Question

The dimensional formula for magnetic induction is:
A. $M{T^{ - 1}}{A^{ - 1}}$
B. $M{T^{ - 2}}{A^{ - 1}}$
C. $ML{A^{ - 1}}$
D. $M{T^{ - 2}}A$

Answer 1

Hint: We know one of the equation for magnetic induction is $B = \dfrac{{{\mu _0}}}{{4\pi }} \times \dfrac{{2m}}{{{r^3}}}$ . We know that the dimensional formula of magnetic moment is $\left[ {{M^0}{L^2}{T^0}{A^1}} \right]$ and the dimensional formula for magnetic susceptibility is $\left[ {{M^1}{L^1}{T^{ - 2}}{A^{ - 2}}} \right]$ and finally the dimensional formula for distance is $(r) = \left[ {{M^0}{L^1}{T^0}{A^0}} \right]$ .

Complete answer:
So, the dimensional formula of magnetic induction \[ = \left[ {{M^1}{T^{ - 2}}{A^{ - 1}}} \right]\]

Dimensional formula – Every physical quantity has a unit assigned for it, for example, the force has a unit \[kgm/{s^2}\]. When we express this unit in terms of the fundamental quantities we get the dimensional formula of that physical quantity. The dimensional formula for force is \[[ML{T^{ - 2}}{A^0}]\] .
Here \[M\] represents mass, \[L\] represents the length, \[T\] represents time, and \[A\] represents current.

We know that the equation for the magnetic induction is
$B = \dfrac{{{\mu _0}}}{{4\pi }} \times \dfrac{{2m}}{{{r^3}}}$
Here, $B = $ Magnetic induction
$m = $ The magnetic moment or the magnetic dipole moment of the magnetic dipole
${u_0} = $ The magnetic permeability of free space
$r = $ The distance of the point from the axis of the dipole
We know that the dimensional formula of magnetic moment is $\left[ {{M^0}{L^2}{T^0}{A^1}} \right]$ and the dimensional formula for magnetic susceptibility is $\left[ {{M^1}{L^1}{T^{ - 2}}{A^{ - 2}}} \right]$ and finally the dimensional formula for distance is $(r) = \left[ {{M^0}{L^1}{T^0}{A^0}} \right]$ .
So, the dimensional formula of magnetic induction \[ = \left[ {{M^1}{L^1}{T^{ - 2}}{A^{ - 2}}} \right] \times \dfrac{{\left[ {{M^0}{L^2}{T^0}{A^1}} \right]}}{{\left[ {{M^0}{L^1}{T^0}{A^0}} \right]}}\]
The dimensional formula of magnetic induction \[ = \left[ {{M^1}{L^0}{T^{ - 2}}{A^{ - 1}}} \right]\]
The dimensional formula of magnetic induction \[ = \left[ {{M^1}{T^{ - 2}}{A^{ - 1}}} \right]\]

Therefore, the dimension formula for magnetic induction is \[\left[ {{M^1}{T^{ - 2}}{A^{ - 1}}} \right]\]

So, the correct answer is “Option B”.

Note:
Dimensional formula is unique, whereas the unit can be measured in the SI unit system, metric system, etc. So, for example, the unit for magnetic induction is Tesla in the SI unit but weber per square meter is also a unit of magnetic induction.