Consider the efficiency of Carnot's engine is given by $\eta=\frac{\alpha \beta}{\sin \theta} \log _{\mathrm{e}} \frac{\beta \mathrm{x}}{\mathrm{kT}}$, 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.

(D) Dimensions of $\alpha$ is same as that of $\beta$

(A) Dimensions of $\beta$ is same as that of force.

(B) Dimensions of $\alpha^{-1} \mathrm{x}$ is same as that of energy.

(C) Dimensions of $\eta^{-1} \sin \theta$ is same as that of $\alpha \beta$

JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Units & Measurement

Consider the efficiency of Carnot's engine is given by

η = \frac{α β}{s i n θ} lo g_{e} \frac{β x}{kT}

, where

α

and

β

are constants. If T is temperature, k is Boltzman constant,

θ

is angular displacement and x has the dimensions of length. Then, choose the incorrect option.

Answer: (D)

Since

η

(efficiency) is dimensionless, and

sin θ

is dimensionless, we have

[α β] = [η] = [sin θ] =

dimensionless. Therefore

[η^{- 1} sin θ] = [α β] =

dimensionless, confirming option (C). For the argument of the logarithm to be dimensionless,

[β x] = [k T]

. Since

[k T] = energy

and

[x] = length

, we get

[β] = \frac{energy}{length} = force

, confirming option (A). Then

[α^{- 1} x] = \frac{[ x ]}{[ α ]} = \frac{length}{dimensionless} = length

, but we need

[α^{- 1} x] = [β x] = energy

, so

[α] = \frac{length}{energy}

. Since

[β] = force

and

[α] = \frac{length}{energy}

, they are not the same. Option (D) is incorrect.

JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Single correct option