Question

Column-I gives physical quantities and Column-II represent their dimensions. Choose the option representing correct matching.

• A. I-S, II-R, III-P, IV-Q
• B. I-Q, II-R, III-P, IV-S
• C. I-R, II-S, III-P, IV-Q
• D. I-S, II-P, III-R, IV-Q

Hint:

Dimensional analysis is a powerful tool. When you need to find the dimensions of a quantity, think of the simplest formula that defines it. For example, using \(F=qvB\) for B is often easier than using the Biot-Savart law. Be prepared for potential typos in exam questions and choose the best fit among the given options.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This is a dimensional analysis problem. We need to find the correct dimensional formula for each physical quantity listed in Column-I and match it with the options in Column-II. The fundamental dimensions are Mass (M), Length (L), Time (T), and Electric Current (A).
Step 2: Key Formula or Approach:
We will derive the dimensional formula for each quantity using a defining physical equation.
- For Magnetic field (B), use \(F = qvB\).
- For Magnetic flux (\(\Phi\)), use \(\Phi = B \cdot Area\).
- For Magnetic permeability (\(\mu\)), use \(B = \mu n I\).
- For Magnetic inductance (L), use \(U = \frac{1}{2} L I^2\).
Step 3: Detailed Explanation:
(I) Magnetic field intensity (B):
From the Lorentz force equation, \(F = qvB\).
\[ [B] = \frac{[F]}{[q][v]} = \frac{[\text{MLT}^{-2}]}{[\text{AT}][\text{LT}^{-1}]} = \frac{[\text{MLT}^{-2}]}{[\text{AL}]} = [\text{MT}^{-2}\text{A}^{-1}] \] This matches with (S). So, I \(\rightarrow\) S.
(II) Magnetic flux (\(\Phi\)):
Flux is defined as \(\Phi = B \cdot A\).
\[ [\Phi] = [B][\text{Area}] = [\text{MT}^{-2}\text{A}^{-1}] \cdot [\text{L}^2] = [\text{ML}^2\text{T}^{-2}\text{A}^{-1}] \] This dimension is not present in the options P, Q, R, S as written. There seems to be a typo in option (R) in the question paper. Assuming (R) was intended to be \([\text{ML}^2\text{T}^{-2}\text{A}^{-1}]\), let's proceed. However, the provided solution maps (II) to (R). This is a known discrepancy in the question paper, as (R) is \([\text{MLT}^{-2}\text{A}^{-1}]\). Following the provided answer key, we match II \(\rightarrow\) R, acknowledging the inconsistency.
(III) Magnetic permeability (\(\mu\)):
From the formula for the magnetic field inside a solenoid, \(B = \mu_0 n I\), where n is turns per unit length.
\[ [\mu] = \frac{[B]}{[n][I]} = \frac{[\text{MT}^{-2}\text{A}^{-1}]}{[\text{L}^{-1}][\text{A}]} = [\text{MLT}^{-2}\text{A}^{-2}] \] This matches with (P). So, III \(\rightarrow\) P.
(IV) Magnetic inductance (L):
The energy stored in an inductor is \(U = \frac{1}{2} L I^2\).
\[ [L] = \frac{[U]}{[I]^2} = \frac{[\text{Energy}]}{[I]^2} = \frac{[\text{ML}^2\text{T}^{-2}]}{[\text{A}^2]} = [\text{ML}^2\text{T}^{-2}\text{A}^{-2}] \] This matches with (Q). So, IV \(\rightarrow\) Q.
Step 4: Final Answer:
Based on our derivations and correcting for the apparent typo in the question for magnetic flux, the matching is: I\(\rightarrow\)S, III\(\rightarrow\)P, IV\(\rightarrow\)Q. The given answer key states I-S, II-R, III-P, IV-Q. We will follow the key. The combination is I-S, II-R, III-P, IV-Q, which is option (A).