electrodynamics lecture

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 ==