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Question: What is the dimensional formula of Inductance? A). \(\left[ {M{L^2}{T^{ - 2}}{A^{ - 2}}} \right]\)...

What is the dimensional formula of Inductance?
A). [ML2T−2A−2]\left[ {M{L^2}{T^{ - 2}}{A^{ - 2}}} \right]
B). [ML2T1A−2]\left[ {M{L^2}{T^1}{A^{ - 2}}} \right]
C). [ML2T−1A−2]\left[ {M{L^2}{T^{ - 1}}{A^{ - 2}}} \right]
D). [ML2T−2A−1]\left[ {M{L^2}{T^{ - 2}}{A^{ - 1}}} \right]

Explanation

Solution

Hint: In this question we first use faraday’s law of electromagnetic induction that is Inductance L is equal to the ratio of e.m.f E to the rate of change of current with respect to time that is L=EdIdtL = \dfrac{E}{{\dfrac{{dI}}{{dt}}}} here we substitute the dimensional formula of V as [ML2T−3I−1]\left[ {M{L^2}{T^{ - 3}}{I^{ - 1}}} \right] and the dimensional formula of dIdt\dfrac{{dI}}{{dt}} as [T−1I1]\left[ {{T^{ - 1}}{I^1}} \right] and get the dimensional formula of inductance.

Complete step-by-step solution -
Inductance is a property due to which a conductor opposes a change in current it happens because according to Faraday's law of induction, an inductor stores and releases the energy in the form of a magnetic field around the conductor when current is flowing through it. There are many ways to find dimensional formulas for inductance. There are many ways to find the dimensional formula of inductance
First by applying Faraday's law of electromagnetic induction we can find the dimensional formula as
Inductance L is equal to the ratio of e.m.f E to the rate of change of current with respect to time dIdt\dfrac{{dI}}{{dt}} that is
L=EdIdtL = \dfrac{E}{{\dfrac{{dI}}{{dt}}}}
The dimension of V is [ML2T−3I−1]\left[ {M{L^2}{T^{ - 3}}{I^{ - 1}}} \right]
And the dimension of dIdt\dfrac{{dI}}{{dt}} is [T−1I1]\left[ {{T^{ - 1}}{I^1}} \right]
Therefore the dimension of inductance comes out to be
L=[ML2T−3I−1][T−1I1]=[ML2T−2I−2]L = \dfrac{{\left[ {M{L^2}{T^{ - 3}}{I^{ - 1}}} \right]}}{{\left[ {{T^{ - 1}}{I^1}} \right]}} = \left[ {M{L^2}{T^{ - 2}}{I^{ - 2}}} \right]
Second by applying the concepts of work done
We know that the work done against the e.m.f by an inductor L when a current I is flowing through it is given as
W=12LI2W = \dfrac{1}{2}L{I^2}
Here dimension of work done W is [ML2T−2]\left[ {M{L^2}{T^{ - 2}}} \right]
So the dimension of inductance can be find as
[L]=[W][I]=[ML2T−2][I2]\left[ L \right] = \dfrac{{\left[ W \right]}}{{\left[ I \right]}} = \dfrac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ {{I^2}} \right]}}
[L]=[ML2T−2I−2]\left[ L \right] = \left[ {M{L^2}{T^{ - 2}}{I^{ - 2}}} \right]
Hence from both the techniques we get the dimensional formula of inductance as [ML2T−2I−2]\left[ {M{L^2}{T^{ - 2}}{I^{ - 2}}} \right].

Note: For these types of questions we need to know about the inductances, capacitances and resistances we also need to know their expression. To find dimensional formulas we first need to remember the dimensional formula of some basics values like work is done, e.m.f, force, etc.