Classical And Quantum Dynamics of the Multispherical Nanostructures

Appendix A: Calculation of Field's Energy in a Sphere

We assume the volume V with field is spherical with radius a. First we calculate the "transverse" part


where k 0 = ?/ c, R( r)and Y( ?, ?) are the radial and angular parts of the Debye potential accordingly. In (A.1) through integrating by parts we calculate the radial integral as the following


where . The "longitude" part of electric energy has the following form


So the full electric part of energy becomes


The magnetic part of energy one can be calculated by a similar way


where (see Appendix B). We perform a change variable k 0 r = y 0, k 0 dr = dy 0 in the integral ? a 0 R 2 dr. Now and are given by



where ? = k 0 a = ? a/c. Remember that R( y 0) satisfies the equation


Equations for the TE fields can be derived by the similar way.

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