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Question

The dimensional formula of permeability of free space $\mu_0$ is given by:

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [Option A]
## CONCEPT FRAMEWORK
	**Primary Concept**: Dimensional Analysis
	**Related Topics**: Magnetic Field, Force on a current carrying conductor
	**Prerequisites**: Understanding of dimensional analysis, force on a current carrying conductor in a magnetic field
## SOLUTION PATHWAY
**Given Information**:
	Permeability of free space is denoted by $\mu_0$
	We need to find its dimensional formula

**Method & Steps**:
	1. We can use the formula for the force between two parallel current-carrying wires to find the dimensions of $\mu_0$.
		Reasoning: The force per unit length between two parallel wires carrying currents $I_1$ and $I_2$, separated by a distance r is given by $F/L = \frac{\mu_0 I_1 I_2}{2\pi r}$.
		Formula/Rule: $F/L = \frac{\mu_0 I_1 I_2}{2\pi r}$
	2. Rearranging the formula to isolate $\mu_0$ gives: $\mu_0 = \frac{2\pi r F}{L I_1 I_2}$. Since $2\pi$ is dimensionless, we can write $\mu_0 = \frac{ r F}{L I_1 I_2}$.
		Working:  We know the dimensions of force (F) as $[MLT^{-2}]$, length (r and L) as $[L]$, and current (I) as $[A]$. So, $\mu_0 = \frac{[L] [MLT^{-2}]}{[L] [A][A]} = [MLT^{-2}A^{-2}]$
		Key Point: Force has dimensions of $[MLT^{-2}]$, current has dimensions of $[A]$, and length has dimensions of $[L]$
	3. Thus, the dimensional formula of $\mu_0$ is $[MLT^{-2}A^{-2}]$.
		Result: $[MLT^{-2}A^{-2}]$
		Verification: The calculated dimension matches with Option A.
 ## OPTION ANALYSIS
[A] ✓: The dimensional formula derived using the force between current-carrying conductors matches with $[MLT^{-2}A^{-2}]$.
[B] :x: This option has $L^{-1}$ term which is incorrect based on the derivation. $\mu_0$ is proportional to length and not inversely proportional.
[C] :x: This option has $A^2$ which is incorrect. The current terms should be in the denominator and thus $A^{-2}$
[D] :x: This option has $M^{-1}$ and $L^{-1}$ which is incorrect based on the derivation.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Incorrectly remembering the formula for force between current carrying wires.
			[How to avoid]: Always derive the formula from the basic principles or have it clearly noted.
		2.[Error 2]: Incorrectly substituting the dimensions of the physical quantities.
			[How to avoid]: Practice dimensional analysis frequently and memorize basic dimensions.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember the force between two current carrying wires and derive the dimensions from it.
		2.[Memory aid]: Think of the permeability of free space as something related to force and magnetic field which can help you recall the formula.

ABDUL JILANI BASHA · Answer

The correct Option is "$[MLT^{-2}A^{-2}]$"

ABDUL JILANI BASHA · Answer

$[ML^{-1}T^{-2}A^{-2}]$

ABDUL JILANI BASHA · Answer

$[MLT^{-2}A^{2}]$

ABDUL JILANI BASHA · Answer

$[M^{-1}L^{-1}T^{2}A^{2}]$

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