Question

The dimensional formula for inductance is:
A. $\left[ {{M^1}{L^2}{T^{ - 2}}{A^{ - 2}}} \right]$
B. $\left[ {{M^1}{L^2}T{A^{ - 2}}} \right]$
C. $\left[ {{M^1}{L^2}{T^{ - 1}}{A^{ - 2}}} \right]$
D. $\left[ {{M^1}{L^1}{T^{ - 2}}{A^{ - 2}}} \right]$

Answer 1

Hint: Inductance is defined as the ratio of Magnetic Flux to the current. For finding the dimensional formula of inductance we have many ways as it is directly linked with voltage and rate of change of current, energy and current etc.

Complete step by step answer:
We will use the basic way to find it as follow:
\[Inductance = \dfrac{{Magnetic{\text{ }}Flux}}{{Current}} - - - - - - - {\text{ }}\left( 1 \right)\]
Magnetic field = N=Magnetic field $ \times $ Area
Magnetic field = Force $/$ current $ \times $ Length............ (Since F = BIL)
Force = Mass $ \times $ Acceleration
Acceleration = Velocity $/$ Time
Velocity = Displacement $/$ Time
Now we will start solving all the from the bottom i.e. from the velocity
Displacement, Length = L , Time = T , Mass = M , Current = A
 Velocity = Displacement $/$ Time
Velocity = \[\left[ {ML{T^{ - 1}}} \right]\]
Acceleration = $\dfrac{{{\text{velocity}}}}{{{\text{time}}}}$ = \[\dfrac{{[{M^0}L{T^{ - 1}}]}}{{\left[ T \right]}}\]
Acceleration = \[[{\text{ }}{{\text{M}}^0}L{T^{ - 2}}]\]
 Force = Mass $ \times $ Acceleration = \[\left[ M \right] \times [{M^0}L{T^{ - 2}}]\]
Force = $\left[ {ML{T^{ - 2}}} \right]$
Magnetic field = Force $/$ (current $ \times $ Length) = \[\dfrac{{[ML{T^{ - 2}}]}}{{[A \times L]}}\]
Magnetic field = \[\left[ {M{L^0}{T^{ - 2}}{A^{ - 1}}} \right]\]
 Magnetic Flux = Magnetic field $ \times $ Area = \[[M{L^0}{T^{ - 2}}{A^{ - 1}}] \times \left[ {{L^2}} \right]\] ........ (Since Area = L $ \times $ B = L2)
Magnetic Flux = \[\left[ {M{L^2}{T^{ - 2}}{A^{ - 1}}} \right]\]
Inductance = Magnetic Flux $/$ Current = \[\dfrac{{[M{L^2}{T^{ - 2}}{A^{ - 1}}]}}{{\left[ A \right]}}\]
Inductance = \[\left[ {M{L^2}{T^{ - 2}}{A^{ - 2}}} \right]\]
Hence the correct option is “A”.

Note:
In order to solve these types of problems first of all you must remember the formula of the quantity whose dimension you want to calculate and the dimensional formula of that quantity. There are many way to find dimensional formula for inductance One can find this by applying Faraday's law of electromagnetic induction as Inductance (L) = the ratio of potential difference to the time rate of change of electric current or in symbolic notation \[L\].