Question

Using dimensional analysis, the resistivity in terms of fundamental constants $h ,m_e, c,e,$ $\varepsilon_{0}$ can be expressed as

• A. $\frac{h}{\varepsilon_{0}m_{e}ce^{2}}$
• B. $\frac{\varepsilon_{0}m_{e}ce^{2}}{h}$
• C. $\frac{h^{2}}{m_{e}ce^{2}}$
• D. $\frac{m_{e}\varepsilon_{0}}{ce^{2}}$

Answer 1

Answer: (C)

Let resistivity depends on given fundamental constants.
$\rho=h^{a} m_{e}^{b} \,c^{c} \,e^{d} \,\varepsilon_{0}^{f}$
where, $k=$ a numeric constant.
Now, substituting dimensions of different physical constants, we have
$\left[ ML ^{3} T ^{-3} A ^{-2}\right]=k\left[\left[ ML ^{2} T ^{-1}\right]^{a}[ M ]^{b}\left[ LT ^{-1}\right]^{c}\right.$
$\left.[ AT ]^{d}\left[ M ^{-1} L ^{-3} T ^{4} A ^{2}\right]^{f}\right]$
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
$\rho=k\left(\frac{h^{2}}{m_{e} c e^{2}}\right)$