NEET · Physics · STD 11 - 1. units,dimensions and measurement

A physical quantity of the dimensions of length that can be formed out of \(c, G\) and \(\frac{e^2}{4\pi \varepsilon _0}\) is \([c\) is velocity of light, \(G\) is the universal constant of gravitation and \(e\) is charge \(] \)

A \(\frac{1}{{{c^2}}}\)\(\sqrt {\frac{{{e^2}}}{{G4\pi \varepsilon_0}}} \)
B \(\frac{1}{{{c^{}}}}\frac{{G{e^2}}}{{4\pi \varepsilon_0}}\)
C \(\frac{1}{{{c^2}}}\)\(\sqrt {\frac{{G{e^2}}}{{4\pi \varepsilon_0}}} \)
D \({c^2}\;\sqrt {\frac{{G{e^2}}}{{4\pi \varepsilon_0}}} \)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{1}{{{c^2}}}\)\(\sqrt {\frac{{G{e^2}}}{{4\pi \varepsilon_0}}} \)

Step-by-step Solution

Detailed explanation

dimensions of
\(\frac{{{e^2}}}{{4\pi {\varepsilon _0}}} = \left[ {F \times {d^2}} \right] = \left[ {M{L^3}{T^{ - 2}}} \right]\)
dimensions of \(G = \left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]\)
dimensions of \(c = \left[ {L{T^{ - 1}}} \right]\)
\( l\, \propto \,{\left( {\frac{{{e^2}}}{{4\pi {\varepsilon _o}}}} \right)^p}{G^q}{c^r}\)
\(\therefore \left[ {{L^1}} \right] = {\left[ {M{L^3}{T^{ - 2}}} \right]^p}{\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]^q}{\left[ {L{T^{ - 1}}} \right]^r}\) On comparing both sides and solving we get

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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