Derive an expression for the radius of `n^(th)` Bohr's orbit of hydrogen atom hence write the expression for the radius of first orbit of hydrogen atom.

Explanation

Consider an electron of mass .m. and charge .e. revolving with a speed .u. around the nucleus of charge.+Ze. in the nth stationary orbit of radius.Y. The necessary centripetal force is provided by the electrostatic force of attracting between the electron and the nucleus.
`(mv^(2))/(r)=(Ze^(2))/(4pi epsi_(0)r^(2))`
`mv^(2)r=(Ze^(2))/(4pi epsi_(0)) " "...1`
From Bohr.s postulates we know
`mvr=(nh)/(2pi)`
Squring we get
`m^(2)v^(2)r^(2)=(n^(2)h^(2))/(4pi^(2))" "....(2)`
Dividing (2) by (1)
`(m^(2)v^(2)r^(2))/(mv^(2)r)= ((n^(2)h^(2))/(4pi^(2)))/((Ze^(2))/(4pie_(0)))`
`r=(epsi_(0)n^(2)h^(2))/(pi m Ze^(2))`
For hydrogen atom Z=1
`:. r=(epsin^(2)h^(2))/(pi me^(2))`