Question

If L denotes the inductance of an inductor through which a current I is flowing, then the dimensional formula of is
A) $\left[ {ML{T^{ - 2}}} \right]$
B) $\left[ {M{L^2}{T^{ - 2}}} \right]$
C) $\left[ {{M^2}{L^2}{T^{ - 2}}} \right]$
D) not expressible in the form of M.L.T

Answer 1

Hint
As we know that the energy possessed by the circuit containing inductor passing current I in it is $\dfrac{1}{2}L{I^2}$, then the dimension of $L{I^2}$ is same as that of energy. And we know that the dimensional formula of energy is $\left[ {M{L^2}{T^{ - 2}}} \right]$, substituting the values we get the desired result.

Complete step by step answer
Let us consider the inductor L and current passing through the inductor is I
Then the energy possessed by the inductor is $\dfrac{1}{2}L{I^2}$
We have to find out the dimensional formula of this energy
As this also same as that of energy so both have same dimensional formula
Now the energy is simply same as that of work and work is the product of force and displacement i.e.
$W = F \times S$ …………….. (1)
As the dimensional formula of the force is $\left[ {ML{T^{ - 2}}} \right]$
And the dimensional formula of length is $\left[ L \right]$
Then the dimensional formula of the work or energy is $\left[ {M{L^2}{T^{ - 2}}} \right]$
hence, the dimensional formula of $L{I^2}$ is $\left[ {M{L^2}{T^{ - 2}}} \right]$
Thus, option (B) is correct.

Note
In these types of questions we just need to remember the formula, as here formula is given or we analyse this formula is the same as that of energy therefore, it has the same dimensional formula. For calculating the dimensional formula we just need to substitute mass as $\left[ M \right]$, length as $\left[ L \right]$ and time as $\left[ T \right]$. After that just substitute the value and we will get the required result. And it must also be noticed that these dimensional formulas are always written in the square bracket.