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Dipole (changes)

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Contents

Idea

The concept of electric - and to a lesser extend - magnetic dipole is important for classical electrodynamics and the understanding of interaction of matter with electromagnetic radiation, like for example the understanding of molecular emission spectra.

Details

Let us assume that there is a time independent distribution ρ\rho of electrical charges contained in a sphere SS around the origin of a cartesian coordinate system. This charge distribution generates an electric potential

Φ(x)=14πϵ 0ρ(y)xydy \Phi(\vec{x}) = \frac{1}{4 \pi \epsilon_0} \int \frac{\rho(\vec{y})}{ \| \vec{x} - \vec{y} \|} d \vec{y}

When we describe the electric potential as a function of spherical coordinates Φ(r,ϕ,θ)\Phi(r, \phi, \theta), far away from the sphere SS, it will decrease proportional to 1r\frac{1}{r}, so that we can express it as a series like

Φ(r,ϕ,θ)= n=1 f(ϕ,θ)1r n \Phi(r, \phi, \theta) = \sum_{n = 1}^{\infty} f(\phi, \theta) \frac{1}{r^n}

For an electric charge flying by the sphere SS, it will be important to know the coefficients of the first summands of this series in order to determine the most important effects that will influence the charge.

When we choose as an orthonormal basis on a sphere the spherical harmonics Y lm(ϕ,θ)Y_{l m}(\phi, \theta) , with proper normalization we get what is called the multipole expansion of the electric potential:

Φ(r,ϕ,θ)=14πϵ 0 l=0 m=l l4π2l+1q lmY lm(ϕ,θ)r l+1 \Phi(r, \phi, \theta) = \frac{1}{4 \pi \epsilon_0} \sum_{l = 0}^{\infty} \sum_{m = -l}^{l} \frac{4 \pi}{2 l +1} q_{l m} \frac{Y_{l m}(\phi, \theta)}{r^{l+1}}

The l=0l = 0 term is called the monopole term, it is proportional to the electric charge qq contained in the sphere SS. So the first term in the expansion tells us if a charge flying by SS will feel a net attractive or repulsive force.

The terms for l=1l = 1 form a vector p\vec{p} which is called the dipole moment. The next terms in the series Q ijQ_ij form the quadrupol tensor. So, for the expansion of the potential we get

Φ(r,ϕ,θ)=14πϵ 0(qr+pxr 3+12Q ijx ix jr 5+) \Phi(r, \phi, \theta) = \frac{1}{4 \pi \epsilon_0} (\frac{q}{r} + \frac{\vec{p} \cdot \vec{x}}{r^3} + \frac{1}{2} \sum Q_{ij} \frac{x_i x_j}{r^5} + \cdot \cdot \cdot)

For atoms and molecules the net charge qq is zero, so the next relevant term in the series expansion of their electric potential is the dipole moment.

The vector p\vec{p} is calculated via

p=xρ(x)dx \vec{p} = \int \vec{x} \rho(\vec{x}) d \vec{x}

which has a very simple geometric interpretation: It is the vector pointing from the center of negative charge to the center of positive charge. As an example, put a negative elementary charge at the origin represented by the density eδ 0(x)-e \delta_0( \vec{x}), and a positive one outside of the origin represented by eδ x 0(x)e \delta_{\vec{x}_0}(x), where δ y\delta_{\vec{y}} is the Dirac delta function supported at y\vec{y}. Then we get

p=xe(δ x 0(x)δ 0(x))dx=x 0 \vec{p} = \int \vec{x} e (\delta_{\vec{x}_0}(x)- \delta_0( \vec{x})) d \vec{x} = \vec{x}_0

References

  • John David Jackson: Classical Electrodynamics (Wiley; 3 edition (August 10, 1998))