Some physical constants are given in List-1 and their dimensional formulae are given in List-2. Match the correct pairs in the lists
`{:("List-I","List-2"),((a)"Planck's constant ",(e)ML^(-1)T^(-2)),((b)"Gravitational constant ", (f) ML^(-1)T^(-1)),((c)"Bulk modulus" ,(g) ML^(-2)T^(-1)),((d) "Coefficient of viscosity " ,(h) M^(-1)L^(3)T^(-2)):}`

Options

A. a - h`" "` b- g `" "`c -f `" "` d-e
B. a - f`" "` b- e `" "`c -g `" "` d-h
C. a - g`" "` b- f `" "`c -e `" "` d-h
D. a - g`" "` b- h `" "`c -e`" "` d-f

Answer

To match the physical constants from List-1 with their corresponding dimensional formulae from List-2, we will analyze each constant and determine its dimensional formula step by step.

### Step 1: Identify the Dimensional Formula of Planck's Constant
- **Planck's constant (h)** is defined as the energy of a photon divided by the frequency of the photon. The formula for energy (E) is given by \( E = m \cdot v^2 \) (kinetic energy) or \( E = h \cdot f \) (where f is frequency).
- The dimensional formula for energy is \( [E] = M L^2 T^{-2} \).
- Frequency (f) has the dimensional formula \( [f] = T^{-1} \).
- Thus, the dimensional formula for Planck's constant is:
\[ [h] = \frac{[E]}{[f]} = \frac{M L^2 T^{-2}}{T^{-1}} = M L^2 T^{-1} \]
- Therefore, **Planck's constant** matches with **(g) \( M L^{-2} T^{-1} \)**.

### Step 2: Identify the Dimensional Formula of Gravitational Constant
- **Gravitational constant (G)** relates the gravitational force between two masses to the product of their masses and the inverse square of the distance between them.
- The formula is given by \( F = \frac{G m_1 m_2}{r^2} \).
- Rearranging gives \( G = \frac{F r^2}{m_1 m_2} \).
- The dimensional formula for force (F) is \( [F] = M L T^{-2} \) and distance (r) is \( [r] = L \).
- Thus, the dimensional formula for G is:
\[ [G] = \frac{[F] \cdot [r]^2}{[m]^2} = \frac{M L T^{-2} \cdot L^2}{M^2} = M^{-1} L^3 T^{-2} \]
- Therefore, **Gravitational constant** matches with **(h) \( M^{-1} L^3 T^{-2} \)**.

### Step 3: Identify the Dimensional Formula of Bulk Modulus
- **Bulk modulus (K)** is defined as the ratio of pressure increase to the fractional change in volume.
- The formula is \( K = -\frac{dP}{dV/V} \).
- The dimensional formula for pressure (P) is \( [P] = M L^{-1} T^{-2} \) and volume (V) is \( [V] = L^3 \).
- Thus, the dimensional formula for bulk modulus is:
\[ [K] = \frac{[P]}{[V]} = \frac{M L^{-1} T^{-2}}{L^3} = M L^{-1} T^{-2} \]
- Therefore, **Bulk modulus** matches with **(e) \( M L^{-1} T^{-2} \)**.

### Step 4: Identify the Dimensional Formula of Coefficient of Viscosity
- **Coefficient of viscosity (η)** is defined as the ratio of shear stress to shear rate.
- The formula is \( \eta = \frac{\text{shear stress}}{\text{shear rate}} \).
- Shear stress has the dimensional formula \( [\text{stress}] = M L^{-1} T^{-2} \) and shear rate is \( [\text{rate}] = T^{-1} \).
- Thus, the dimensional formula for coefficient of viscosity is:
\[ [\eta] = \frac{[\text{stress}]}{[\text{rate}]} = \frac{M L^{-1} T^{-2}}{T^{-1}} = M L^{-1} T^{-1} \]
- Therefore, **Coefficient of viscosity** matches with **(f) \( M L^{-1} T^{-1} \)**.

### Final Matching
Now, we can summarize the matches:
- (a) Planck's constant → (g) \( M L^{-2} T^{-1} \)
- (b) Gravitational constant → (h) \( M^{-1} L^3 T^{-2} \)
- (c) Bulk modulus → (e) \( M L^{-1} T^{-2} \)
- (d) Coefficient of viscosity → (f) \( M L^{-1} T^{-1} \)

Thus, the correct pairs are:
- (a, g), (b, h), (c, e), (d, f)