Question

Match the physical quantities given in List-I with dimensions expressed in terms of mass (M), length (L), time (T) and electric current (A) given in List-II.

Codes:

• A. a - iv, b - ii, c - iii, d - i
• B. a - iv, b - iii, c - i, d - ii
• C. a - iv, b - i, c - iii, d - ii
• D. a - ii, b - i, c - iii, d - iv

Hint:

To quickly derive dimensions, relate quantities to Energy or Work. For instance, Torque and Energy have identical dimensions (\( ML^2T^{-2} \)). Similarly, Planck's constant has the same dimensions as Angular Momentum (\( ML^2T^{-1} \)).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The objective is to derive the dimensional formulas for each physical quantity and match them with the corresponding dimensions given in List-II.
Step 3: Detailed Explanation:
(a) Torque (\( \tau \)):
Formula: \( \tau = \text{Force} \times \text{perpendicular distance} \).
Dimensions: \( [MLT^{-2}] \times [L] = [ML^2T^{-2}] \).
This matches with (iv).
(b) Gravitational constant (G):
Formula: \( F = G \frac{m_1 m_2}{r^2} \Rightarrow G = \frac{F r^2}{m_1 m_2} \).
Dimensions: \( \frac{[MLT^{-2}] \times [L^2]}{[M^2]} = [M^{-1}L^3T^{-2}] \).
This matches with (iii).
(c) Capacitance (C):
Formula: \( C = \frac{Q}{V} = \frac{Q^2}{W} \) (since Potential \( V = \text{Work } W / \text{Charge } Q \)).
Dimensions of Charge \( Q = [AT] \).
Dimensions of Work \( W = [ML^2T^{-2}] \).
Dimensions of \( C = \frac{[AT]^2}{[ML^2T^{-2}]} = [M^{-1}L^{-2}T^4A^2] \).
This matches with (i).
(d) Planck's constant (h):
Formula: \( E = h \nu \Rightarrow h = \frac{E}{\nu} \) (where \( \nu \) is frequency).
Dimensions: \( \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}] \).
This matches with (ii).
Step 4: Final Answer:
The correct matching is a-iv, b-iii, c-i, d-ii. This corresponds to code (2).