Matching Physical Quantities and Dimensions

This solution identifies the correct dimensional formulas for the physical quantities listed in List-I and matches them with the dimensions provided in List-II.

Dimensional Analysis of List-I Quantities

A. Magnetic Induction (B):
The magnetic induction (B) can be determined from the Lorentz force formula \( F = qvB \). Rearranging for B, \( [B] = \frac{[F]}{[q][v]} = \frac{[M L T^{-2}]}{[A T][L T^{-1}]} = [M T^{-2} A^{-1}] \). This dimension is equivalent to \( [M L^0 T^{-2} A^{-1}] \), matching dimension III.

B. Magnetic Flux ($\Phi$):
Magnetic flux is defined as \( \Phi = BA \), where A is the area.
Using the dimension of magnetic induction: \( [\Phi] = [B][A] = [M L^0 T^{-2} A^{-1}] [L^2] = [M L^2 T^{-2} A^{-1}] \). This matches dimension IV.

C. Magnetic Permeability ($\mu$):
Magnetic permeability ($\mu$) relates magnetic field strength to the current producing it, often expressed via Ampere's Law or formulas like \( B = \mu n I \) for a solenoid (n = turns per unit length).
The dimension is calculated as: \( [\mu] = \frac{[B]}{[n][I]} = \frac{[M L^0 T^{-2} A^{-1}]}{[L^{-1}][A]} = [M L T^{-2} A^{-2}] \). This matches dimension I.

D. Self Inductance (L):
Self inductance (L) is related to the induced EMF (V) by \( V = -L \frac{dI}{dt} \).
The dimension is derived as: \( [L] = \frac{[V]}{[dI/dt]} = \frac{[M L^2 T^{-3} A^{-1}]}{[A T^{-1}]} = [M L^2 T^{-2} A^{-2}] \). This matches dimension II.

Conclusion: Correct Matching

The derived dimensions lead to the following matches:

• Magnetic Induction (A) matches with III. \( [M L^0 T^{-2} A^{-1}] \)
• Magnetic Flux ($\Phi$) (B) matches with IV. \( [M L^2 T^{-2} A^{-1}] \)
• Magnetic Permeability ($\mu$) (C) matches with I. \( [M L T^{-2} A^{-2}] \)
• Self Inductance (L) (D) matches with II. \( [M L^2 T^{-2} A^{-2}] \)

Therefore, the correct combination is A-III, B-IV, C-I, D-II.