Dimensionless Quantities - Practice Questions & MCQ

The quantities which do not have dimension are known as Dimensionless Quantities.

Because these quantities are the ratio of two similar quantities.

Example.

1. Strain

2. Refractive index

3. Relative density

4. Poisson's ratio  

So all these quantities are dimensionless.

Or they have a dimensional formula which is equal to $M^0 L^0 T^0$.

And all these quantities are unitless.

1. Temperature-

It is fundamental quantity.

Dimensional formula- $M^0 L^0 T^0 K^1$ (where K represents Kelvin)
Sl unit- Kelvin

2. Heat

Dimensional formula- $M L^2 T^{-2}$
SI unit- Joule

3. Latent heat

Its dimensional formula is equal to $M^0 L^2 T^{-2}$
And its SI unit is equal to $\mathrm{m}^2 \mathrm{~s}^{-2}$ or $\mathrm{J} / \mathrm{kg}$

4. Specific heat capacity

Dimensional formula- $M^0 L^2 T^{-2} K^{-1}$
Sl unit- $\frac{J}{k g K}$

Surface tension- Dimensional formula- $M^1 L^0 T^{-2}$
Sl unit- $\mathrm{kgs}^{-2}$
Surface energy(per unit area) - Dimensional formula- $M^1 L^0 T^{-2}$
Sl unit- $\mathrm{kgs}^{-2}$
But (Surface tension) and (surface energy per unit area) will have the same dimensional formula and SI units.

The real gas equation is given as

$$
\left(P+\frac{n^2 a}{V^2}\right)(V-n b)=n R T
$$

Where a and b are called Vander waal's constant.
1) Vander Waal's constant (a)

Dimension- $M L^5 T^{-2}$
Unit-Newton $-m^4$
2) Vander waal 's constant (b)

Dimension- $M^0 L^3 T^0$
Unit- $m^3$

1) Voltage (V)

Dimension- $M L^2 T^{-3} A^{-1}$
Unit- Volt
2) Resistance (R)

Dimension- $M L^2 T^{-3} A^{-2}$
Unit- Ohm
3) Resistivity ( $\rho$ )

Dimension- $M L^3 T^{-3} A^{-2}$
Unit- Ohm - meter

1) The permittivity of free space ( $\epsilon_o$ )

Dimension- $M^{-1} L^{-3} T^4 A^2$
Unit- $C^{-2} N^1 m^{-2}$ or farad/metre
2) dielectric constant ( $k$ )

Dimension- $M^0 L^0 T^0$
Unit- Unitless

1) Magnetic Field (B)

Dimension- $M^1 L^0 T^{-2} A^{-1}$
Unit- $\frac{\text { newton }}{\text { ampere }- \text { metre }}$ or $\frac{\text { volt }- \text { second }}{\text { metre }^2}$
2)Permeability of free space

The dimension of permeability of free space $\left(\mu_o\right)-M^1 L^1 T^{-2} A^{-2}$
Sl unit- $\frac{\text { newton }}{\text { ampere }}$ or $\frac{\text { henry }}{\text { metre }}$
3) Magnetic flux ( $\varphi$ )

Dimension- $M L^2 T^{-2} A^{-1}$
Unit- Weber or Volt-second
4) Coefficient of self-induction (L)

Dimension- $M L^2 T^{-2} A^{-2}$
Unit- Henry