Answer Happy

could you please help me with question 3 (a) and (b)
Question 3 A small metal ellipsoid with dielectric constant & and dimensions a, b, and c along the principal axes, is embedded in a dielectric with relative permittivity & ,. Under an applied electric field, the electrical polarizability along the l'h axis is given by a = 478, (abc) 38, +31 (6. - €) where L is a geometrical factor describing the shape of the ellipsoid and its orientation with respect to the electric field (i=1, 2 or 3), and the sum of all L, is 1. The dielectric response of the metal is described by a Drude model in which the real part of the dielectric constant is given by €.(@)=1=(Q),/w) ' where w, is the plasmon frequency, and the imaginary part can be neglected. 0 (a) For a spherical particle, all three dimensions along the principal axes are equal to the radius R. For what range of values of R is the above description valid? [2 marks] (ii) State the general condition corresponding to a resonant enhancement of the light scattered by the metal particle. [2 marks] (iii) Derive an expression for ως the frequency at resonance. [3 marks] (b) A spheroid particle has one longitudinal axis (dimension a) and two transverse axes (dimension b). The spheroid is characterised by two geometrical factors, longitudinal one , and a transverse one, L. (1) Derive expressions for the longitudinal and transverse resonant frequencies, in terms of the longitudinal geometric factor L. [7 marks] (ii) as If the deviation from sphericity is small, then the geometrical factor can be approximated L =(b/a) /3. Find an approximate expression for the difference between longitudinal and transverse resonant frequencies. [6 marks]