Question

A physical quantity of the dimensions of length that can be formed out of c, G and \(\frac {e^2}{4πε_0}\) is
[c is velocity of light, G is universal constant of gravitation and e is charge]

• A. \(\frac {1}{c^2}[G\frac {e^2}{4πε_0}]^{\frac 12}\)
• B. \(c^2[G\frac {e^2}{4πε_0}]^{\frac 12}\)
• C. \(\frac {1}{c^2}[\frac {e^2}{G4πε_0}]^{\frac 12}\)
• D. \(\frac {1}{c}G\frac {e^2}{4πε_0}\)

Answer 1

The Correct Option is A

To find a physical quantity of the dimensions of length that can be formed from the constants \(c\) (velocity of light), \(G\) (universal gravitational constant), and \(\frac{e^2}{4\pi \varepsilon_0}\) (related to electric force), we need to analyze the dimensions of each component:

1. Velocity of Light \(c\): The dimensions are \([L T^{-1}]\), where \(L\) is length and \(T\) is time.
2. Gravitational Constant \(G\): The dimensions are \([M^{-1}L^3T^{-2}]\).
3. \(\frac{e^2}{4\pi \varepsilon_0}\): This term represents electric force and has dimensions \([M L^3 T^{-2} Q^{-2}]\), where \(Q\) is electric charge.

The goal is to form a dimensional expression equivalent to length \([L]\). Let's consider the expression given in the options:

• \(\frac{1}{c^2}\left[G \frac{e^2}{4\pi \varepsilon_0}\right]^{\frac{1}{2}}\)

Breaking down the dimensions of this expression:

1. \(c^2\): The dimensions are \([L^2 T^{-2}]\).
2. \(\left[G \frac{e^2}{4\pi \varepsilon_0}\right]^{\frac{1}{2}}\): Let's calculate the dimensions:
3. \(G \frac{e^2}{4\pi \varepsilon_0}\) has dimensions \([M^{-1}L^3T^{-2}][M L^3 T^{-2} Q^{-2}]\).
4. Simplifying: \([M^0 L^6T^{-4} Q^{-2}]\).
5. Taking the square root: \([M^0 L^3T^{-2} Q^{-1}]\).

Substitute back into the expression:

• \(\frac{1}{c^2} \times [M^0 L^3T^{-2} Q^{-1}]\).

The dimensions of \(\frac{1}{c^2}\) are \([L^{-2}T^{2}]\). Multiplying these with the dimensions of the square-rooted term gives:

• \([L^{-2}T^{2}] \times [M^0 L^3T^{-2} Q^{-1}] = [L]\).

This leads to the final dimensional formula of \([L]\), representing length. Therefore, the correct expression that has the dimensions of length is:

\(\frac{1}{c^2}\left[G \frac{e^2}{4\pi \varepsilon_0}\right]^{\frac{1}{2}}\)