Electrostatic Boundary Value Problem for a Metal Sphere

Electrostatic Boundary Value Problem for a Metal Sphere 

One source of the electromagnetic (EM) enhancement comes from the interaction of the uniform exciting field Eq with the metallic particle. The metallic particle has a high conductivity which is expressed by its internal 

The problem of finding the electric field at r is treated by solving the Laplace equation, Vd = 0, for the electrostatic potential, 4 , with the appropriate boundary conditions. The electric field is then found from the negative gradient of the electrostatic potential. This is a correct procedure for the problem if the time-varying field of the incident light is suitably averaged as it will be in our case. The boundary conditions are  

The solutions are given in a standard electrodynamics textbook, and so the potential outside the sphere is found to be  

The potential generated outside the sphere contains two contributions, i.e., one from the incident field (first term) and another from the field of an electric dipole located at the center of the sphere with polarizability ga oriented in the direction of the incident field (second term). Thus the local electric potential at position r, in the small-sphere approximation, is equivalent to that generated by the applied incident field and a reflected field produced by the metal sphere. 

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