Question

The dimensions of the coefficient of self-inductance are:

• A. \( [M L^2 T^{-2} A^{-2}] \)
• B. \( [M L^2 T^{-2} A^{-1}] \)
• C. \( [M L T^{-2} A^{-2}] \)
• D. \( [M L T^{-2} A^{-1}] \)

Hint:

Self-inductance is the property of an inductor that determines how much EMF is induced per unit rate of change of current. Its dimensional formula is derived using the energy equation.

Answer 1

The Correct Option is A

Step 1: {Define self-inductance}
The formula for the energy stored in an inductor is: \[ U = \frac{1}{2} L I^2 \] In this equation, \( L \) represents self-inductance, and \( I \) represents the current.
Step 2: {Rearrange for \( L \)}
The equation can be rearranged to solve for \( L \): \[ L = \frac{2U}{I^2} \]
Step 3: {Determine the dimensional formula of \( L \)}
The dimensional formula for energy (\( U \)) is: \[ [U] = [M L^2 T^{-2}] \] The dimensional formula for current (\( I \)) is: \[ [I] = [A] \] Substituting these into the rearranged equation yields: \[ [L] = \frac{[M L^2 T^{-2}]}{[A^2]} \] Which simplifies to: \[ = [M L^2 T^{-2} A^{-2}] \]
Step 4: {Match with options}
Comparing this derived dimensional formula with the provided options, the correct answer is (A) \( [M L^2 T^{-2} A^{-2}] \).