PDF ρ being the - National Institute of Science Education and Research

Instead of discrete charges forming dipole, if we have a distribution of charges, ρbeing the charge density, then the potential is,

Vd = 1 4πǫ0

1 r²

Z

r^′ρcosθdτ^′ (11)

The integration here is performed over source coordinate, where the charge distribution is, and not over the field coordinates. Compare the dipole potential with our usual “monopole”

potential,

Vm(r) = 1 4πǫ0

q

r, Vd(r) = 1 4πǫ0

~ p·ˆr

r² The dipole potential is related to monopole potential for unit charge as,

Vd(r) = 1 4πǫ0

~ p· rˆ

r² = 1 4πǫ0

~ p·

−∇

1 r

= −p~· ∇ 1

4πǫ0

1 r

⇒Vd(r) = −p~· ∇Vm(r). (12)

The electric field due to the dipole at point P is, E~^d = −∇V^d = − 1

4πǫ0

∇ p~·~r

r³

= − 1 4πǫ0

∇(~p·~r)

r³ + (~p·~r)∇ 1

r³

Let us simplify the two terms by applying indentities of vector calculus,

∇ 1

r³

= −3~r r⁵

∇(~p·~r) = ~p×(∇ ×~r) +~r×(∇ ×p) + (~~ p· ∇)~r+ (~r· ∇)~p

Since ∇ ×~r= 0 and ~pis constant, the first, second and last terms in above last expression are zero, and therefore ∇(~p·~r) =~p since the only surviving term (~p· ∇)~r = 1. Hence, the electric field due to dipole is,

E~d = 1 4πǫ0

3(~p·~r)~r r⁵ − ~p

r³

(13) Specifically, in spherical polar coordinate the electric field is

E~ = −∇Vd =

ˆ r ∂

∂r + ˆθ1 r

∂

∂θ + ˆφ 1 r sinθ

∂

∂φ 1

4πǫ0

pcosθ

r² = 1 4πǫ0

p r³

2 cosθrˆ+ sinθθˆ . (14) As discussed before, there is no reason that the polarized or polar object be di-polar only, it can have higher moment. For some general distribution of charge, the potential is,

V = K0

r +K2

r² +K4

r³ +K8

r⁴ +· · · (15)

where, the constantsK’s corresponds to

K0 = monopole term K2 = dipole term K4 = quadrupole term K8 = octopole term

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Hydrogen bondingand Van der Waal’sforces originate from quarupole and octopole terms.

The origin of the multi-polar terms follows from multipole expansion of the potential of a localized charge distribution, which is given as

V = 1 4πǫ0

∞

X

n=0

1 rⁿ⁺¹

Z

(r^′)ⁿPn(cosθ^′)ρ(r^′)dτ^′ (16) where the prime denotes source coordinate andPⁿ(x) is the Legendre polynomials of order n. Then= 0 term is the monopole contribution (∼1/r),n= 1 term is the dipole (∼1/r²), n= 2 is the quadrupole (∼1/r³), n= 3 is the octopole (∼1/r⁴) etc.

Example 20. A spherical shell of radius R carries a surface charge σ = kcosθ. Calculate the dipole moment of this charge distribution.

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