Units & Dimensions

JEE Main level, single correct, medium

84 of 149

Question 84 of 149

Consider the efficiency of Carnot's engine is given by η=αβsin⁡θlog⁡eβxkT\eta =\dfrac{\alpha \beta }{\sin \theta }{\log }_{e}\dfrac{\beta x}{kT}, where α\alpha and β\beta are constants. If TT is temperature, kk is Boltzman constant, θ\theta is angular displacement and xx has the dimensions of length. Then, choose the incorrect option.

Show answer and solution

Answer: (D) Dimensions of α\alpha is same as that of β\beta

Given that, η=αβsin⁡θlog⁡(βxkT)\eta =\dfrac{\alpha \beta }{\sin \theta }\log \left(\dfrac{\beta x}{kT}\right). Since expression inside the logarithm is dimensionless
[βx]=[kT]\left[\beta x\right]=\left[kT\right]
Since the dimensions of kTkT are equivalent to that of energy,
[β]=[kT][x]=[E][x]=[ML2T−2][L]=[MLT−2]\left[\beta \right]=\dfrac{\left[kT\right]}{\left[x\right]}=\dfrac{\left[E\right]}{\left[x\right]}=\dfrac{\left[{ML}^{2}{T}^{-2}\right]}{\left[L\right]}=\left[{MLT}^{-2}\right]
Therefore, β\beta has dimensions of force.
Since efficiency is a dimensionless ratio,
[α][β]=[M0L0T0]\left[\alpha \right]\left[\beta \right]=\left[{M}^{0}{L}^{0}{T}^{0}\right]
⇒[β]=1[α]\Rightarrow \left[\beta \right]=\dfrac{1}{\left[\alpha \right]}
Therefore, dimensions of α\alpha is not same as that of β\beta. Hence option D is incorrect statement and the answer.
[x][α]=[β][x]=[ML2T−2]\dfrac{\left[x\right]}{\left[\alpha \right]}=\left[\beta \right]\left[x\right]=\left[{ML}^{2}{T}^{-2}\right]. Therefore, α−1x{\alpha }^{-1}x has dimensions of energy.

JEE Main question paper PDF

Open a shift to read the full paper on this page, or download it.

JEE Main 2025 Apr 2 Shift 1 question paper

Open Download PDF

Opening paper…

Explain it as a whiteboard videoStep-by-step, in Hinglish or English. Your first video needs no login.Solved it on paper but got it wrong?Upload a photo of your working. Mistake Doctor marks the exact line that broke.

More Physics practice

JEE Main Physics practice question on Units & Dimensions (Dimensions), with the answer and a step-by-step solution. Scan a question from your own book or paper to get a whiteboard video.