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ELECTRIC DIPOLES

In these notes, I write down the electric field of a dipole, and also the net force and the
torque on a dipole in the electric field of other charges. For simplicity, I focus on ideal dipoles
— also called pure dipoles — where the distance a between the positive and the negative
charges is infinitesimal, but the charges are so large that the dipole moment p is finite.

Electric Field of a Dipole

The potential due to an ideal electric dipole p is

p ·b
r
V (r) = , (1)
4πǫ0 r 2

or in terms of spherical coordinates where the North pole (θ = 0) points in the direction of
the dipole moment p,
p cos θ
V (r, θ) = . (2)
4πǫ0 r 2

Taking (minus) gradient of this potential, we obtain the dipole’s electric field

 
p 2 cos θ sin θ p 1 b

E = 3
∇r + 2 ∇θ = 2 cos θ b
r + sin θ θ . (3)
4πǫ0 r r 4πǫ0 r 3

r and b
In this formula, the unit vectors b θ themselves depend on θ and φ. Translating them
to Cartesian unit vectors, we have

b b + sin θ sin φ y
r = sin θ cos φ x b + cos θ b
z,
(4)
b
θ = cos θ cos φ x
b + cos θ sin φ y
b − sin θ b
z,

hence

r + sin θ b
2 cos θ b θ = 3 sin θ cos θ(cos φ xb + sin φ yb) + (2 cos2 θ−sin2 θ = 3 cos2 θ−1) bz, (5)

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and therefore
p 3 sin θ cos θ cos φ
Ex (r, θ, φ) = ,
4πǫ0 r3
p 3 sin θ cos θ sin φ
Ey (r, θ, φ) = ,
4πǫ0 r3
p 3 cos2 θ − 1
Ez (r, θ, φ) = .
4πǫ0 r3
In terms of the (x, y, z) coordinates
p 3xz
Ex (x, y, z) = ,
4πǫ0 (x + y 2 + z 2 )5/2
2

p 3yz
Ey (x, y, z) = , (6)
4πǫ0 (x + y 2 + z 2 )5/2
2

p 2z 2 − x2 − y 2
Ez (x, y, z) = ,
4πǫ0 (x2 + y 2 + z 2 )5/2
or in vector notations,
3(p · b
r)b
r−p
E(r) = . (7)
4πǫ0 r 3

Note that along the dipole axis the electric field points in the direction of the dipole
moment p, while in the plane ⊥ to the dipole axis the field points in the opposite direction
from the dipole moment. To get a more general pocture of the dipole’s electric field, here is
the diagram of the electric field lines in the xz plane:

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Force and Torque on a Dipole
Now consider an ideal dipole p placed in an electric field E(x, y, z) due to some other
sources. If this electric field is uniform, there is no net force on the dipole but there is a net
torque. Indeed, the force F+ = +qE acting on the positive charge cancels the opposite force
F− = −qE = −F+ acting on the negative charge — so the net force is zero — but the two
forces are acting at different points, which causes a torque. Specifically, the net torque of
the two forces is

or in terms of the dipole moment p = q(r+ − r− ),

~τ = p × E. (9)

This torque vanishes when the dipole moment p is parallel to the electric field E. Otherwise,
the torque twists the dipole trying to make it align with the field, p → p′ ↑↑ E.

When the electric field E(x, y, z) is not uniform, the two charges of the dipole feel slightly
different electric fields, so the net force on the dipole does not quite vanish:

Fnet = q E(r+ ) − E(r− ) 6= 0. (10)

but for small displacements a = r+ − r− between the charges, we may expand the difference
between the electric fields acting on them into a power series in a. Let r± = r ± 12 a where r
is the center of the dipole; then

E(r± ) = E(r) ± ( 21 a · ∇)E + 1 1
2(2a · ∇)2 E ± 1 1
6(2a · ∇)3 E + ···, (11)
@r @r @r

hence the difference

1 3
E(r+ ) − E(r− ) = (a · ∇)E + 24 (a · ∇) E @r + ···, (12)
@r

so the net force on the dipole is

q

Fnet = q(a · ∇)E + (a · ∇)3 E + ···. (13)
@r 24 @r

For a physical dipole with a finite distance a between the two charges, we must generally

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take into account all the subleading terms in this expansion. But for an ideal dipole we take
the limit a → 0 while p = q × a stays finite, so in this limit q × an → 0 for any n > 1.
Consequently, the leading term q(a · ∇)E in eq. (13) stays finite, but all the subleading terms
q(a · ∇)n E vanish in the pure dipole limit. Thus, the net force on an ideal dipole is simply

Fnet = (p · ∇)E(r). (14)

Similar to the net force, the net potential energy of a dipole obtains as

where

V (r± ) = V (r) ± ( 21 a · ∇)V (r) + 1 1

2(2a · ∇)2 V (r) ± 61 ( 12 a · ∇)3 V (r) + · · · (16)

and therefore

U net = q(a · ∇)V (r) + 1

24 q(a · ∇)3 V (r) + · · · (17)

Again, for a real dipole with a finite distance a between the two charges we should generally
take into account all terms in this series, but for a pure dipole with a → 0 (but finite p = qa)

qa · ∇V (r) = −p · E(r). (18)

Thus, an ideal dipole with moment p located at point r has net potential energy

U(r, p) = −p · E(r). (19)

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The potential energy (19) accounts for the mechanical work of the force (14) when the
dipole is moved around and also for the work of the torque (9) when the dipole is rotated;
thus, both the force (14) and the torque (9) are conservative. To see how this works, consider
infinitesimal dosplacements and rotations of the dipole,

r → r + α
~, p → p + ϕ
~×p (20)

for some infinitesimal vectors α

~ and ϕ
~ . The work of the force (14) and the torque (9) due
to such combined displacement and rotation is

   
δW = α
~ ·F + ϕ
~ · ~τ = α
~ · (p · ∇)E(r) + ϕ
~ · p × E(r) , (21)

so let’s check that the infinitesimal variation of the energy (19) agrees with

δW = −δU(r, p). (22)

Indeed,
 
−δU = +δp · E(r) + p · δE(r) = (δr · ∇)E(r)
  (23)
= (~
ϕ × p) · E(r) + p · (~
α · ∇)E(r) ,

where the first term has form (a × b) · c = c · (a × b) = b · (c × a), thus

1st term = (~
ϕ × p) · E(r) = ϕ
~ · (p × E(r)) ≡ ϕ
~ × τ, (24)

which is precisely the torque term in the work (21). As to the second term in eq. (23),
   
2nd term = p · (~α · ∇)E(r) = −p · (~ α · ∇)∇V (r)
 
= −(~α · ∇)(p · ∇)V (r) = −~ α · (p · ∇)∇V (r) (25)
 
= +~α · (p · ∇)E(r) ≡ α ~ · F,

which is precisely the force term in the work (21). And this proves that the force (14) and
the torque (9) on the dipole are indeed conservative and their work is accounted by the
potential energy (19).

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To be precise, the torque (9) is the torque relative to the dipole center r. In a non-uniform
electric field, the torque relative to some other point r0 has an extra term due to the net
force (14) on the dipole, thus

~τ net = (r − r0 ) × Fnet + ~τ relative to r = (r − r0 ) × (p · ∇)E(r) + p × E(r). (26)

This net torque may also be obtained from the potential energy U — or rather its infinitesimal
variation under simultaneous rotations of the dipole moment vector p and of radius vector
r − r0 of the dipole from the reference point r0 — but I am not going to work it out in these
notes.

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