JEE Mains · Physics · STD 11 - 1. units,dimensions and measurement

The workdone by a gas molecule in an isolated system is given by, \(W =\alpha \beta^{2} e ^{-\frac{ x ^{2}}{\alpha kT }},\) where \(x\) is the displacement, \(k\) is the Boltzmann constant and \(T\) is the temperature, \(\alpha\) and \(\beta\) are constants. Then the dimension of \(\beta\) will be

A \(\left[ M L ^{2} T ^{-2}\right]\)
B \(\left[ M L T ^{-2}\right]\)
C \(\left[ M ^{2} L T ^{2}\right]\)
D \(\left[ M ^{0} L T ^{0}\right]\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\left[ M L T ^{-2}\right]\)

Step-by-step Solution

Detailed explanation

\(\frac{ x ^{2}}{\alpha kT } \rightarrow\) dimensionless \(\Rightarrow[\alpha]=\frac{\left[ x ^{2}\right]}{[ kT ]}=\frac{ L ^{2}}{ ML ^{2} T ^{-2}}= M ^{-1} T ^{2}\) Now \([ W ]=[\alpha][\beta]^{2}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match List\(-I\) with List\(-II\)

List\(-I\)	List\(-II\)
\((a)\) Phase difference between current and voltage in a purely resistive \(AC\) circuit	\((i)\) \(\frac{\pi}{2} ;\) current leads voltage
\((b)\) Phase difference between current and voltage in a pure inductive \(AC\) circuit	\((ii)\) zero
\((c)\) Phase difference between current and voltage in a pure capacitive \(AC\) circuit	\((iii)\) \(\frac{\pi}{2} ;\) current lags voltage
\((d)\) Phase difference between current and voltage in an \(LCR\) series circuit	\((iv)\) \(\tan ^{-1}\left(\frac{X_{C}-X_{L}}{R}\right)\)

Choose the most appropriate answer from the options given below:

JEE Mains 2021 Medium

List - I	List - II
(A) Heat capacity of body	\(\text{(I) } \mathrm{J} \mathrm{kg}^{-1}\)
(B) Specific heat capacity of body	\(\text{(II) } \mathrm{JK}^{-1}\)
(C) Latent heat	\(\text{(III) } \mathrm{J} \mathrm{kg}^{-1} \mathrm{~K}^{-1}\)
(D) Thermal conductivity	\(\text{(IV) } \mathrm{Jm}^{-1} \mathrm{~K}^{-1} \mathrm{~s}^{-1}\)

Choose the correct answer from the options given below:

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