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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
V (r) = (^4) πǫp^ ·^ ̂r 0 r^2
or in terms of spherical coordinates where the North pole (θ = 0) points in the direction of the dipole moment p,
V (r, θ) = (^4) πǫp 0
cos θ r^2.^ (2)
Taking (minus) gradient of this potential, we obtain the dipole’s electric field
E = (^4) πǫp 0
(2 cos θ r^3 ∇r^ +
sin θ r^2 ∇θ
= (^4) πǫp 0
r^3
2 cos θ ̂r + sin θ ̂θθ
In this formula, the unit vectors ̂r and ̂θθ themselves depend on θ and φ. Translating them
to Cartesian unit vectors, we have
̂ r = sin θ cos φ ̂x + sin θ sin φ ̂y + cos θ ̂z,
̂θ θ = cos θ cos φ ̂x + cos θ sin φ ̂y − sin θ ̂z, (4)
hence
2 cos θ ̂r + sin θ ̂θθ = 3 sin θ cos θ(cos φ ̂x + sin φ ̂y) + (2 cos^2 θ−sin^2 θ = 3 cos^2 θ−1) ̂z, (5)
and therefore
Ex(r, θ, φ) = (^4) πǫp 0
3 sin θ cos θ cos φ r^3 , Ey(r, θ, φ) = (^4) πǫp 0
3 sin θ cos θ sin φ r^3 , Ez(r, θ, φ) = (^4) πǫp 0
3 cos^2 θ − 1 r^3. In terms of the (x, y, z) coordinates
Ex(x, y, z) = (^4) πǫp 0
3 xz (x^2 + y^2 + z^2 )^5 /^2 , Ey(x, y, z) = (^4) πǫp 0
3 yz (x^2 + y^2 + z^2 )^5 /^2 , Ez(x, y, z) = (^4) πǫp 0 2 z
(^2) − x (^2) − y 2 (x^2 + y^2 + z^2 )^5 /^2 ,
or in vector notations,
E(r) = 3(p 4 ·πǫ^ ̂r)̂^ r^ −^ p 0 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:
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 · ∇)nE 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
Unet^ = qV (r+) − qV (r−) (15)
where
V (r±) = V (r) ± (^12 a · ∇)V (r) + 12 (^12 a · ∇)^2 V (r) ± 16 (^12 a · ∇)^3 V (r) + · · · (16)
and therefore
Unet^ = q(a · ∇)V (r) + 241 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) the subleading terms become negligible compared to the leading term
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)
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)
]
[
p × E(r)
]
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)
]
= (ϕ~ × p) · E(r) + p ·
[
(~α · ∇)E(r)
]
where the first term has form (a × b) · c = c · (a × b) = b · (c × a), thus
1 st^ 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),
2 nd^ term = p · [(~α · ∇)E(r)]^ = −p · [(α~ · ∇)∇V (r)] = −(~α · ∇)(p · ∇)V (r) = −~α · [(p · ∇)∇V (r)] = +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).
