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 $R_{H}=kh ^{a}e^{b} \ldots\left(i\right)$
As, R$=\frac{V}{I} $
$\therefore \left[R_{H}\right]=\left[ML^{2}T^{-3}A^{-2}\right]$
$h=E.t$
$\Rightarrow \left[h\right]=\left[ML^{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
$\left[ML^{2}T^{-3}A^{-2}\right]=k\left[ML^{2}T^{-1}\right]^{a}\left[AT\right]^{b}$
$=k\left[M^{a}L^{2a}T^{-a+b}A^{b}\right]$
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 $\frac{h}{e^{2}}$