Question

From the given following combinations of physical constants which is expressed through their usual symbols, then the only combination, that would have the identical value in different system of units will be given as,
$\begin{align}
  & A.\dfrac{ch}{2\pi {{\varepsilon }_{0}}^{2}} \\
 & B.\dfrac{{{e}^{2}}}{2\pi {{\varepsilon }_{0}}G{{m}_{e}}^{2}}\left( {{m}_{e}}=\text{mass of electron} \right) \\
 & C.\dfrac{{{\mu }_{0}}{{\varepsilon }_{0}}}{{{c}^{2}}}\dfrac{G}{h{{e}^{2}}} \\
 & D.\dfrac{2\pi \sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}}{c{{e}^{2}}}\dfrac{h}{G} \\
\end{align}$

Answer 1

Hint: Coulomb's law is otherwise called Coulomb's inverse-square law. This is basically an experimental law in physics which is defining the measure of the force between two stationary electrically charged bodies. The electric force between the charged objects at the stationary position will be conventionally known as Coulomb force. This will help you in answering this question.

Complete answer:
As mentioned in the question, the combination of the physical constants that will be having the identical value in various systems of units must be a dimensionless quantity. This is because of the dimensions that are something which is varying the magnitude of a particular quantity.
According to the Newton's law of gravitation and the Coulomb's law of forces on charges, we can write that,
$F=\dfrac{GMm}{{{r}^{2}}}=\dfrac{kQq}{{{r}^{2}}}$
Now, Observing the above expressions will be suggested such that the dimensions of $\dfrac{{{e}^{2}}}{2\pi {{\varepsilon }_{0}}}$ are that of $N{{m}^{2}}$.
Also, the dimensions of $G{{m}^{2}}$ are that of $N{{m}^{2}}$.
Therefore, we can assume that a classification of these quantities will be a dimensionless quantity and its value will not be varying depending on the system of units we choose. Therefore the question has been answered.

Hence the correct answer is given as option B.

Note:
Newton's law of universal gravitation is said as that each and every object which attracts every other object in the universe with a force which is directly proportional to the product of their masses and also inversely proportional to the square of the distance between the centres of the objects.