Calculate the mutual inductance between two coils when a current of `4 A` changes to `12 A` in `0.5 s` in primary and induces an emf of `50 mV` in the secondary. Also, calculate the induced emf in the secondary if current in the primary changes from `3 A` to `9 A` is `0.02 s`.

Explanation

To solve the problem of calculating the mutual inductance between two coils and the induced emf in the secondary coil, we can follow these steps:

### Step 1: Calculate the Rate of Change of Current in the Primary Coil

The current in the primary coil changes from \( I_1 = 4 \, A \) to \( I_1' = 12 \, A \) over a time interval of \( t = 0.5 \, s \).

The rate of change of current (\( \frac{dI_1}{dt} \)) can be calculated using the formula:
\[
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