Correct Option: (D) [M L2 T-2 A-2]
Step 1 (Formula / Governing Law & Given Data): [1/3 Mark]
The self-inductance ($L$) of a coil is defined by the relationship between the induced electromotive force ($\varepsilon$) and the rate of change of current ($I$) flowing through it, given by Faraday's Law:
$$\varepsilon = -L \frac{dI}{dt} \implies L = \frac{\varepsilon}{dI/dt}$$Where:
- $\varepsilon$ (EMF) = Work done per unit charge = $\frac{[M L^2 T^{-2}]}{[A T]} = [M L^2 T^{-3} A^{-1}]$
- $I$ (Current) = $[A]$
- $t$ (Time) = $[T]$
Step 2 (Step-by-Step Mathematical Working): [1/3 Mark]
Substituting the dimensional formulas into the expression for $L$:
$$[L] = \frac{[\text{EMF}]}{[I] / [t]}$$ $$[L] = \frac{[M L^2 T^{-3} A^{-1}]}{[A] / [T]}$$ $$[L] = [M L^2 T^{-3} A^{-1}] \cdot [A^{-1} T]$$Combining the powers of the base dimensions:
$$[L] = [M L^2 T^{-3+1} A^{-1-1}]$$ $$[L] = [M L^2 T^{-2} A^{-2}]$$Step 3 (Final Concluded Answer): [1/3 Mark]
The dimensions of self-inductance are [M L2 T-2 A-2]. Therefore, the correct option is (D).
💡 EL10 Concept Insight: Think of self-inductance as the "electrical inertia" of a circuit; just as mass resists changes in motion, self-inductance resists changes in current. Because it relates energy stored in a magnetic field ($\frac{1}{2}LI^2$) to the square of the current, its dimensions are simply the dimensions of Energy divided by Current squared.