Question

Using dimensional analysis, the resistivity in terms of fundamental constants $h,m_e,c,e,$ ${\epsilon }_0$ 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

Answer: (C)

Let resistivity depends on given fundamental constants.
$\rho =h^a{m}_e^b{c}^c{e}^d{\epsilon }_0^f$
where, $k=$ a numeric constant.
Now, substituting dimensions of different physical constants, we have
$\left[M{L}^3{T}^{-3}{A}^{-2}\right]=k[{\left[M{L}^2{T}^{-1}\right]}^a[M{]}^b{\left[L{T}^{-1}\right]}^c$
$[AT{]}^d{\left[M^{-1}{L}^{-3}{T}^4{A}^2\right]}^f]$
Equating dimensions, we have
$1=a+b-f$
$3=2a+c-Sf$
$-3=-a-c+d+4f$
$-2=d+2f$
Solving these, we get
$a=2$
$b=-1$
$c=-1$
$d=-2$
$f=0$
So, resistivity can be expressed as
$\rho =k\left(\frac{h^2}{m_{e}c{e}^2}\right)$