If μ_0 , e and h represent the permeability of free space,...

First, write the dimensional formulas for each physical quantity:

Permeability of free space ($\mu_0$): From the force per unit length between parallel currents $\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi r}$, we get $[\mu_0] = \dfrac{[\text{M}\text{L}\text{T}^{-2}][\text{L}]}{[\text{L}][\text{A}^2]} = [\text{M}\text{L}\text{T}^{-2}\text{A}^{-2}]$

Elementary charge ($e$): From $q = It$, $[e] = [\text{A}\text{T}]$

Planck's constant ($h$): From $E = h\nu$, $[h] = \dfrac{[\text{M}\text{L}^2\text{T}^{-2}]}{[\text{T}^{-1}]} = [\text{M}\text{L}^2\text{T}^{-1}]$

Now, substitute these into the given expression $\dfrac{\mu_0 e^2}{h}$:

\left[\dfrac{\mu_0 e^2}{h}\right] = \dfrac{[\text{M}\text{L}\text{T}^{-2}\text{A}^{-2}] \times [\text{A}\text{T}]^2}{[\text{M}\text{L}^2\text{T}^{-1}]}$<br><br>\left[\dfrac{\mu_0 e^2}{h}\right] = \dfrac{[\text{M}\text{L}\text{T}^{-2}\text{A}^{-2}] \times [\text{A}^2\text{T}^2]}{[\text{M}\text{L}^2\text{T}^{-1}]}$

\left[\dfrac{\mu_0 e^2}{h}\right] = \dfrac{[\text{M}\text{L}]}{[\text{M}\text{L}^2\text{T}^{-1}]}$<br><br>\left[\dfrac{\mu_0 e^2}{h}\right] = [\text{L}^{-1}\text{T}]$

Answer: $[\text{L}^{-1}\text{T}]$