Q 11-01-154JEE MainJEE Main 2022 (26 Jun, Shift 1)Medium
An expression for a dimensionless quantity $P$ is given by $P = \dfrac\alpha\beta\log_e\left(\dfrac{kT}{\beta x}\right)$, 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
Answer: (C) $[\text{M}\text{L}\text{T}^{-2}]$
The argument of the logarithm is dimensionless, so $[\beta] = \dfrac{[kT]}{[x]} = \dfrac{[\text{M}\text{L}^2\text{T}^{-2}]}{[\text{L}]} = [\text{M}\text{L}\text{T}^{-2}]$.
$P$ is dimensionless, so $\dfrac\alpha\beta$ is dimensionless and $[\alpha] = [\beta] = [\text{M}\text{L}\text{T}^{-2}]$.
Solution by Sreeraj P, M.Sc Physics