# Near-field radiative energy transfer between sphere-sphere and sphere-flat surfaces

Near-field thermal radiation between two spheres was first reported by Volokitin and Persson [1] by assuming them as dipoles. When there are two spherical nanoparticles whose dielectric constants are ε1 and ε2 , the spectral power dissipated in particle 2 by the electromagnetic field induced by particle 1 can be written using the dipolar approximation as

(1)

where x2 is the position of the particle 2 and α2 = 4πR32 − 1) / (ε2 + 2) is the polarizability of a sphere of radius R having the relative permittivity of ε2. The electric field incident on the particle 2, Einc(x2,ω), is created by the thermal fluctuating dipole of particle 1 at Temperature T1 :

(2)

where $\rm \overline {\overline G}_e (x_2,x_1,\omega)$ is the electric dyadic Green’s function between two dipoles in vacuum and expressed as [2]

(3)

where