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1. units,dimensions and measurement — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysics1. units,dimensions and measurement1 Mark
MCQ
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
  • A
    $[ M ^{0} L ^{-1} T ^{0} ]$
  • B
    $[ ML ^{0} T ^{-2}]$
  • ✓
    $[ MLT ^{-2}]$
  • D
    $[ ML ^{2} T ^{-2}]$
✓

Answer

Correct option: C.
$[ MLT ^{-2}]$
c
$P=\frac{\alpha}{\beta} \log _{ e }\left(\frac{ kt }{\beta x }\right)$

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

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

$As \;  P$ is dimensionless

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

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