r

=•

Vector Field / Vector Function

Vector Calculus

jxiyF ˆsinˆsin +=r

A vector field describing the velocity of a flow in a pipe

Velocity vector field of a flow around a aircraft wing

Circular flow in a tub

kzyxFjzyxFizyxFzyxF ˆ),,(ˆ),,(ˆ),,(),,( 321 ++=r

kxjzxixyzzyxF ˆˆˆ),,( 42 +−=r

dx

dfh

xfhxfh

)()(lim

0

−+→

Scalar field),,( zyxTT =

h

zyxfzyhxfzyxf

hx

),,(),,(lim),,(

0

−+=→

h

zyxfzhyxfzyxf

hy

),,(),,(lim),,(

0

−+=→

h

zyxfhzyxfzyxf

hz

),,(),,(lim),,(

0

−+=→

Scalar Field

Vector Field

T(x,y)

Ty

Yx

XT

∂∂+

∂∂=∇ ˆˆ

∇⋅≠⋅∇ VVrr

In many cases, the divergence of a vector function at point P may be predicted by considering a closed surface surrounding P and analyzing the flow over the boundary, keeping in mind that at P:

=⋅∇ Fr

outflow – inflow

Paddle wheel analysis

0ˆ)],([ 00 <⋅×∇ kyxFr

jxiyyxF ˆˆ),( +−=r

2=×∇ Fr

iyyxF ˆ),( =r

1−=×∇ Fr

jxiyyxF ˆˆ),( +=r

jyixyxF ˆˆ),( +=r

0=×∇ Fr

Laplacian oparator

Vector Line Integration

http://upload.wikimedia.org/wikipedia/commons/d/d8/Line-Integral.gif

Vector surface Integral

Volume Integral

Difference of function’s value at b and a

The integral of a derivative over a region is equal to the value of the function at the boundary

0=∇×∇ Trr

0=×∇ Frr

F conservative field

=⋅∇ Vr

outflow – inflow +ve (source)-ve (sink)

The integral of a derivative over a region is equal to the value of the function at the boundary

The integral of a derivative over a region is equal to the value of the function at the boundary

Stokes’ theorem

rotational force field / non-conservative force field

Cylindrical and spherical co-ordinate system

φρ

φr

(ρ,φ,z)

z

Y

X

zz

y

x

aa

aaa

aaa

=

+=

−=

φφφφ

φρ

φρ

cossin

sincos

zz

yx

yx

aa

aaa

aaa

=

+−=

+=

φφφφ

φ

ρ

cossin

sincos

−=

zz

y

x

a

a

a

a

a

a

ϕ

ρ

φφφφ

100

0cossin

0sincos

−=

z

y

x

z a

a

a

a

a

a

100

0cossin

0sincos

φφφφ

φ

ρ

zayaxaA zyx ˆˆˆ ++=r

zaaaA z ˆˆˆ ++= φρ φρ

r

ρρρρφφφφρρρρρρρρφφφφ ˆˆ3 dzddIdIda z ==

zddzdIdIda ˆˆ2 ρρρρφφφφρρρρρρρρφφφφ ==

φφφφρρρρφφφφρρρρˆˆ

1 dzddIdIda z ==

The infinitesimal surface elements

dφ

dρρ

ρdφdz

dφ

dρρ

ρdφdz

Volume element in cylindrical coordinate system

dz

dρ

y

x

φ

φ

z

dz

dρ

y

x

φ

φ

z

φφφφρρρρφφφφρρρρˆˆ

1 dzddIdIda z ==

dρ

ρdφy

x

φ

dρ

ρdφy

x

φ

zddzdIdIda ˆˆ2 ρρρρφφφφρρρρρρρρφφφφ ==

dz

ρdφ

y

x

φ

φ

z

dz

ρdφ

y

x

φ

φ

z

ρρρρφφφφρρρρρρρρφφφφ ˆˆ3 dzddIdIda z ==

(r,θ,φ)

r

θ

φ

θθφφθφθφφθφθ

θ

φθ

φθ

sincos

cossincossinsin

sincoscoscossin

aaa

aaaa

aaaa

rz

ry

rx

−=

++=

−+=

−

−=

φ

θ

θθφφθφθφφθφθ

a

a

a

a

a

a r

z

y

x

0sincos

cossincossinsin

sincoscoscossin

−−=

z

y

xr

a

a

a

a

a

a

0cossin

sinsincoscoscos

cossinsincossin

φφθφθφθ

θφθφθ

φ

θ

zayaxaA zyx ˆˆˆ ++=r

φθ φθˆˆˆ aaraA r ++=

r

r

z

y

x

dr

rdθ

rsinθdφ

dφφ

θ

The infinitesimal surface elements

φφφφθθθθφφφφθθθθˆˆ

3 rdrddIdIda r ==