The radii of two soap bubbles are $$r_1$$ and $$r_2$$. In isothermal condition they combine with each other to form a single bubble. The radius of resultant bubble is

The rays of different colours fail to converge at a point after passing through a thick converging lens. This defect is called

The ratio of potential difference that must be applied across parallel and series combination of two capacitors $$C_1$$ and $$C_2$$ with their capacitance in the ratio $$1: 2$$ so that energy stored in these two cases becomes same is

The potential energy of charged parallel plate capacitor is $$v_0$$. If a slab of dielectric constant $$\mathrm{K}$$ is inserted between the plates, then the new potential energy will be