Potential Energy in an Electric Field

Electric multipole moments also play an important role in the description of interactions between molecules and external electric fields. The potential energy E of a distribution of charges p f) immersed in an external inhomogeneous static electric field is given as  

The integrals over p r) are again the electric moments defined in Eqs. (4.4)-(4.6). 

The energy E , ) of the interaction between a charge distribution and a static but inhomogeneous electric field can therefore be expressed in terms of the electric 

Exercise 4.3 Show that the contribution J2ap Qap Ro) ap Ro) of the second-order electric moment to the interaction energy in Eq. (4.17) is unchanged if an arbitrary constant C is added to the diagonal elements Qaa of the second-order eiectric moment. 

Hint Recall that we only use the scalar potential j Ro) far away from the charges that originally generated it and that therefore it satisfies Laplace s equation V 4 (.Ro) = 0. This leads to an equation that can be used in the solution of this exercise. 

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