Two full turns of the circular scale of a screw gauge cover a distance of $$1 \,mm$$ on its main scale. The total number of divisions on the circular scale is $$50.$$ Further, it is found that the screw gauge has a zero error of $$- 0.03 \,mm.$$ While measuring the diameter of a thin wire, a student notes the main scale reading of $$3 \,mm$$ and the number of circular scale divisions in line with the main scale as $$35.$$ The diameter of the wire is-
A quantity $$X$$ is given by $${\varepsilon _0}L\frac{{\Delta V}}{{\Delta t}}$$ where $${ \in _0}$$ is the permittivity of the free space, $$L$$ is a length, $$\Delta V$$ is a potential difference and $$\Delta t$$ is a time interval. The dimensional formula for $$X$$ is the same as that of-
Pressure depends on distance as, $$P = \frac{\alpha }{\beta }exp\left( { - \frac{{\alpha z}}{{k\theta }}} \right),$$ where $$\alpha ,$$ $$\beta $$ are constants, $$z$$ is distance, $$k$$ is Boltzman’s constant and $$\theta $$ is temperature. The dimension of $$\beta $$ are-