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 relation for mutual inductance.
The mutual inductance \( M \) is defined as the ratio of the induced emf (electromotive force) in a coil to the rate of change of current in another coil. The general relation is: \[ \mathcal{E} = M \cdot \frac{dI}{dt} \] Where: - \( \mathcal{E} \) is the induced emf, with the dimension of \( [\mathcal{E}] = \text{ML}^2 \text{T}^{-3} \text{A}^{-1} \). - \( \frac{dI}{dt} \) is the rate of change of current, which has the dimension \( [\frac{dI}{dt}] = \text{A} \text{T}^{-1} \).

Step 2: Deriving the dimension of \( M \).
Using the formula \( \mathcal{E} = M \cdot \frac{dI}{dt} \), we can rearrange to find the dimension of mutual inductance \( M \): \[ [M] = \frac{[\mathcal{E}]}{[dI/dt]} = \frac{\text{ML}^2 \text{T}^{-3} \text{A}^{-1}}{\text{A} \text{T}^{-1}} = \text{ML}^2 \text{T}^{-2} \text{A}^{-2} \]

Step 3: Conclusion.
Therefore, the correct dimension for mutual inductance is \( \text{ML}^2 \text{T}^{-2} \text{A}^{-2} \), which corresponds to option (B).

Final Answer: (B) \( \text{ML}^2 \text{T}^{-2} \text{A}^{-2} \)