Match List I with List II 
- A-I, B-III, C-II, D-IV
- A-IV, B-III, C-II, D-I
- A-III, B-I, C-II, D-IV
- A-IV, B-II, C-III, D-I
The Correct Option is B
Approach Solution - 1
The equations and dimensional analysis are as follows:
The torque (\(\tau\)) is given by: \[ \tau = \mathbf{r} \times \mathbf{F} \implies [\tau] = [ML^2T^{-2}] \]
The magnetic field (\(\mathbf{B}\)) is derived as: \[ \mathbf{F} = [q \mathbf{v} \times \mathbf{B}] \implies [\mathbf{B}] = \frac{[\mathbf{F}]}{[q][\mathbf{v}]} = \frac{MLT^{-2}}{ATL^{-1}} = [MA^{-1}T^{-2}] \]
The magnetic moment (\(\mathbf{M}\)) has the dimensions: \[ [\mathbf{M}] = [\mathbf{I} \times \mathbf{A}] = [AL^2] \]
Using Biot-Savart's Law: \[ B = \frac{\mu_0 I dl \sin \theta}{r^2} \]
The permeability of free space (\(\mu\)) is derived as: \[ \mu = \frac{B r^2}{I dl} \implies \mu = \frac{MT^{-2}A^{-1} \times L^2}{AL} = [MLT^{-2}A^{-2}] \]
Thus, the correct matching is:
- Torque \(\to [ML^2T^{-2}]\)
- Magnetic field \(\to [MA^{-1}T^{-2}]\)
- Magnetic moment \(\to [AL^2]\)
- Permeability of free space \(\to [MLT^{-2}A^{-2}]\)
Approach Solution -2
To solve the problem of matching List I with List II, we need to understand the dimensional formulas related to each physical quantity.
- Torque: It is the rotational equivalent of linear force. The dimensional formula for torque is \([M^1L^2T^{-2}]\). This matches with List II – IV.
- Magnetic Field: The dimensional formula for magnetic field (B) is \([M^1T^{-2}A^{-1}]\). This matches with List II – III.
- Magnetic Moment: The dimensional formula is \([L^2A^1]\). This matches with List II – II.
- Permeability of Free Space: The dimensional formula for permeability \((\mu_0)\) is \([M^1L^1T^{-2}A^{-2}]\). This matches with List II – I.
Thus, the correct matching is:
- A-IV: Torque matches with \([M^1L^2T^{-2}]\).
- B-III: Magnetic field matches with \([M^1T^{-2}A^{-1}]\).
- C-II: Magnetic moment matches with \([L^2A^1]\).
- D-I: Permeability of free space matches with \([M^1L^1T^{-2}A^{-2}]\).
Therefore, the correct answer is A-IV, B-III, C-II, D-I.
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