Question

Using dimensional analysis the resistivity in terms of fundamental constants $h,m_e,c,e,{\epsilon }_o$ can be expressed as.

• A. $\frac{h}{{\epsilon }_0{m}_{e}c{e}^2}$
• B. $\frac{{\epsilon }_0{m}_{e}c{e}^2}{h}$
• C. $\frac{h^2}{m_{e}c{e}^2}$
• D. $\frac{m_e{\epsilon }_0}{c{e}^2}$

Answer 1

The correct option is C $\frac{h^2}{m_{e}c{e}^2}$

SI base units of resistivity are written as $Kg{m}^3{s}^{-1}{C}^{-2}$.

For plank constant $h$ SI base units are $Kg{m}^2{s}^{-1}$.

For $e$ SI unit is $C$.

For $c$ SI unit is $m{s}^{-1}$.

For $m_e$ SI unit is $Kg$.

For ${\epsilon }_o$ SI unit is $C^2{Kg}^{-1}{m}^{-3}{s}^2$

Solving the dimensions of constants in option $C$,

$\frac{(Kg{m}^2{s}^{-1}{)}^2}{Kgm{s}^{-1}{C}^2}=Kg{m}^3{s}^{-1}{C}^{-2}$

We get the dimensions of resistivity hence option $C$ is correct.