Q. 7

 

 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 :

Your Answer:

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

 

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}

Correct Answer:

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

Answers (1)

The combination of physical constants that will have the same value in different systems of units should be a dimensionless quantity, since dimensions are something that change the magnitude of a particular quantity.
Form Newton's law of gravitation and Coulomb's law of forces on charges, 

\mathrm{F}=\frac{G M m}{r^{2}}=\frac{k Q q}{r^{2}}

Now, Observing the above expressions suggests that, the dimensions of \frac{e^{2}}{2 \pi \epsilon_{o}}$  are that of N m^{2}$ 

Also, the dimensions of Gm2 are that of Nm2. Hence, we conclude that a division of these quantities will be a dimensionless quantity and its value will not change depending on the system of units chosen.

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