Match List - I with List - II.

Choose the correct answer from the options given below :

Permeability of free space:

Using relation:

$$B=\mu_0\frac{I}{L}$$

$$\mu_0=\frac{BL}{I}$$

Substitute dimensions:

$$[B]=[MT^{-2}A^{-1}]$$

$$[\mu_0]=\frac{(MT^{-2}A^{-1})\cdot L}{A}=[MLT^{-2}A^{-2}]$$

So, Permeability → (III)

Magnetic field:

From Lorentz force,

$$F=qvB$$

$$B=\frac{F}{qv}$$

Now substitute dimensions:

$$[q]=[AT]$$

$$\quad[v]=[LT^{-1}]$$

$$[B]=\frac{MLT^{-2}}{(AT)(LT^{-1})}=[MT^{-2}A^{-1}]$$

So, Magnetic field → (II)

Magnetic moment:

Magnetic moment is given by:

$$μ=IA$$

$$[\mu]=[A]\cdot[L^2]=[L^2A]$$

So, Magnetic moment → (IV)

Torsional constant (torsion constant) is defined from the relation:

$$C\theta\ =\tau$$

where $$τ=torque$$ and $$\theta$$ = angular displacement (dimensionless).

So,

$$C=\frac{\tau}{\theta}$$

Since θ\thetaθ is dimensionless:

$$[C]=[\tau]$$

Now, torque:

$$\tau=r\times F$$

$$[F]=[MLT^{-2}]$$

$$[\tau]=[L]\cdot[MLT^{-2}]=[ML^2T^{-2}]$$

So Torsional Constant →(I)

Final matching:

(A)−(III), (B)−(II), (C)−(IV), (D)−(I)