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JEE Main 2022 (28 Jul, Shift 2) · Applications of Dimensional Analysis

Applications of Dimensional Analysis question from JEE Main 2022 (28 Jul, Shift 2)

Q37JEE Main 2022 · 28 JUL · SHIFT 2Medium

Applications of Dimensional Analysis

Consider the efficiency of Carnot's engine is given by η=αβsin⁡θlog⁡eβxkT\eta = \frac{\alpha\beta}{\sin\theta}\log_e\frac{\beta x}{kT}, where α\alpha and β\beta are constants. If T is temperature, k is Boltzmann constant, θ\theta is angular displacement and x has the dimensions of length. Then, choose the incorrect option:

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Answer: (d)

  1. The log argument is dimensionless: [βx]=[kT]=[\beta x] = [kT] = energy, so [β]=[\beta] = energy/length == force. (a) is correct.
  2. η\eta and sin⁡θ\sin\theta are dimensionless, so αβ\alpha\beta is dimensionless: [α]=1/force[\alpha] = 1/\text{force}.
  3. [α−1x]=[\alpha^{-1}x] = force ×\times length == energy, so (b) is correct; η−1sin⁡θ\eta^{-1}\sin\theta and αβ\alpha\beta are both dimensionless, so (c) is correct.
  4. [α]=1/force≠[β][\alpha] = 1/\text{force} \neq [\beta], so (d) is the incorrect option.
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