Mathematical Methods of Physics/The multipole expansion

Tensors are useful in all physical situations that involve complicated dependence on directions. Here, we consider one such example, the multipole expansion of the potential of a charge distribution.

Introduction

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Consider an arbitrary charge distribution  . We wish to find the electrostatic potential due to this charge distribution at a given point  . We assume that this point is at a large distance from the charge distribution, that is if   varies over the charge distribution, then  

Now, the coulomb potential for a charge distribution is given by  

Here,  , where  

Thus, using the fact that   is much larger than  , we can write  , and using the binomial expansion,

  (we neglect the third and higher order terms).

The multipole expansion

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Thus, the potential can be written as  

We write this as  , where,

 

 

 

and so on.

Monopole

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Observe that   is a scalar, (actually the total charge in the distribution) and is called the electric monopole. This term indicates point charge electrical potential.

Dipole

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We can write  

The vector   is called the electric dipole. And its magnitude is called the dipole moment of the charge distribution. This terms indicates the linear charge distribution geometry of a dipole electrical potential.

Quadrupole

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Let   and   be expressed in Cartesian coordinates as   and  . Then,  

We define a dyad to be the tensor   given by  

Define the Quadrupole tensor as  

Then, we can write   as the tensor contraction   this term indicates the three dimensional distribution of a quadruple electrical potential.