The energy (E), angular momentum (L), and uni | vedclass

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(A) Let the Planck's constant $h$ be expressed as $h = k G^x L^y E^z$, where $k$ is a dimensionless constant.Dimensional formulas are:$[h] = [M^1 L^2

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Zero

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(A) Let the Planck's constant $h$ be expressed as $h = k G^x L^y E^z$, where $k$ is a dimensionless constant.Dimensional formulas are:$[h] = [M^1 L^2 T^{-1}]$$[G] = [M^{-1} L^3 T^{-2}]$$[L] = [M^1 L^2 T^{-1}]$$[E] = [M^1 L^2 T^{-2}]$Substituting these into the equation:$[M^1 L^2 T^{-1}] = [M^{-1} L^3 T^{-2}]^x [M^1 L^2 T^{-1}]^y [M^1 L^2 T^{-2}]^z$$[M^1 L^2 T^{-1}] = [M^{-x+y+z} L^{3x+2y+2z} T^{-2x-y-2z}]$Comparing the powers of $M, L,$ and $T$ on both sides:$(i)$ $-x + y + z = 1$(ii) $3x + 2y + 2z = 2$(iii) $-2x - y - 2z = -1$Adding $(i)$ and (iii):$(-x + y + z) + (-2x - y - 2z) = 1 - 1$$-3x - z = 0 \implies z = -3x$Substitute $z = -3x$ into $(i)$:$-x + y - 3x = 1 \implies y - 4x = 1 \implies y = 1 + 4x$Substitute $y$ and $z$ into (ii):$3x + 2(1 + 4x) + 2(-3x) = 2$$3x + 2 + 8x - 6x = 2$$5x + 2 = 2 \implies 5x = 0 \implies x = 0$Thus, the dimension of $G$ in the formula for $h$ is $0$.

The energy $(E)$, angular momentum $(L)$, and universal gravitational constant $(G)$ are chosen as fundamental quantities. The dimensions of the universal gravitational constant in the dimensional formula of Planck's constant $(h)$ is

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