If energy E = G^p h^q c^r,where G is the univ | vedclass

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(A) The dimensional formula for energy $E$ is $[M^1 L^2 T^{-2}]$.The dimensional formula for $G$ is $[M^{-1} L^3 T^{-2}]$.The dimensional formula for

Vedclass · Accepted Answer

$-\frac{1}{2}, \frac{1}{2}, \frac{5}{2}$

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(A) The dimensional formula for energy $E$ is $[M^1 L^2 T^{-2}]$.The dimensional formula for $G$ is $[M^{-1} L^3 T^{-2}]$.The dimensional formula for $h$ is $[M^1 L^2 T^{-1}]$.The dimensional formula for $c$ is $[L^1 T^{-1}]$.Given $E = G^p h^q c^r$,we have:$[M^1 L^2 T^{-2}] = [M^{-1} L^3 T^{-2}]^p [M^1 L^2 T^{-1}]^q [L^1 T^{-1}]^r$$[M^1 L^2 T^{-2}] = [M^{-p+q} L^{3p+2q+r} T^{-2p-q-r}]$Comparing the powers of $M, L$,and $T$ on both sides:For $M$: $-p + q = 1 \implies q = 1 + p$For $T$: $-2p - q - r = -2 \implies 2p + q + r = 2$For $L$: $3p + 2q + r = 2$Subtracting the $T$ equation from the $L$ equation: $(3p + 2q + r) - (2p + q + r) = 2 - 2 \implies p + q = 0 \implies q = -p$Substituting $q = -p$ into $q = 1 + p$: $-p = 1 + p \implies 2p = -1 \implies p = -\frac{1}{2}$.Then $q = -(-\frac{1}{2}) = \frac{1}{2}$.Using $2p + q + r = 2$: $2(-\frac{1}{2}) + \frac{1}{2} + r = 2 \implies -1 + \frac{1}{2} + r = 2 \implies r = 2 + 1 - \frac{1}{2} = \frac{5}{2}$.Thus,$p = -\frac{1}{2}, q = \frac{1}{2}, r = \frac{5}{2}$.

If energy $E = G^p h^q c^r$,where $G$ is the universal gravitational constant,$h$ is Planck's constant,and $c$ is the speed of light,find the values of $p, q$,and $r$ respectively.

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