NEET · Physics · STD 11 - 1. units,dimensions and measurement
Planck's constant \((h),\) speed of light in vacuum \((c)\) and Newton's gravitational constant \((G)\) are three fundamental constants. Which of the following combinations of these has the dimension of length \(?\)
- A \(\sqrt {\frac{{hc}}{G}} \)
- B \(\;\sqrt {\frac{{Gc}}{{{h^{\frac{3}{2}}}}}} \)
- C \(\frac{{\sqrt {hG} }}{{{c^{\frac{3}{2}}}}}\)
- D \(\;\frac{{\sqrt {hG} }}{{{c^{\frac{5}{2}}}}}\)
Answer & Solution
Correct Answer
(C) \(\frac{{\sqrt {hG} }}{{{c^{\frac{3}{2}}}}}\)
Step-by-step Solution
Detailed explanation
According to questions
\(l \propto \,{h^p}{c^q}{G^r}\)
\(l = k\,\,{h^p}\) ..............(\(i\))
Writting dimensions of physical quantities on both sides
\(\left[ {{M^0}L{T^0}} \right] = {\left[ {M{L^2}{T^{ - 1}}} \right]^p}{\left[ {L{T^{ - 1}}} \right]^q}{\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]^r}\)
Applying the principle of homogeneity of dimensions we get \(P - r = 0\) .........(\(ii\))
\(l \propto \,{h^p}{c^q}{G^r}\)
\(l = k\,\,{h^p}\) ..............(\(i\))
Writting dimensions of physical quantities on both sides
\(\left[ {{M^0}L{T^0}} \right] = {\left[ {M{L^2}{T^{ - 1}}} \right]^p}{\left[ {L{T^{ - 1}}} \right]^q}{\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]^r}\)
Applying the principle of homogeneity of dimensions we get \(P - r = 0\) .........(\(ii\))
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