Filament of an incandescent bulb has length $l$ and radius of cross-section $r$. Rating of the bulb is ($P$ watt, $V$ volt). Emissivity of the filament is $\epsilon$ and Wien's constant is $b$. Wavelength of the most intense radiation emitted by the filament, if the bulb is operated at half of the rating voltage, is $b{\left(\frac{n\pi \sigma \epsilon rl}{P}\right)}^{\frac{1}{4}}$. Value of $n$, is $(\sigma =$ Stefan's constant)

Three rods $AB, BC, CA$ are joined to form a triangle $ABC$. Length of the rods $AB$ and $AC$ each are equal to $b$ and their coefficient of linear expansion is ${\alpha }_1$ whereas length of $\text{rod} BC$ is $a$ and its coefficient of linear expansion is ${\alpha }_2.$ Ratio $\frac{b}{a}$, such that the height $h$ does not change upon heating the rods by same temperature, is $\frac{1}{n}\sqrt{\frac{{\alpha }_2}{{\alpha }_1}}$. Value of $n$, is

The figure shows a composite rod made of two different material of thermal conductivity $\left(\frac{K}{4}\right)$ and $K$ respectively. The length of two rods are shown in figure. If the end temperature are $T_1$ and $T_2$ respectively, then the temperature of interface is $\frac{T_1+n{T}_2}{n+1}·$ Value of $n$, is