Question

Match List I with List II

Choose the correct answer from the options given below :

• A. A-I, B-III, C-II, D-IV
• B. A-IV, B-III, C-II, D-I
• C. A-III, B-I, C-II, D-IV
• D. A-IV, B-II, C-III, D-I

Answer 1

The Correct Option is B

The following equations and dimensional analyses are presented:

The torque (\(\tau\)) is defined as: \[ \tau = \mathbf{r} \times \mathbf{F} \implies [\tau] = [ML^2T^{-2}] \]

The magnetic field (\(\mathbf{B}\)) is determined 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 (\(\mathbf{M}\)) possesses the dimensions: \[ [\mathbf{M}] = [\mathbf{I} \times \mathbf{A}] = [AL^2] \]

According to Biot-Savart's Law: \[ B = \frac{\mu_0 I dl \sin \theta}{r^2} \]

The permeability of free space (\(\mu\)) is calculated as: \[ \mu = \frac{B r^2}{I dl} \implies \mu = \frac{MT^{-2}A^{-1} \times L^2}{AL} = [MLT^{-2}A^{-2}] \]

The corresponding matches are as follows:

• Torque \(\to [ML^2T^{-2}]\)
• Magnetic field \(\to [MA^{-1}T^{-2}]\)
• Magnetic moment \(\to [AL^2]\)
• Permeability of free space \(\to [MLT^{-2}A^{-2}]\)