JEE Main 2022 · 28 Jul · Shift 2 · Physics Q1

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Consider the efficiency of carnot's engine is given by $\eta=\dfrac{\alpha\beta}{\sin\theta}\log_e\dfrac{\beta x}{kT}$ , where $\alpha$ and $\beta$ are constants. If T is temperature, k is Boltzmann constant, θ is angular displacement and $x$ has the dimensions of length. Then, choose the incorrect option :

Dimensions of β is same as that of force.

Dimensions of

\alpha^{-1}x

is same as that of energy.

Dimensions of

\eta^{-1}\sin\theta

is same as that of

\alpha\beta

Dimensions of α is same as that of β.

Answer and worked solution

Official answer: D

Dimensionless pieces. η and the log are dimensionless, so $\frac{\beta x}{kT}$ is too: β has the dimensions of energy per length, i.e. force. Then $\alpha\beta$ is dimensionless, so $\alpha=\beta^{-1}$ , which is length per energy.

Check. (A) is true. (B): $\alpha^{-1}x$ is energy per length times length, energy, so true. (C): $\eta^{-1}\sin\theta$ and $\alpha\beta$ are both dimensionless, true. (D) is false.

(D)

Topics:Dimensional analysis

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JEE Main 2022 · 28 Jul · Shift 2, Physics question 1