If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express length in terms of dimensions of these quantities.
If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express length in terms of dimensions of these quantities.
Let $l \propto c^{x} y^{y} G^{z} ; 1=k c^{x} h^{y} G^{z}$
where $k$ is a dimensionless constant and $x , y$ and $z$ are the exponents.
Equating dimensions on both sides, we get
${\left[ M ^{0} LT ^{0}\right]=\left[ LT ^{-1}\right]^{ x }\left[ ML ^{2} T ^{-1}\right]^{y}\left[ M ^{-1} L ^{3} T ^{-2}\right]^{z}}$
$=\left[ M ^{y-z} L ^{ x +2 y +3 z } T ^{- x - y -2 z }\right]$
Applying the principle of homogeneity of dimensions, we get
$y-z=0$
$x+2 y+3 z=1$
$-x-y-2 z=0$
$x =\frac{-3}{2}, y =\frac{1}{2} z =\frac{1}{2}$
$l=\sqrt{\frac{ hG }{ c ^{3}}}$
Similar Questions
Which of the following is dimensionally incorrect?
Which of the following is dimensionally incorrect?
Position of a body with acceleration '$a$' is given by $x = K{a^m}{t^n},$ here $t$ is time. Find dimension of $m$ and $n$.
Position of a body with acceleration '$a$' is given by $x = K{a^m}{t^n},$ here $t$ is time. Find dimension of $m$ and $n$.
The quantity $X = \frac{{{\varepsilon _0}LV}}{t}$: ${\varepsilon _0}$ is the permittivity of free space, $L$ is length, $V$ is potential difference and $t$ is time. The dimensions of $X$ are same as that of
The quantity $X = \frac{{{\varepsilon _0}LV}}{t}$: ${\varepsilon _0}$ is the permittivity of free space, $L$ is length, $V$ is potential difference and $t$ is time. The dimensions of $X$ are same as that of
- [IIT 2001]
In a new system of units energy $(E)$, density $(d)$ and power $(P)$ are taken as fundamental units, then the dimensional formula of universal gravitational constant $G$ will be .......
In a new system of units energy $(E)$, density $(d)$ and power $(P)$ are taken as fundamental units, then the dimensional formula of universal gravitational constant $G$ will be .......
If momentum $[ P ]$, area $[ A ]$ and time $[ T ]$ are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :
If momentum $[ P ]$, area $[ A ]$ and time $[ T ]$ are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :
- [JEE MAIN 2022]