Using dimensional analysis to calculate bohr radius

Question

Derive bohr radius using dimensional analusis

Answer 1

Bohr Radius Formula

The formula for Bohr Radius is

Where,

a₀ is Bohr radius

m_e is the rest mass of electron

∈₀ permittivity of free space

ℏ is reduced planck’s constant

c is the velocity of light

a is the fine structure constant

e is the elementary charge on the particle

Bohr Radius Derivation

For a hydrogen atom, an electron moving in its respective orbit with a definite nucleus,

The centripetal force is,

\(C_p=\frac{mv^2}{r}\)

The electrostatic force is,

\(E=\frac{1}{4\pi \epsilon _0}.\frac{ze^2}{r^2}\)

In a hydrogen atom,

Electrostatic force = Centripetal force

From Bohr’s second postulate,

Angular momentum,

L = mvr = nh

Hence,

v = nh / mr – substituting in equation (1)

As we know,

Radius of an atom with n = 1

m = 9.11 × 10⁻³¹kg

z = 1

e = 1.6 × 10⁻¹⁹C

Substituting these values, in the equation, r = \(\frac{4\pi \epsilon _0(nh)^2}{mze^2}\)

We get,

r = 5.2917721067 x 10⁻¹¹m which is the radius of the first Bohr Orbit.