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 ==