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  •  From the following combinations of  physical constants (expressed through their usual symbols) the only combination, that would have the same value in different systems of units, is :</p

 From the following combinations of  physical constants (expressed through their usual symbols) the only combination, that would have the same value in different systems of units, is :

Option: 1

\frac{ch}{2\pi \epsilon _{0}^{2}}


Option: 2

\frac{e^{2}}{2\pi \epsilon _{0}\; Gm_{e}\; ^{2}}\; (m_{e}=mass\: of\: electron)


Option: 3

\frac{\mu _{0}\; \epsilon _{0}}{c^{2}}\frac{G}{he^{2}}


Option: 4

\frac{2\pi \sqrt{\mu _{0}\epsilon _{0}}}{ce^{2}}\frac{h}{G}


Answers (1)

best_answer

As we have learnt,

Physical quantity, n u =constant -

n_{1}u_{1}=n_{2}u_{2}= constant

The permittivity of free space -

\epsilon_{o}=\ M^{1}L^{3}T^{-4}A^{2}

- wherein its unit is

C^{-2}N^{1}m^{-2}

 

 

Permeability of free space -

Dimension of permeability of free space \mu_{o}-M^{1}L^{1}T^{-2}A^{-2} 

- wherein its unit is

\dpi{100} \frac{newton}{ampere^{2}} \ or \ \frac{henry}{metre}

Plank's Constant (h) -

M^{1}L^{2}T^{-1}

- wherein its unit is

joule-sec

\\ \text{The combination of physical constants that will have the same } \\ \text{value in different systems of units should be a dimensionless quantity,} \\ \text{since dimensions are something that change the magnitude of a particular} \\ \text{quantity. Form Newton's law of gravitation and Coulomb's law of forces on charges,} \\ \\ F =\frac{ GMm }{ r ^{2}}=\frac{ kQq }{ r ^{2}}$ \\ Now, Observing the above expressions suggests that, the dimensions of $\frac{ e ^{2}}{2 \pi \epsilon_{ o }}$ are that of $Nm ^{2}$. Also, the dimensions of $Gm ^{2}$ are that of $Nm ^{2}. \\ \text{Hence, we conclude that a division of these quantities will be a dimensionless} \\ \text{quantity and its value will not change depending on the system of units chosen.}

This expression is similar to option 2, hence answer is 2

Posted by

seema garhwal

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