If velocity of light c , Planck’s constant h and gravitational constant G are taken as fundame

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1.Units, Dimensions and Measurement

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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.

Option A

Option B

Option C

Option D

Solution

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

$y-z=0$

$x+2 y+3 z=1$

$-x-y-2 z=0$

$x =\frac{-3}{2}, y =\frac{1}{2} z =\frac{1}{2}$

$l=\sqrt{\frac{ hG }{ c ^{3}}}$

Standard 11

Physics

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Similar Questions

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $E]$ and $B$ stand for dimensions of electric and magnetic fields respectively, while $\left[\varepsilon_0\right]$ and $\left[\mu_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $L$ and $T$ are dimensions of length and time respectively. All the quantities are given in $SI$ units.
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$(D)$ $[\mathrm{E}]=[\mathrm{B}][\mathrm{L}]^{-1}[\mathrm{~T}]^{-1}$
($2$) The relation between $\left[\varepsilon_0\right]$ and $\left[\mu_0\right]$ is
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$(D)$ $\left[\mu_0\right]=\left[\varepsilon_0\right]^{-1}[\mathrm{~L}]^{-2}[\mathrm{~T}]^2$
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