An expression dimensionless quantity|HEATTRANSFERJEEMain2022

JEE MAIN 2022

26-06-2022 S1

Question

An expression for a dimensionless quantity P is given by $P=\frac{\alpha}{\beta} \log _e\left(\frac{k t}{\beta x}\right)$ ; 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 :

Select the correct option:

$\left[\mathrm{M}^0 \mathrm{~L}^{-1} \mathrm{~T}^0\right]$

$\left[\mathrm{ML}^0 \mathrm{~T}^{-2}\right]$

$\left[\mathrm{MLT}^{-2}\right]$

$\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$

✓ Correct! Well done.

✗ Incorrect. Try again or view the solution.

Solution

$$ \begin{aligned} & P=\frac{\alpha}{\beta} \log _e\left(\frac{k t}{\beta x}\right) \\ & \frac{k t}{\beta x}=1 \Rightarrow \beta=\frac{k t}{x}=\frac{M L^2 T^{-2}}{L} \\ & \left(\because E=\frac{1}{2} k t\right) \end{aligned} $$ As P is dimensionless $$ \Rightarrow[\alpha]=[\beta]=\left[M L T^{-2}\right] $$

Question Tags

JEE Main

Physics

Easy

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