Question

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 :

• A. $\frac{ch}{2\pi\epsilon^{2}_{0}}$
• B. $\frac{e^{2}}{2\pi\epsilon_{0}Gm_{e}^{2}}$ ($m_e$ = mass of electron)
• C. $\frac{\mu_{0}\,\epsilon_{0}}{c^{2}} \frac{G}{he^{2}}$
• D. $\frac{2\pi\sqrt{\mu_{0}\epsilon_{0}}}{ce^{2}} \frac{h}{G}$

Answer 1

The Correct Option is B

The problem asks us to identify which of the given combinations of physical constants have a value that remains the same across different systems of units. This property is known as being "dimensionless." A dimensionless quantity has no physical dimensions and is a pure number, which implies that its value remains constant regardless of the system of units used.

Let us analyze the given options:

1. \(\frac{ch}{2\pi\epsilon^{2}_{0}}\)
   Here, \(c\) is the speed of light, \(h\) is Planck's constant, and \(\epsilon_0\) is the permittivity of free space. The dimensions of this combination are not dimensionless because both \(c\) and \(\epsilon_0\) have specific dimensions related to length, time, and electric charge.
2. \(\frac{e^{2}}{2\pi\epsilon_{0}Gm_{e}^{2}}\) (\(m_e\) = mass of electron)
   In this combination, \(e\) is the elementary charge, \(G\) is the gravitational constant, \(m_e\) is the mass of the electron, and \(\epsilon_0\) is the permittivity of free space. The dimensions of this combination reduce to a dimensionless form when analyzed, meaning it remains the same across different systems of units.
3. \(\frac{\mu_{0}\,\epsilon_{0}}{c^{2}} \frac{G}{he^{2}}\)
   Here, \(\mu_0\) is the permeability of free space, \(\epsilon_0\) is the permittivity, \(c\) is the speed of light, \(h\) is Planck's constant, \(G\) is the gravitational constant, and \(e\) is the elementary charge. The analysis shows that it does not yield a dimensionless combination, as the individual components have distinct dimensions that do not wholly cancel out.
4. \(\frac{2\pi\sqrt{\mu_{0}\epsilon_{0}}}{ce^{2}} \frac{h}{G}\)
   Similar to the previous options, this involves various constants each with specific dimensions for magnetism, electromagnetism, speed, charge, etc. These dimensions do not cancel out to yield a dimensionless number.

Upon reviewing all options, it is clear that the only option that results in a dimensionless quantity is the second one: \(\frac{e^{2}}{2\pi\epsilon_{0}Gm_{e}^{2}}\). Therefore, this combination will have the same numerical value across various systems of units.