Question 4: Units , Dimensions and Error Analysis

Question

$$ 	ext { Match List - I with List - II. } $$
$$
	ext { List - I }
$$
$$
	ext { List - II }
$$
A.
Boltzmann constant
I.
$$
\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}ight]
$$
B.
Stefan's constant
II.
$$
\left[\mathrm{M} \mathrm{~L}^2 \mathrm{~T}^{-1}ight]
$$
C.
Planck's constant
III.
$$
\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}ight]
$$
D.
Gravitational constant
IV.
$$
\left[\mathrm{M} \mathrm{~L}^0 \mathrm{~T}^{-3} \mathrm{~K}^{-4}ight]
$$
Choose the correct answer from the options given below :

Accepted Answer

For the dimensional equation of Boltzmann Constant $\left(k_Bight)$
The average kinetic energy (E) of a gas molecule is related to temperature (T) by the equation :
$$ \mathrm{E}=\frac{3}{2} \mathrm{k}_{\mathrm{B}} \mathrm{~T} $$
To find the dimensions of $\mathrm{k}_{\mathrm{B}}$ :
$$ \left[\mathrm{k}_{\mathrm{B}}ight]=[\mathrm{E}][\mathrm{T}] $$
Using the formula Work $=$ Force ÃÂ Distance, dimensions of energy is $[\mathrm{E}]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight]$
The dimensions of temperature is $[\mathrm{T}]=[\mathrm{K}]$
So,
$$ \left[\mathrm{k}_{\mathrm{B}}ight]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight][\mathrm{K}]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}ight] $$
Thus, (A) Boltzmann constant Ã¢ÂÂ (III) [ $\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}$ ]
For the dimensions of Stefan's Constant ( $\sigma$ )
According to Stefan-Boltzmann law, the energy radiated per unit area per unit time $(\mathrm{P})$ is:
$$ \mathrm{P}=\sigma \mathrm{T}^4 $$
Where $\mathrm{P}=\frac{	ext { Energy }}{	ext { AreaÃÂ Time }}$
Dimensions of
$$ [\mathrm{P}]=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight]}{\left[\mathrm{L}^2ight][\mathrm{T}]}=\left[\mathrm{MT}^{-3}ight] $$
Dimensions of $\mathrm{T}^4$ is $\left[\mathrm{K}^4ight]$
So, the dimensions of Stefan's constant is
$$ [\sigma]=\frac{[\mathrm{P}]}{\left[\mathrm{T}^4ight]}=\frac{\left[\mathrm{MT}^{-3}ight]}{\left[\mathrm{K}^4ight]}=\left[\mathrm{MT}^{-3} \mathrm{~K}^{-4}ight]=\left[\mathrm{ML}^0 \mathrm{~T}^{-3} \mathrm{~K}^{-4}ight] $$
Thus, (B) Stefan's constant Ã¢ÂÂ (IV)[ $\mathrm{ML}^0 \mathrm{~T}^{-3} \mathrm{~K}^{-4}$ ]
For the dimensions of Planck's Constant (h):
The energy (E) of a photon is given by $E=h 
u$, where $
u$ is frequency (Time ${ }^{-1}$ ).
So, the dimension of Planck's constant is
$[\mathrm{h}]=\frac{[\mathrm{E}]}{[v]}=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight]}{\left[\mathrm{T}^{-1}ight]}$
$$\Rightarrow $$ $$ [\mathrm{h}]=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}ight] $$
Thus, (C) Planck's constant Ã¢ÂÂ (II)[ $\mathrm{ML}^2 \mathrm{~T}^{-1}$ ]
For the dimensions of Gravitational Constant (G):
From Newton's law of gravitation $F=\frac{G m_1 m_2}{r^2} \Rightarrow G=\frac{F r^2}{m_1 m_2}$
The dimensions of Force is $[\mathrm{F}]=\left[\mathrm{MLT}^{-2}ight]$
The dimensions of Distance squared is $\left[\mathrm{r}^2ight]=\left[\mathrm{L}^2ight]$
The dimensions of mass squared is $\left[\mathrm{m}_1 \mathrm{~m}_2ight]=\left[\mathrm{M}^2ight]$
So, the dimensions of Gravitational constant is
$$ [\mathrm{G}]=\frac{\left[\mathrm{MLT}^{-2}ight]\left[\mathrm{L}^2ight]}{\left[\mathrm{M}^2ight]}=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}ight] $$
Thus (D) Gravitational constant $ightarrow$ (I) $\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}ight]$
$$ \mathrm{A} ightarrow \mathrm{III}, \mathrm{~B} ightarrow \mathrm{IV}, \mathrm{C} ightarrow \mathrm{II}, \mathrm{D} ightarrow \mathrm{I} $$
Therefore, the correct option is (C).

Answer

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

Answer

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

Answer

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

Answer

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

Units , Dimensions and Error Analysis

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