A physical quantity of the dimensions of leng | vedclass

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(C) Let the physical quantity of length $l$ be expressed as $l = k \left( \frac{e^2}{4\pi \varepsilon_0} \right)^p G^q c^r$.Dimensions of $\frac{e^2}{

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$\frac{1}{c^2} \sqrt{\frac{Ge^2}{4\pi \varepsilon_0}}$

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(C) Let the physical quantity of length $l$ be expressed as $l = k \left( \frac{e^2}{4\pi \varepsilon_0} \right)^p G^q c^r$.Dimensions of $\frac{e^2}{4\pi \varepsilon_0} = [F \cdot d^2] = [ML^3T^{-2}]$.Dimensions of $G = [M^{-1}L^3T^{-2}]$.Dimensions of $c = [LT^{-1}]$.Equating dimensions: $[L^1] = [ML^3T^{-2}]^p [M^{-1}L^3T^{-2}]^q [LT^{-1}]^r$.Comparing powers of $M$: $p - q = 0 \implies p = q$.Comparing powers of $T$: $-2p - 2q - r = 0 \implies -4p = r$.Comparing powers of $L$: $3p + 3q + r = 1 \implies 6p - 4p = 1 \implies 2p = 1 \implies p = 1/2$.Thus,$q = 1/2$ and $r = -2$.Therefore,$l = \frac{1}{c^2} \sqrt{\frac{Ge^2}{4\pi \varepsilon_0}}$.

$A$ physical quantity of the dimensions of length that can be formed out of $c, G$ and $\frac{e^2}{4\pi \varepsilon_0}$ is $[c$ is velocity of light,$G$ is the universal gravitational constant and $e$ is charge$]$.

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