Problem $4$:

Assuming the quasi

$-$static limit$,$ answer the following questions: $[10$ pts$]$

$($a$)$ What is the resonance wavelength of a silver nano

$-$sphere inside water? Suppose the

relative permittivity of silver is described by a lossless Drude model $?$

$(?)=$

$5\mathrm{.}7?$

$!$

$/?$

$"$

$!$

$,$

where the $?$

$"$

$=138??.$ The refractive index of water is $1\mathrm{.}33.[5$pts$]$

$($b$)$ The sensitivity of a plasmonic sensor is defined as the derivative of the plasmonic

resonance wavelength with respect to the refractive change of the surrounding medium, with

a unit of nm$/$RIU$,$ where RIU is short for refractive index unit. For the silver nanosphere in

water, calculate the sensitivity$.[$