Question

If $A$ represents density, $B$ represents velocity, $C$ represents specific heat capacity and $D$ represents wavelength, then the quantity having the dimensions of product of $A, B, C$ and $D$ is

• A. Stefan's constant
• B. Boltzmann's constant
• C. thermal conductivity
• D. universal gas constant

Answer 1

Answer: (C)

According to the question,
Given, $A=$ density, $B=$ velocity, $C=$ specific heat capacity and $D=$ wavelength
So, dimensions of $A, B, C$ and $D$ are
$A=\left[M L^{-3} \,T^{0}\right] B=\left[M^{0} \,L\, T^{-1}\right], C=\left[M^{0}\, L^{2}\, T^{-2} \,K^{-1}\right]$
and $D=\left[ M ^{0} \,L\,T ^{0}\right]$
Hence, dimension of $A B C D=$ $\left[ ML ^{-3} \,T ^{0}\right]\left[ M ^{0} \,L\,T ^{-1}\right]\left[ M ^{0}\, L ^{2} \,T ^{-2} \,K ^{-1}\right]\left[ M ^{0} \,LT ^{0}\right]$
$=\left[ ML ^{(-3+1+2+1)} \,T ^{(-1-2)} \,K ^{-1}\right]=\left[ MLT ^{-3} \,K ^{-1}\right]$
or $A B C D=\left[ MLT ^{-3} \,K ^{-1}\right] \,\,\,\ldots(i)$
Now, dimension of each given option,
(a) Stefan's constant $=\left[ ML ^{0} \,T ^{-3} \,K ^{-4}\right]$
(b) Boltzmann's constant $=\left[ ML ^{2}\, T ^{2} \,K ^{-1}\right]$
(c) Thermal conductivity $=\left[ MLT ^{-3} \,K ^{-1}\right]$
(d) Universal gas constant $=\left[ ML ^{2}\, T ^{-2} \,K ^{-1}\right]$
So, the dimension of product of $A, B, C$ and $D$ is same as dimension of thermal conductivity.