The definition of electric dipole moment of an electric dipole
Electric dipole is the configuration of two equal and opposite charges separated by a distance.
d p q =
The direction of the electric dipole moment is from the negative
charge to positive charge. The characteristic of a dipole is completely determined by its dipole moment p .
The electric field and electric potential produced by an electric dipole
The electric field produced by a dipole (All the following formula are valid for the approximation of d x >>). The electric field at any point P (U -∇=E )
3
0ˆ)ˆ(341r p r r
p E -⋅=
πε Where r
ˆ means unit vector. While in the spherical coordinate system, if the direction of p is in z axis
)sin cos 2(413
0θθθπεe e E r +=r p
There are two special cases: (1) Along dipole axis (P 1 shown in the figure)
3
021r p
E πε=
And (2) In dipole ’s median plane (P 2 shown in the figure)
3
041r p
E πε-=
The field at any distant points varies with the distance r from the dipole as 3/1r . But the electric field of an electric quadrupole changes as 4/1r .
The electric potential produced by a dipole (Choosing the potential at infinity is zero) (P 3 shown in the figure)
2
020cos 41ˆ41r p r U θ
πεπε=
⋅=r p An electric dipole in an external electric field
The net torque acted on the
dipole about the center of the dipole due to external field 'E p ⨯=τ The potential energy of a
dipole in an external field
(Define the reference angle to be 90o and choose the potential energy to be zero at that angle)
'E p U ⋅-=
The above formulas are valid for intrinsic moment. On the other hand, if the induced moment
of a particle is E p '=
αi , then the energy of the particle in external field is
2i
2
d )(E E p '-='⋅-⎰α ?? Th
e net force acted on an electric dipole due to an external field
The net force acted on a dipole due to a uniform external electric field is zero.
-
0net =F
The net force acted on a dipole due to a nonuniform external electric field is
-+-==F F F F z net This force is along the z axis.
z
E p pE z z U
F z ∂∂=∂∂=∂∂-=')cos ()cos '(θθ
It depends on the gradient of the external field.
This force can be calculated from potential energy. The general relation between conservative force and potential energy is
E p E p
F '∇⋅='⋅--∇=-∇=)(U
(because ()()
''''E p E p E p E p
∇⋅=⨯∇⨯+∇⋅=⋅∇)
[1] David Halliday, Robert Resnick, Kenneth S. Krane. Physics. Fifth Edition. New York: John Wiley & Sons, Inc., 2005.
[2] 姚志欣. 偶极子在非均匀外场中的受力. 大学物理, 1990, 9(10): 10-13 [3]尹真.电动力学（第二版）.科学出版社，2005
磁偶极矩的定义
载流平面线圈的电流强度I 和线圈面积S 的乘积叫做载流线圈
的磁矩。用m 表示线圈的磁矩，n
ˆ表示线圈平面法线方向的单位矢量，则
n IS I ˆd ='=⎰S m ，
磁偶极矩产生的磁场
3
0ˆ)ˆ(34r m r r
m B -⋅=
πμ
若取m 方向与z 轴一致，在球坐标中有（r r <<'）
)sin cos 2(430θθθπμe e B r +=r
m
也可以用毕—萨定律计算P 点的磁感应强度
⎰⨯=3ˆ4R R l Id B O
π
μ
2
/5222
2
0)'(4
'3r z x xz
Ir B x ++⋅=
μ
2/52222
202/322220)
'(4'3)'(12'r z x x Ir r z x Ir B z ++⋅-++⋅=μμ
特别的，当0=x 时
2
/3222
0)
'(1
2' 0r z Ir B B z x +⋅=
=μ， )'( '2320r z z r I B z >>⋅=μ 外磁场中的磁偶极子
利用电流与外场的互作用能公式⎰⋅=v U d A J ，得磁偶极子在外磁场B '中的能量是
-
