Derive the expression for the fringe width in a Young's double slit experiment. How will the fringe width change if (i) separation between the slits is increased (ii) screen is moved away from the plane of the slits.

In Young's double slit experiment, fringes of width $\beta$ are produced on a screen kept at a distance of $1m$ from the slit. When the screen is moved away by $5\times {10}^{-2}m$, fringe width changes by $3\times {10}^{-5}m$. The separation between the slits is $1\times {10}^{-3}m$. The wavelength of the light used is ________ $nm$.