Dimensional Formula Matching Shortcut

 

❓ Question

Match the following physical quantities in List-I with their correct dimensional formulae in List-II.

List-I

(A) Permeability of free space $\mu_0$
(B) Magnetic field $B$
(C) Magnetic moment
(D) Torsional constant

List-II

$$(1)\ [ML^2T^{-2}]$$
$$(2)\ [MT^{-2}A^{-1}]$$
$$(3)\ [MLT^{-2}A^{-2}]$$
$$(4)\ [L^2A]$$

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✍️ Solution

(A) Permeability of free space $\mu_0$

We know

$$B=\mu_0H$$

Therefore,

$$[\mu_0]=\frac{[B]}{[H]}$$

Using

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

and

$$[H]=[AL^{-1}]$$

we get

$$[\mu_0] = \frac{[MT^{-2}A^{-1}]}{[AL^{-1}]}$$
$$\boxed{[\mu_0]=[MLT^{-2}A^{-2}]}$$

Hence,

$$\boxed{A\rightarrow 3}$$

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(B) Magnetic field $B$

Using magnetic force,

$$F=qvB$$

Therefore,

$$[B]=\frac{[F]}{[q][v]}$$
$$[B] = \frac{[MLT^{-2}]} {[AT][LT^{-1}]}$$
$$\boxed{[B]=[MT^{-2}A^{-1}]}$$

Hence,

$$\boxed{B\rightarrow 2}$$

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(C) Magnetic moment

Magnetic moment is

$$\mu=IA$$

where $A$ here denotes area.

Therefore,

$$[\mu]=[A][L^2]$$
$$\boxed{[\mu]=[L^2A]}$$

Hence,

$$\boxed{C\rightarrow 4}$$

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(D) Torsional constant

For a torsion wire,

$$\tau=C\theta$$

Therefore,

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

Since angle $\theta$ is dimensionless,

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

Now,

$$[\tau]=[\text{Force}\times\text{distance}]$$
$$=[MLT^{-2}][L]$$
$$\boxed{[C]=[ML^2T^{-2}]}$$

Hence,

$$\boxed{D\rightarrow 1}$$

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✅ Final Matching

$$\boxed{A\rightarrow3,\quad B\rightarrow2,\quad C\rightarrow4,\quad D\rightarrow1}$$

Final Answer

$$\boxed{(A)-(3),\ (B)-(2),\ (C)-(4),\ (D)-(1)}$$