Question 270: Units , Dimensions and Error Analysis

Question

Match
List I
with
List II
and select the correct answer using the codes given below the lists:
List I
P. Boltzmann Constant
Q. Coefficient of viscosity
R. Plank Constant
S. Thermal conductivity
List II
1. [ML
2
T
-1
]
2. [ML
-1
T
-1
]
3. [MLT
-3
K
-1
]
4. [ML
2
T
-2
K
-1
]

Accepted Answer

Let's break down the dimensional analysis of each physical quantity:
P. Boltzmann Constant ($k_B$)
The Boltzmann constant relates energy to temperature. Its dimensional formula is derived from the relationship: $E = k_B T$.
Energy ($E$) has dimensions of [ML
2
T
-2
]. Temperature ($T$) has dimensions of [K]. Therefore:
$$[k_B] = \frac{[E]}{[T]} = \frac{[ML^2T^{-2}]}{[K]} = [ML^2T^{-2}K^{-1}]$$
Q. Coefficient of Viscosity ($\eta$)
Viscosity is a measure of a fluid's resistance to flow. It is defined as the ratio of shear stress to shear rate. Shear stress has dimensions of [ML
-1
T
-2
] (force per unit area), and shear rate has dimensions of [T
-1
] (velocity gradient). Therefore:
$$[\eta] = \frac{[ML^{-1}T^{-2}]}{[T^{-1}]} = [ML^{-1}T^{-1}]$$
R. Plank Constant ($h$)
Planck's constant relates energy to frequency. Its dimensional formula is derived from the relationship: $E = h
u$.
Energy ($E$) has dimensions of [ML
2
T
-2
]. Frequency ($
u$) has dimensions of [T
-1
]. Therefore:
$$[h] = \frac{[E]}{[
u]} = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}]$$
S. Thermal Conductivity ($k$)
Thermal conductivity is a measure of a material's ability to conduct heat. It is defined as the rate of heat transfer per unit area per unit temperature gradient. Heat transfer rate has dimensions of [ML
2
T
-3
]. Area has dimensions of [L
2
], and temperature gradient has dimensions of [K/L]. Therefore:
$$[k] = \frac{[ML^2T^{-3}]}{[L^2][K/L]} = [MLT^{-3}K^{-1}]$$
Matching the Dimensions:
Based on the dimensional analysis, we can match the quantities in List I with the dimensions in List II as follows:
List I
List II
P. Boltzmann Constant ($k_B$)
4. [ML
2
T
-2
K
-1
]
Q. Coefficient of Viscosity ($\eta$)
2. [ML
-1
T
-1
]
R. Plank Constant ($h$)
1. [ML
2
T
-1
]
S. Thermal conductivity ($k$)
3. [MLT
-3
K
-1
]
Therefore, the correct answer is
Option C
.

Answer

$$\eqalign{
  & P\,\,\,\,Q\,\,\,\,R\,\,\,\,S  \cr 
  & 3\,\,\,\,\,\,1\,\,\,\,\,2\,\,\,\,\,4 \cr} $$

Answer

$$\eqalign{
  & P\,\,\,\,Q\,\,\,\,R\,\,\,\,S  \cr 
  & 3\,\,\,\,\,\,2\,\,\,\,\,1\,\,\,\,\,4 \cr} $$

Answer

$$\eqalign{
  & P\,\,\,\,Q\,\,\,\,R\,\,\,\,S  \cr 
  & 4\,\,\,\,\,\,2\,\,\,\,\,1\,\,\,\,\,3 \cr} $$

Answer

$$\eqalign{
  & P\,\,\,\,Q\,\,\,\,R\,\,\,\,S  \cr 
  & 4\,\,\,\,\,\,1\,\,\,\,\,2\,\,\,\,\,3 \cr} $$

Units , Dimensions and Error Analysis

Match List I with List II and select the correct answer using the codes given below the lists: List I P. Boltzmann Constant Q. Coefficient of viscosity R. Plank Constant S. Thermal conductivity List II 1. [ML 2 T -1 ] 2. [ML -1 T -1 ] 3. [MLT -3 K -1 ] 4. [ML 2 T -2 K -1 ]

Choose the correct answer: