Introduction to Electrostatics. The Electric Dipole
and Multipoles
1 The electric dipole
A dipole is defined as two equal and opposite charges that are positioned at a short distance
from each other.
The figure shows such a system of charges lined along z axis.
The potential at P(r, θ , φ ) is
Q 1 1
V= − (1)
4πε0 r1 r2
The two charges are separated by distance d.
When r >> d,
r2 − r1 = dcosθ (2)
1
,and
r1 r2 = r2 (3)
Therefore,
Q r2 − r1 Q dcosθ
V= = (4)
4πε0 r1 r2 4πε0 r2
From the figure, dcosθ = d.r̂, since d = d r̂ It is important to observe that the vector d is ori-
ented along the z-axis, extending from the negative charge to the positive charge. By defining
a vector p = Qd, we can express the potential as follows.
p.r̂
V= (5)
4πε0 r2
This vector, whose magnitude is equal to the product of any one of the charges comprising
the dipole, and the direction from negative to positive charge, is called the dipole moment. Its
unit is Coulomb meter (C m) in the SI system of units.
If the dipole’s center is located at a point represented by the position vector r0 , rather than at
the origin, the potential generated by this dipole at a point indicated by the position vector r
can be expressed as follows.
p.(r − r0 )
V= (6)
4πε0 |r − r0 |3
From Eq.(4), we see that the potential in the z=0 plane (θ = 900 ), is zero.
We can find the electric field produced by a dipole using the relation
∂V 1 ∂V 1 ∂V
E = −∇V = − r̂ + θ̂ + φ̂ (7)
∂r r ∂θ rsinθ ∂ φ
we obtain
Q 2dcosθ Q dsinθ
E=− − r̂ − θ̂ (8)
4πε0 r3 4πε0 r3
i.e.,
Qd
E= (2 cosθ r̂ + sinθ θ̂ ) (9)
4πε0 r3
i.e.,
p
E= (2 cosθ r̂ + sinθ θ̂ ) (10)
4πε0 r3
The resultant electric field at P thus has a magnitude.
p p
q
E = E2r + E2θ = 3cos2 θ + 1 (11)
4πε0 r3
The streamlines representing the potential and electric field generated by the dipole can be
visualized. This is achieved by expressing them in the format r = f (θ ). The potential can be
written as follows.
cosθ
r2 = (12)
V
2
, or r
cosθ
r=± (13)
V
The electric field has only r and θ components.
Eθ sinθ r dθ
= = (14)
Er 2 cosθ dr
or
dr
= 2cotθ dθ (15)
r
i.e,
r = C1 sin2 θ (16)
A polar graph of these gives the streamlines of the potential and the field.
A mathematica code to plot the polar graph of the electric field and potential is given below.
3
and Multipoles
1 The electric dipole
A dipole is defined as two equal and opposite charges that are positioned at a short distance
from each other.
The figure shows such a system of charges lined along z axis.
The potential at P(r, θ , φ ) is
Q 1 1
V= − (1)
4πε0 r1 r2
The two charges are separated by distance d.
When r >> d,
r2 − r1 = dcosθ (2)
1
,and
r1 r2 = r2 (3)
Therefore,
Q r2 − r1 Q dcosθ
V= = (4)
4πε0 r1 r2 4πε0 r2
From the figure, dcosθ = d.r̂, since d = d r̂ It is important to observe that the vector d is ori-
ented along the z-axis, extending from the negative charge to the positive charge. By defining
a vector p = Qd, we can express the potential as follows.
p.r̂
V= (5)
4πε0 r2
This vector, whose magnitude is equal to the product of any one of the charges comprising
the dipole, and the direction from negative to positive charge, is called the dipole moment. Its
unit is Coulomb meter (C m) in the SI system of units.
If the dipole’s center is located at a point represented by the position vector r0 , rather than at
the origin, the potential generated by this dipole at a point indicated by the position vector r
can be expressed as follows.
p.(r − r0 )
V= (6)
4πε0 |r − r0 |3
From Eq.(4), we see that the potential in the z=0 plane (θ = 900 ), is zero.
We can find the electric field produced by a dipole using the relation
∂V 1 ∂V 1 ∂V
E = −∇V = − r̂ + θ̂ + φ̂ (7)
∂r r ∂θ rsinθ ∂ φ
we obtain
Q 2dcosθ Q dsinθ
E=− − r̂ − θ̂ (8)
4πε0 r3 4πε0 r3
i.e.,
Qd
E= (2 cosθ r̂ + sinθ θ̂ ) (9)
4πε0 r3
i.e.,
p
E= (2 cosθ r̂ + sinθ θ̂ ) (10)
4πε0 r3
The resultant electric field at P thus has a magnitude.
p p
q
E = E2r + E2θ = 3cos2 θ + 1 (11)
4πε0 r3
The streamlines representing the potential and electric field generated by the dipole can be
visualized. This is achieved by expressing them in the format r = f (θ ). The potential can be
written as follows.
cosθ
r2 = (12)
V
2
, or r
cosθ
r=± (13)
V
The electric field has only r and θ components.
Eθ sinθ r dθ
= = (14)
Er 2 cosθ dr
or
dr
= 2cotθ dθ (15)
r
i.e,
r = C1 sin2 θ (16)
A polar graph of these gives the streamlines of the potential and the field.
A mathematica code to plot the polar graph of the electric field and potential is given below.
3
