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University Electromagnetism: Electric field calculated from potential distribution
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E-field of a dipole 1
The Electric Field of a Dipole
© Frits F.M. de Mul
E-field of a dipole 2
The Electric Field of a Dipole
Given:
Electric Dipole, situated in O, with dipole moment p [Cm]
P
Op
Question:
Calculate E-field in arbitrary points P around the dipole
E-field of a dipole 3
The Electric Field of a Dipole
• Analysis and symmetry• Approach to solution• Calculations• Conclusions • Field lines in XY-plane
E-field of a dipole 4
Analysis and Symmetry
P
Op
Coordinate axes: X,Y, Z
Y-axis = dipolar axis
X
Z
Y
Symmetry: cylindrical around Y-axis
Cylindrical coordinates:
r, θ and ϕ (perpendicular’
rθ
er
eθ
er
ϕ
eϕ
All points at equal r and θ are equivalent, even if at different ϕ
E-field of a dipole 5
Approach to solution
Several approaches possible:
1. E with “Coulomb”;
2. E from V with: E = - grad V
rθ
P
O
er
p
eθ
er
X Y
Z
ϕ
eϕChoose option 2.
Necessary: calculate V(r,θ,ϕ) first !
E-field of a dipole 6
Intermezzo: V(r,θ,ϕ) -calculation
P
-Q +Q
p=Qa
a
Calculate: Potential in P : (reference in inf.)
r+r-
θ+θ−
+
+=
r
QPV
04)(
πε −
−+
r
Q
04πε
−+
−+ −=
rr
rrQ
04πε
If a << r+ ,r- , then
P
-Q +Q
r- r+
θ
r
a
a cos θ
20
cos
4),(
r
aQrV
θπε
θ =
Contributions to V from +Q and –Q :
E-field of a dipole 7
Intermezzo: V(r,θ,ϕ) -calculation
P
-Q +Q
r- r+
θ
er
r
a
a cos θ2
02
0 4
coscos
4),(
r
p
r
aQrV
πεθθ
πεθ ==
2),(
rrV
0
r
4
ep
πεθ •=
This approximation is called:
Far-field approximation
E-field of a dipole 8
Calculation of E (1)
Calculate E from V using :
E = - grad V
X Y
Z zy z
V
y
V
x
VeeeE x ∂
∂−∂∂−
∂∂−=
In general :
UV
W
ww
vvu s
V
s
V
s
VeeeE u ∂
∂−
∂∂
−∂∂
−=
with su ,sv and sw line elements
E-field of a dipole 9
Calculation of E (2)
rθ
POp
eθ
er
X Y
Z
ϕ
eϕ
ww
vvu s
V
s
V
s
VeeeE u ∂
∂−
∂∂
−∂∂
−=
here u=r, v=θ ,w=ϕ
rsu ∂=∂
r∂
204
cos),(with
r
prV
πεθθ =
rrr eeE3
04
cos2
r
p
r
V
πεθ=
∂∂−=
E-field of a dipole 10
Calculation of E (3)
θ∂=∂ rsv
θ∂r
θ∂
204
cos),(with
r
prV
πεθθ =
θθθ π εθ
θeeE
304
sin
r
p
r
V =∂
∂−=
here u=r, v=θ ,w=ϕ
rθ
POp
eθ
er
X Y
Z
ϕ
eϕ
r∂ ww
vvu s
V
s
V
s
VeeeE u ∂
∂−
∂∂
−∂∂
−=
E-field of a dipole 11
Calculation of E (4)
rθ
Op
eθ
er
X Y
Z
ϕ
eϕ
r∂ ww
vvu s
V
s
V
s
VeeeE u ∂
∂−
∂∂
−∂∂
−=
here u=r, v=θ ,w=ϕ
θ∂r
ϕθ ∂=∂ sinrsv
ϕ∂ϕθ ∂sinr
204
cos),(with
r
prV
πεθθ =
ϕϕϕ ϕθeeE 0
sin=
∂∂−=
r
V
E-field of a dipole 12
Calculation of E (5)
rθ
O
er
p
eθ
er
X Y
Z
eϕ
ϕ
rr eE3
04
cos2
r
p
πε
θ=
Er
θθ πεθ
eE3
04
sin
r
p=
Eθ
ϕϕ eE 0=
E
E-field of a dipole 13
Conclusions (1)
rθ
O
er
p
eθ
er
X Y
Z
ϕ
eϕ
204
cos),(
r
prV
πεθθ =
2),(
rrV
0
r
4
ep
πεθ •=
( )θθθπε
eeE r sincos24 3
0
+=r
p
Er
EθE
Er
E-field of a dipole 14
Conclusions (2)
( )θθθπε
eeE r sincos24 3
0
+=r
p
θ
r
ereθ
ereθ
er
eθ
er
eθ
III
III IV
I II III IV
cos + - - + sin + + - -
Er and Eθ -comp.
E
E -directions in the 4 quadrants:
E-field of a dipole 15
Field lines in XY-plane (1)
rθ
O
p
eθ
er
X
Yrr eE
304
cos2
r
p
πε
θ=
θθ πεθ
eE3
04
sin
r
p=
Field lines in
XY-plane : x
y
E
E
dx
dy=
E-field of a dipole 16
Field lines in XY-plane (2)
Parameter: distance a between -Q and +Q
a = 20
a = 10
a = 1
a = 0.1
far field
E-field of a dipole 17
Field lines in XY-plane (3)
Parameter: distance a between -Q and +Q
a = 20
a = 10
a = 1
a = 0.1
far field
E-field of a dipole 18
Field lines in XY-plane (4)
Parameter: distance a between -Q and +Q
a = 20
a = 10
a = 1
a = 0.1
far field
E-field of a dipole 19
Field lines in XY-plane (5)
Parameter: distance a between -Q and +Q
a = 20
a = 10
a = 1
a = 0.1
far field
E-field of a dipole 20
Field lines in XY-plane (6)
Parameter: distance a between -Q and +Q
a = 20
a = 10
a = 1
a = 0.1
far field
