Question

A lenght- scale depends on the permittivity of a dielectric material, Boltzmann ,the absolute temperature,the number per unit volume of certain particles and the charge carried by each of the particles . which of the following expression for length(l) is dimensionally correct​

Answer 1

[Dimension of permittivity , ∈ = dimension of {q²/Fr²}

= [A²T²]/[MLT⁻²][L²]

= [M⁻¹L⁻³T⁴A²]

dimension of Boltzmann's Constant and temperature = dimension of energy [∵ Kinetic energy of gases = Boltzmann's Constant × temperature in Kelvin]

so, dimension of KB × T = [ML²T⁻²]

[note :- here I took KB and T together because we see all the given options are included KB and T together ]

dimension of q = [AT]

Dimension of n = [L⁻³]

Now, let's check ,

dimension of {∈KBT/nq²} =[M⁻¹L⁻³T⁴A²][ML²T⁻²]/[L⁻³A²T²]

= [L⁻¹T²A²]/[L⁻³A²T²]

= [L²]

Hence, dimension of {εKBT/nq²} = {dimension of length}²

So, dimension of length = dimension of √{εKBT/nq²}