The Bohr radius is given by a_0 = ( _0 h^2)/(π m e^2) . Determine...

The dimensions of the physical quantities are:

Permittivity of free space, $[\varepsilon_0] = [M^{-1} L^{-3} T^4 A^2]$

Planck's constant, $[h] = [M L^2 T^{-1}]$

Mass, $[m] = [M]$

Charge, $[e] = [A T]$

Substituting these dimensions into the right-hand side of the given equation:

$[\text{RHS}] = \dfrac{[\varepsilon_0] [h]^2}{[m] [e]^2}$

$[\text{RHS}] = \dfrac{[M^{-1} L^{-3} T^4 A^2] [M L^2 T^{-1}]^2}{[M] [A T]^2}$

$[\text{RHS}] = \dfrac{[M^{-1} L^{-3} T^4 A^2] [M^2 L^4 T^{-2}]}{[M A^2 T^2]}$

$[\text{RHS}] = \dfrac{[M L T^2 A^2]}{[M A^2 T^2]} = [L]$

Alternatively, since $a_0$ represents the Bohr radius, its dimension must be that of length, which is $[L]$.