The potential energy (U) of a diatomic molecule is a function dependent on r (interatomic distance) as

$$U = {\alpha \over {{r^{10}}}} - {\beta \over {{r^5}}} - 3$$

where, $$\alpha$$ and $$\beta$$ are positive constants. The equilibrium distance between two atoms will be $${\left( {{{2\alpha } \over \beta }} \right)^{{a \over b}}}$$, where a = ___________.

The electric field in a region is given by $$\overrightarrow E = \left( {{3 \over 5}{E_0}\widehat i + {4 \over 5}{E_0}\widehat j} \right){N \over C}$$. The ratio of flux of reported field through the rectangular surface of area 0.2 m² (parallel to y $$-$$ z plane) to that of the surface of area 0.3 m² (parallel to x $$-$$ z plane) is a : b, where a = __________ [Here $${\widehat i}$$, $${\widehat j}$$ and $${\widehat k}$$ are unit vectors along x, y and z-axes respectively.]

In the given circuit of potentiometer, the potential difference E across AB (10 m length) is larger than E₁ and E₂ as well. For key K₁ (closed), the jockey is adjusted to touch the wire at point J₁ so that there is no deflection in the galvanometer. Now the first battery (E₁) is replaced by second battery (E₂) for working by making K₁ open and K₂ closed. The galvanometer gives then null deflection at J₂. The value of $${{{E_1}} \over {{E_2}}}$$ is $${a \over b}$$, where a = _________.

In a certain thermodynamical process, the pressure of a gas depends on its volume as kV³. The work done when the temperature changes from 100$$^\circ$$C to 300$$^\circ$$C will be ___________ nR, where n denotes number of moles of a gas.