Question

The dimension of mutual inductance is (Denote dimension of current as A)

• A. \( \text{ML}^2 \text{T}^2 \text{A}^{-2} \)
• B. \( \text{ML}^2 \text{T}^{-2} \text{A}^{-2} \)
• C. \( \text{ML}^{-2} \text{T}^2 \text{A}^{-2} \)
• D. \( \text{ML}^2 \text{T}^{-3} \text{A}^{-3} \)
• E. \( \text{ML}^2 \text{T}^{-3} \text{A}^{-2} \)

Hint:

The mutual inductance dimension can be derived by considering the dimensions of emf and the rate of change of current. Be sure to handle dimensions carefully, especially for terms involving time and current.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Mutual inductance is the property of two coils where a changing current in one induces an electromotive force in the other.
It shares the exact same dimensional formula as self-inductance.
Step 2: Key Formula or Approach:
The dimension can be derived from the energy stored in an inductor, given by \( U = \frac{1}{2} M I^{2} \).
Rearranging this formula gives \( M = \frac{2U}{I^{2}} \).
Step 3: Detailed Explanation:
The dimensional formula for energy (or work), \( U \), is \([\text{M}^{1}\text{L}^{2}\text{T}^{-2}]\).
The dimension for current, \( I \), is given as \([\text{A}]\).
Substituting these into the rearranged equation yields:
\[ [M] = \frac{[\text{M}^{1}\text{L}^{2}\text{T}^{-2}]}{[\text{A}]^{2}} \] \[ [M] = [\text{M}^{1}\text{L}^{2}\text{T}^{-2}\text{A}^{-2}] \] This matches the second option provided in the choices.
Step 4: Final Answer:
The dimension of mutual inductance is \(\text{ML}^{2}\text{T}^{-2}\text{A}^{-2}\).