What is the derivation of the radius of the n^th orbit in bohr's model?

the formula is R_n = (4πε₀)n²h² / 4π²me²

According to Bohr’s model, the electron revolves revolve in stationary orbits where the angular momentum of electron is an integral multiple of h/2π. 

 mvr = nh/2π  ------------------(1)

Here, h is Planck's constant.

Now, when an electron jumps from an orbit of higher energy E2 to an orbit of lower energy E1, it emits a photon. The energy of the photon is E2-E1.The relation between wavelength of the emitted radiation and energy of photon is given by the Einstein - Planck equation.

E2-E1= hν = hc/λ  -------------(2)

For an electron of hydrogen moving with a constant speed v along a circle of radius R with the center at the nucleus, the force acting on the electron according to Coulomb’s law is:

F = e²/4πε₀R²

The acceleration of the electron is given by v²/r. If m is the mass of the electron, then according to Newton’s law:

e²/4πε₀r² = mv²/R  ---------(3)

mv = [(1/4πε₀) (m e²/R)]^1/2  ----------------(4)

From equation (1) and (4) we get,

{[(1/4πε₀) (m e²/R)]^1/2}² x R² = n²h²/2²π²

R = (4πε₀)n²h² / 4π²me²

For nth orbit,

R_n = ε₀ n²h² / πme²