Which of the following is dimensionless
(a) Boltzmann constant (b) Planks constant
(c) Poissons ratio (d) Relative density

Options

A. Both A & B
B. Both B &C
C. Both C & D
D. Both D &A

Answer

To solve the question of which of the given quantities is dimensionless, we will analyze each option step by step.

### Step 1: Analyze the Boltzmann Constant (k)
The Boltzmann constant is defined as:
\[ k = \frac{PV}{T \cdot n} \]
Where:
- \( P \) is pressure
- \( V \) is volume
- \( T \) is absolute temperature
- \( n \) is the number of molecules

**Dimensional Analysis:**
- Pressure \( P \) has dimensions: \( [P] = M L^{-1} T^{-2} \)
- Volume \( V \) has dimensions: \( [V] = L^3 \)
- Temperature \( T \) has dimensions: \( [T] = K \) (Kelvin)
- The number of molecules \( n \) is dimensionless.

Substituting these into the equation for \( k \):
\[ [k] = \frac{M L^{-1} T^{-2} \cdot L^3}{K \cdot 1} = M L^2 T^{-2} K^{-1} \]

Thus, the Boltzmann constant has dimensions and is not dimensionless.

### Step 2: Analyze Planck's Constant (h)
Planck's constant is defined as:
\[ h = \frac{E}{\nu} \]
Where:
- \( E \) is energy
- \( \nu \) is frequency

**Dimensional Analysis:**
- Energy \( E \) has dimensions: \( [E] = M L^2 T^{-2} \)
- Frequency \( \nu \) has dimensions: \( [\nu] = T^{-1} \)

Substituting these into the equation for \( h \):
\[ [h] = \frac{M L^2 T^{-2}}{T^{-1}} = M L^2 T^{-1} \]

Thus, Planck's constant also has dimensions and is not dimensionless.

### Step 3: Analyze Poisson's Ratio (σ)
Poisson's ratio is defined as the ratio of lateral strain to longitudinal strain:
\[ \sigma = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} \]

**Dimensional Analysis:**
Strain is defined as the change in length divided by the original length, which is dimensionless.

Thus, Poisson's ratio is dimensionless:
\[ [\sigma] = \frac{\text{Dimensionless}}{\text{Dimensionless}} = \text{Dimensionless} \]

### Step 4: Analyze Relative Density (ρd)
Relative density is defined as:
\[ \rho_d = \frac{\text{Density of substance}}{\text{Density of reference}} \]

**Dimensional Analysis:**
Density has dimensions:
\[ [\text{Density}] = M L^{-3} \]

Thus, relative density is:
\[ [\rho_d] = \frac{M L^{-3}}{M L^{-3}} = \text{Dimensionless} \]

### Conclusion
From the analysis:
- Boltzmann constant (k) is not dimensionless.
- Planck's constant (h) is not dimensionless.
- Poisson's ratio (σ) is dimensionless.
- Relative density (ρd) is dimensionless.

**Final Answer:**
The dimensionless quantities are:
- (c) Poisson's ratio
- (d) Relative density

Thus, the correct options are both C and D.