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1.Units, Dimensions and Measurement
hard
In the relation $P = \frac{\alpha }{\beta }{e^{ - \frac{{\alpha Z}}{{k\theta }}}}$ $P$ is pressure, $Z$ is the distance, $k$ is Boltzmann constant and $\theta$ is the temperature. The dimensional formula of $\beta$ will be
A
$[{M^0}{L^2}{T^0}]$
B
$[{M^1}{L^2}{T^1}]$
C
$[{M^1}{L^0}{T^{ - 1}}]$
D
$[{M^0}{L^2}{T^{ - 1}}]$
(IIT-2004)
Solution
(a) In given equation, $\frac{{\alpha z}}{{k\theta }}$ should be dimensionless
$\therefore \alpha = \frac{{k\theta }}{z} \Rightarrow [\alpha ] = \frac{{[M{L^2}{T^{ – 2}}{K^{ – 1}} \times K]}}{{[L]}} = [ML{T^{ – 2}}]$
and $P = \frac{\alpha }{{ \beta }} \Rightarrow [\beta ] = \left[ {\frac{\alpha }{p}} \right] = \frac{{[ML{T^{ – 2}}]}}{{[M{L^{ – 1}}{T^{ – 2}}]}} = [{M^0}{L^2}{T^0}]$.
Standard 11
Physics
