Question

The dimensions of angular momentum, latent heat, and capacitance are, respectively:
A. $M{{L}^{2}}{{T}^{1}}{{A}^{2}},\,{{L}^{2}}{{T}^{-2}},\,{{M}^{-1}}{{L}^{-2}}{{T}^{2}}$
B. $ML{{T}^{2}},\,{{L}^{2}}{{T}^{2}},{{M}^{-1}}{{L}^{-2}}{{T}^{4}}{{A}^{2}}$
C. $M{{L}^{2}}{{T}^{-1}},\,{{L}^{2}}{{T}^{-2}},M{{L}^{2}}T{{A}^{2}}$
D. $M{{L}^{2}}{{T}^{-1}},\,{{L}^{2}}{{T}^{-2}},{{M}^{-1}}{{L}^{-2}}{{T}^{4}}{{A}^{2}}$

Answer 1

- Hint: The physical quantities angular momentum, latent heat, and capacitance are derived quantities. The dimensions can be obtained from the definitions.

Complete step-by-step solution -
Angular momentum: It is the rotational counterpart of linear momentum and defined as,
$\overrightarrow{J}=\overrightarrow{r}\times \overrightarrow{p}$
Here a particle rotates about a point (named the origin) with momentum $\overrightarrow{p}$ having a distance $\overrightarrow{r}$ from the origin. The dimension of angular momentum in the product of the dimension of distance and the dimension of linear momentum.
Dimension of distance $=L$
Dimension of linear momentum $=ML{{T}^{-1}}$
Hence, dimension of $J=L\cdot ML{{T}^{-1}}=M{{L}^{2}}{{T}^{-1}}$
Latent heat: It is defined as the amount of heat (energy) required to change the state of unit mass of a substance at constant temperature. Hence, its dimension is that of energy per unit mass.
Dimension of energy $=M{{L}^{2}}{{T}^{-2}}$ such that the dimension of latent heat is,
$\dfrac{M{{L}^{2}}{{T}^{-2}}}{M}={{L}^{2}}{{T}^{-2}}$
Capacitance: It is defined as the ratio of charge to electric potential i.e. $C=\dfrac{Q}{V}$
Dimension of charge is the product of the dimensions of current and time which is$AT$. Dimension of electric potential is the ratio of dimension energy to that of charge. Hence, dimension of capacitance is given by,
\[\dfrac{{{\left( dimension\text{ }of\text{ }charge \right)}^{2}}}{\left( dimension\text{ }of\text{ }energy \right)}\]
$=\dfrac{{{(AT)}^{2}}}{M{{L}^{2}}{{T}^{-2}}}$
$={{M}^{-1}}{{L}^{-2}}{{T}^{4}}{{A}^{2}}$
Hence, the correct option is: (D) $M{{L}^{2}}{{T}^{-1}},\,{{L}^{2}}{{T}^{-2}},{{M}^{-1}}{{L}^{-2}}{{T}^{4}}{{A}^{2}}$

Note: All the quantities should be expressed in terms of base quantities. The required base quantities in the question are mass, length, time and current. As we know charge is a product of current and time. So the dimensional formula of charge is $\left[ AT \right]$.