Question

Match List - I with List - II :
List - I: (a) h (Planck's constant), (b) E (kinetic energy), (c) V (electric potential), (d) P (linear momentum)
List - II: (i) [M L T$^{-1}$], (ii) [M L$^2$ T$^{-1}$], (iii) [M L$^2$ T$^{-2}$], (iv) [M L$^2$ T$^{-3}$ I$^{-1}$]
Choose the correct answer from the options given below :

• (a) → (i), (b) → (ii), (c) → (iv), (d) → (iii)
• (a) → (iii), (b) → (iv), (c) → (ii), (d) → (i)
• (a) → (ii), (b) → (iii), (c) → (iv), (d) → (i)
• (a) → (iii), (b) → (ii), (c) → (iv), (d) → (i)

Hint:

Planck's constant ($h$) has the same dimensions as angular momentum ($L = mvr$). Linear momentum is mass $\times$ velocity, while energy is always $[ML^2T^{-2}]$.

Answer 1

The Correct Option is C

To solve this problem, we need to match the physical quantities in List-I with their respective dimensional formulas in List-II. Let's take a look at each one:

1. Planck's constant (h): The dimensional formula of Planck's constant is derived from the equation for energy of a photon \( E = h \cdot \nu \), where \( E \) is energy and \( \nu \) is frequency. The dimensional formula for energy is \([M L^2 T^{-2}]\) and for frequency \([T^{-1}]\). Thus, the dimensional formula for Planck's constant \( h \) is: [h] = \frac{[E]}{[\nu]} = [M L^2 T^{-2}] \div [T^{-1}] = [M L^2 T^{-1}]. This matches with option (ii).
2. Kinetic energy (E): Kinetic energy is given by the formula \( KE = \frac{1}{2} mv^2 \), where \( m \) is mass and \( v \) is velocity. The dimensional formula for mass is \([M]\) and for velocity \([L T^{-1}]\), so for kinetic energy we have: [E] = [M][L T^{-1}]^2 = [M L^2 T^{-2}]. This matches with option (iii).
3. Electric potential (V): Electric potential \( V \) is defined as work done per unit charge. Its dimensional formula is derived as follows: Work done (or energy) has a dimensional formula \([M L^2 T^{-2}]\), and charge has a dimensional formula \([I T]\). Thus, the dimensional formula for electric potential is: [V] = \frac{[M L^2 T^{-2}]}{[I T]} = [M L^2 T^{-3} I^{-1}]. This matches with option (iv).
4. Linear momentum (P): Linear momentum is defined as the product of mass and velocity. Hence, its dimensional formula is: [P] = [M][L T^{-1}] = [M L T^{-1}]. This matches with option (i).

By matching each item from List-I with its corresponding dimensional formula from List-II, we obtain the correct pairing as follows:

• (a) - Planck's constant (h) → (ii) [M L² T^-1]
• (b) - Kinetic energy (E) → (iii) [M L² T^-2]
• (c) - Electric potential (V) → (iv) [M L² T^-3 I^-1]
• (d) - Linear momentum (P) → (i) [M L T^-1]

Therefore, the correct answer is: (a) → (ii), (b) → (iii), (c) → (iv), (d) → (i).