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$(A)$ The dimensions are matched as follows:$(A)$ Entropy $(S)$: $S = \frac{\Delta Q}{T}$. The unit is $J/K$, which is the same as the unit of the Bol

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$A-II, B-III, C-IV, D-I$

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$(A)$ The dimensions are matched as follows:$(A)$ Entropy $(S)$: $S = \frac{\Delta Q}{T}$. The unit is $J/K$, which is the same as the unit of the Boltzmann constant $(k_B)$. Thus, $(A) ightarrow (II)$.$(B)$ Young's modulus $(Y)$: $Y = \frac{	ext{Stress}}{	ext{Strain}} = \frac{	ext{Force}}{	ext{Area}}$. The unit is $N/m^2$ or $J/m^3$, which is the same as the unit of energy density $(U/V)$. Thus, $(B) ightarrow (III)$.$(C)$ Angular momentum $(L)$: $L = mvr$. The unit is $kg \cdot m^2/s$, which is the same as the unit of Planck's constant $(h)$. Thus, $(C) ightarrow (IV)$.$(D)$ Decay constant $(\lambda)$: $\lambda = \frac{1}{	ext{time}}$. The unit is $s^{-1}$, which is the same as the unit of angular velocity $(\omega)$. Thus, $(D) ightarrow (I)$.Therefore, the correct matching is $(A-II, B-III, C-IV, D-I)$.

$A$. Entropy	$I$. Angular velocity
$B$. Young's modulus of elasticity	$II$. Boltzmann constant
$C$. Angular momentum	$III$. Energy density
$D$. Decay constant	$IV$. Planck's constant

List-$I$	List-$II$
$A$. Spring constant	$I$. $ML^2 T^{-2} K^{-1}$
$B$. Thermal conductivity	$II$. $ML^0 T^{-2}$
$C$. Boltzmann constant	$III$. $ML^2 T^{-3} A^{-2}$
$D$. Inductive reactance	$IV$. $MLT^{-3} K^{-1}$

Column-$I$	Column-$II$
$(a)$ Wien's constant	$(i)$ $W m^{-2} K^{-4}$
$(b)$ Stefan-Boltzmann's constant	$(ii)$ $W m^{-1} K^{4}$
	$(iii)$ $m K$

Similar Questions

Units for constants in Column-$I$ and $SI$ unit is given in Column-$II$. Match the followings:

Column-$I$ Column-$II$

$(a)$ Wien's constant $(i)$ $W m^{-2} K^{-4}$

$(b)$ Stefan-Boltzmann's constant $(ii)$ $W m^{-1} K^{4}$

$(iii)$ $m K$

Which of the following pairs of quantities do not have the same dimensions?

Select the pair from the following that does not have the same dimensions.

Which relation is wrong?

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Match the physical quantities in Column-$I$ with those in Column-$II$ having the same dimensions. The correct matching is:$A$. Entropy$I$. Angular velocity$B$. Young's modulus of elasticity$II$. Boltzmann constant$C$. Angular momentum$III$. Energy density$D$. Decay constant$IV$. Planck's constant