Question

Match List - I with List - II : 

Choose the most appropriate answer from the options given below : 
 

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

Hint:

Dimensional analysis is a fundamental tool in physics. Memorizing the dimensions of key quantities like Force, Energy, Charge, and Potential can speed up derivations for other related quantities. Always start from a defining formula for the quantity in question.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question requires us to match the physical quantities in List-I with their corresponding dimensional formulas in List-II. We need to derive the dimensional formula for each quantity.
Step 2: Key Formula or Approach:
We will use fundamental physical equations involving these quantities to derive their dimensions. The fundamental dimensions are Mass (M), Length (L), Time (T), and Electric Current (A).
Step 3: Detailed Explanation:
(a) Magnetic Induction (B):
The force on a charge q moving with velocity v in a magnetic field B is given by \(F = qvB\).
So, the dimension of B is:
\[ [B] = \frac{[F]}{[q][v]} = \frac{[MLT^{-2}]}{[AT][LT^{-1}]} = [MLT^{-2}A^{-1}T^{-1}L^{-1}] = [MT^{-2}A^{-1}] \] This matches with (iii) in List-II.
(b) Magnetic Flux (\(\Phi\)):
Magnetic flux is defined as the product of magnetic induction and area (\(\Phi = B \cdot A\)).
So, the dimension of \(\Phi\) is:
\[ [\Phi] = [B][A] = [MT^{-2}A^{-1}][L^2] = [ML^2T^{-2}A^{-1}] \] This matches with (i) in List-II.
(c) Magnetic Permeability (\(\mu\)):
From Ampere's law, the magnetic field due to a long straight wire is \(B = \frac{\mu I}{2\pi r}\).
So, the dimension of \(\mu\) is:
\[ [\mu] = \frac{[B][r]}{[I]} = \frac{[MT^{-2}A^{-1}][L]}{[A]} = [MLT^{-2}A^{-2}] \] This matches with (iv) in List-II.
(d) Magnetization (M):
Magnetization is defined as the magnetic moment per unit volume. Magnetic moment is current times area.
So, the dimension of M is:
\[ [M] = \frac{[\text{Magnetic Moment}]}{[\text{Volume}]} = \frac{[I \cdot A]}{[V]} = \frac{[A][L^2]}{[L^3]} = [AL^{-1}] = [M^0L^{-1}A] \] This matches with (ii) in List-II.
Step 4: Final Answer:
Based on the derivations, the correct matching is: (a) \(\rightarrow\) (iii)
(b) \(\rightarrow\) (i)
(c) \(\rightarrow\) (iv)
(d) \(\rightarrow\) (ii)
This corresponds to option (C).