Question

where e is the magnitude of charge of electron. Assuming
If potential energy of an electron and a proton in a hypothetical situation is given by U--
Bohr's model to be applicable, the radius of oth Bohr's orbit is​

Answer 1

Q -> If potential energy between a proton and an electron is given by ∣U∣=ke²/2R³ , where e is the charge of electron and R is the radius of atom, then radius of Bohr's orbit is given by (h = Planck's constant, k = constant)

given, potential energy, |U| = ke²/2R³

differentiating with respect to R,

d|U|/dR = -3ke²/2R⁴

we know, F = -dU/dR

so, F = 3Ke²/2R⁴

from Bohr's model of atom,

centripetal force is balanced by electrostatic force.

so, F = 3Ke²/2R⁴ = mv²/R .........(1)

from Bohr's postulation , mvR = nh/2π

v = nh/2πmR ...,...(2)

from equations (1) and (2) we get,

3Ke²/2R⁴ = m(nh/2πmR)²/R

⇒3Ke²/2R⁴ = n²h²/4π²mR³

⇒6Ke²π²m/R = n²h²

⇒R = 6Ke²π²m/n²h²

therefore radius of nth Bohr's orbit is 6Ke²π²m/n²h²