An expression for a dimensionless quantity P is given by P=α/β e ( kt /β x); where α and β are constants, x is distance; k is Boltzmann constant and t is the temperature. Then the dimensions of α will be

JEE Mains · Physics · STD 11 - Units,dimensions and measurement

An expression for a dimensionless quantity \(P\) is given by \(P=\frac{\alpha}{\beta} \log _{e}\left(\frac{ kt }{\beta x }\right)\); where \(\alpha\) and \(\beta\) are constants, \(x\) is distance ; \(k\) is Boltzmann constant and \(t\) is the temperature. Then the dimensions of \(\alpha\) will be

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Correct Answer

(C) \([ MLT ^{-2}]\)

Detailed explanation

\(P=\frac{\alpha}{\beta} \log _{ e }\left(\frac{ kt }{\beta x }\right)\)

\(\frac{ kt }{\beta x }=1 \Rightarrow \beta=\frac{ kt }{ x }=\frac{ ML ^{2} T ^{-2}}{ L }\)

\(\left(\because E =\frac{1}{2} kt \right)\)

\(As \; P\) is dimensionless

\(\Rightarrow[\alpha]=[\beta]=\left[ MLT ^{-2}\right]\)

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