Question

Consider the efficiency of Carnot's engine is given by $\eta =\frac{\alpha \beta }{\sin \theta }{\log }_e\frac{\beta x}{kT}$, where $\alpha$ and $\beta$ are constants. If $T$ is temperature, $k$ is Boltzman constant, $\theta$ is angular displacement and $x$ has the dimensions of length. Then, choose the incorrect option.

• A. Dimensions of $\beta$ is same as that of force.
• B. Dimensions of ${\alpha }^{-1}x$ is same as that of energy.
• C. Dimensions of ${\eta }^{-1}\sin \theta$ is same as that of $\alpha \beta$
• D. Dimensions of $\alpha$ is same as that of $\beta$

Answer 1

Answer: (D)

$[\alpha \beta ]=[\eta ]=[\sin \theta ]=\text{ Dimensionless }$
$[{\eta }^{-1}\sin \theta ]=[\alpha \beta ]=\text{ D.L. }$