Classical And Quantum Dynamics of the Multispherical Nanostructures

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.