Question

The quantum hall resistance $R_H$ is a fundamental constant with dimensions of resistance. If $h$ is Planck’s constant and $e$ is the electron charge, then the dimension of $R_H$ is the same as

• A. $\frac{e^2}{h}$
• B. $\frac{h}{e^2}$
• C. $\frac{h^2}{e}$
• D. $\frac{e}{h^2}$

Answer 1

Answer: (B)

Let $R_H=k{h}^a{e}^b\dots \left(i\right)$
As, R$=\frac{V}{I}$
$\therefore \left[R_H\right]=\left[M{L}^2{T}^{-3}{A}^{-2}\right]$
$h=E.t$
$\Rightarrow \left[h\right]=\left[M{L}^2{T}^{-1}\right]$
$e=I.t\Rightarrow \left[e\right]=\left[A.T\right]$
Substituting above values in Eq. $\left(i\right),$we have
$\left[M{L}^2{T}^{-3}{A}^{-2}\right]=k{\left[M{L}^2{T}^{-1}\right]}^a{\left[AT\right]}^b$
$=k\left[M^a{L}^{2a}{T}^{-a+b}{A}^b\right]$
Equating dimensions, we get
$a=1$ and $b=-2$
Hence,$R_H=k\left(\frac{h}{e^2}\right)$
So, dimensions of hall resistance aresame as that of $\frac{h}{e^2}$