A length -scale (l) depends on the permittivity `(epsilon)` of a dielectric material, Boltzmann constant `(k_(B))`, the absolute temperature (T), the number pr unit volume (n) of certain charged paticles, and the charge (q) carried by each of the particles. Which of the following expression for l is dimensionally correct?

Options

A. `l=sqrt(((nq^(2))/(epsilonk_(B)T)))`
B. `l=sqrt(((epsilonk_(B)T)/(nq^(2))))`
C. `l=sqrt(((q^(2))/(epsilonn^(2//3)k_(B)T)))`
D. `l=sqrt(((q)/(epsilonn^(1//3)k_(B)T)))`

Answer

B

Explanation

We know that, dimensionally `in=(q^(2))/(l^(2)F)`,
`k_(B)T=(RT)/(N_(A))=PV=Fxxl`
Now `sqrt((epsilonk_(B)T)/(nq^(2)))=[(q^(2))/(l^(2)F)xx(Fxxl)/(l^(-3)q^(2))]^(1//2)=l`
`"Also, "sqrt((q^(2))/(in n^(1//3)k_(B)T))=[(l^(2)Fxxq^(2))/(q^(2)l^(*-1)xxFxxl)]^(1//2)=l`