Question

A length-scale (l) depends on the permittivity (ε) of a dielectric material, Boltzmann constant (kB), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression for I 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}
Hence , option (b) is correct

Answer 2

Answer:

option (B)

Explanation:

please see the attachment