A liquid cold finger is a device that is used to deeply cool a device$--$often times an electronic detector like an infrared imaging detector, in order to reduce the background noise on the detector $($because of it's high temperature$)$ and achieve higher sensitivity$.$ The device is essentially a fin dipped in liquid nitrogen, as shown below. A cold finger is used to cool a camera detector emitting $\backslash ($ Q$\_\{\backslash$text $\{$detector $\}\}=75\backslash$mathrm$\{$~W$\}\backslash )$ of heat. The cold finger is completely insulated, and is dipped in a vat of saturated liquid nitrogen at $1$ atm pressure $(\backslash ($ T$\_\{\backslash$text $\{$boil $\}\}=77\backslash$mathrm$\{$~K$\}\backslash )).$ The detector is held within a vacuum chamber to prevent condensation and eliminate convective heat transfer. The detector's dimensions are $\backslash ($ W $\backslash )$ and $\backslash ($ L $\backslash )$ and negligible thickness, as shown in the diagram. The back and sides of the detector as well as the sides of the cold finger in the vacuum chamber are thermally insulated. The cold finger is a single copper pin fin of diameter $\backslash ($ D $\backslash )$ and length $\backslash ($ L $\backslash ).$ The portion of the cold finger that is insulated is negligible compared to the portion submerged in liquid nitrogen. The liquid nitrogen is held at saturate, atmospheric conditions $($boiling at $\backslash ($ T$\_\{$L N $2\}=77\backslash$mathrm$\{$~K$\}\backslash )$ and produces a convective heat transfer coefficient of $\backslash ($ h$=100\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}/\backslash$mathrm$\{$K$\}\backslash ).$ You should assume negligible heat leaves or enters the detector from the front $($viewing$)$ side.

FIND:

a$)$ Using the $1-$D heat transfer solutions, determine the minimum temperature at which the back side of the detector chip can be held $($assume no thermal contact resistance between the cold finger and the chip$).$