1. c²[Ge²/4πε₀]^(1/2) 2. 1/c²[e²/G 4πε₀]^(1/2) 3. 1/c Ge²/4πε₀ 4. 1/c²[Ge²/4πε₀]^(1/2)

Answer: 4. 1/c²[Ge²/4πε₀]^(1/2) Key idea: match dimensions. e²/4πε₀ is force × length ² : [ML³T⁻²] . G=[M⁻¹L³T⁻²] . Product Ge²/4πε₀=[L⁶T⁻⁴] , square root =[L³T⁻²] . Dividing by c²=[L²T⁻²] gives [L] . So dfrac1c²[Ge²/4πε₀]^(1/2) has dimension of length.

Topic: Dimensions and dimensional analysis· Physics and Measurement

A physical quantity of the dimensions of length that can be formed out of c, G and e^2/4πε_0 is [c is velocity of light, G is universal constant of gravitation and e is charge] :

1. \(c^2\left[G\frac{e^2}{4\pi\epsilon_0}\right]^{1/2}\)

2. \(\frac{1}{c^2}\left[\frac{e^2}{G\,4\pi\epsilon_0}\right]^{1/2}\)

3. \(\frac1c\,G\frac{e^2}{4\pi\epsilon_0}\)

4. \(\frac{1}{c^2}\left[G\frac{e^2}{4\pi\epsilon_0}\right]^{1/2}\)

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Answer: 4. \(\frac{1}{c^2}\left[G\frac{e^2}{4\pi\epsilon_0}\right]^{1/2}\)

Key idea: match dimensions.

\(\dfrac{e^2}{4\pi\epsilon_0}\) is force \(\times\) length\(^2\): \([ML^3T^{-2}]\). \(G=[M^{-1}L^3T^{-2}]\). Product \(G\dfrac{e^2}{4\pi\epsilon_0}=[L^6T^{-4}]\), square root \(=[L^3T^{-2}]\).

Dividing by \(c^2=[L^2T^{-2}]\) gives \([L]\). So \(\dfrac1{c^2}\left[G\dfrac{e^2}{4\pi\epsilon_0}\right]^{1/2}\) has dimension of length.

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A physical quantity of the dimensions of length that can be formed out of c, G and e^2/4πε_0 is [c is velocity of light, G is universal constant of gravitation and e is charge] :

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