A composite slab consists of two materials having coefficient of thermal conductivity $$\mathrm{K}$$ and $$2 \mathrm{~K}$$, thickness $$\mathrm{x}$$ and $$4 \mathrm{x}$$ respectively. The temperature of the two outer surfaces of a composite slab are $$\mathrm{T}_2$$ and $$\mathrm{T}_1\left(\mathrm{~T}_2 > \mathrm{T}_1\right)$$. The rate of heat transfer through the slab in a steady state is $$\left[\frac{\mathrm{A}\left(\mathrm{T}_2-\mathrm{T}_1\right) \mathrm{K}}{\mathrm{x}}\right] \cdot \mathrm{f}$$ where '$$\mathrm{f}$$' is equal to

A black sphere has radius '$$R$$' whose rate of radiation is '$$E$$' at temperature '$$T$$'. If radius is made $$R / 3$$ and temperature '$$3 T$$', the rate of radiation will be

The potential on the plates of capacitor are $$+20 \mathrm{~V}$$ and $$-20 \mathrm{~V}$$. The charge on the plate is $$40 \mathrm{C}$$. The capacitance of the capacitor is

A thin uniform circular disc of mass '$$\mathrm{M}$$' and radius '$$R$$' is rotating with angular velocity '$$\omega$$', in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of same radius but of mass $$\left(\frac{M}{2}\right)$$ is placed gently on the first disc co-axially. The new angular velocity will be