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(A) Let the length $l$ be expressed as $l = k c^x h^y G^z$,where $k$ is a dimensionless constant and $x, y, z$ are exponents.The dimensions of the qua

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) Let the length $l$ be expressed as $l = k c^x h^y G^z$,where $k$ is a dimensionless constant and $x, y, z$ are exponents.The dimensions of the quantities are:$[l] = [M^0 L^1 T^0]$$[c] = [L T^{-1}]$$[h] = [M L^2 T^{-1}]$$[G] = [M^{-1} L^3 T^{-2}]$Substituting these into the equation:$[M^0 L^1 T^0] = [L T^{-1}]^x [M L^2 T^{-1}]^y [M^{-1} L^3 T^{-2}]^z$$[M^0 L^1 T^0] = [M^{y-z} L^{x+2y+3z} T^{-x-y-2z}]$Equating the powers of $M, L,$ and $T$ on both sides:$1$) $y - z = 0 \implies y = z$$2$) $x + 2y + 3z = 1$$3$) $-x - y - 2z = 0 \implies x = -y - 2z$Substituting $y = z$ into $(3)$: $x = -z - 2z = -3z$Substituting $x = -3z$ and $y = z$ into $(2)$:$-3z + 2z + 3z = 1 \implies 2z = 1 \implies z = 1/2$Thus,$y = 1/2$ and $x = -3/2$.Therefore,$l \propto c^{-3/2} h^{1/2} G^{1/2} = \sqrt{\frac{hG}{c^3}}$.

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.

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