Step 1: Understanding the Concept:
Mutual induction is the phenomenon where a changing current in one coil (primary) induces an electromotive force (e.m.f.) in a nearby second coil (secondary).
The magnitude of the induced e.m.f. is proportional to the rate of change of current in the primary coil.
Step 2: Key Formula or Approach:
The formula relating these quantities is Faraday's Law adapted for mutual inductance:
\[ |e| = M \left| \frac{\Delta I}{\Delta t} \right| \]
where $e$ is the induced e.m.f. in the secondary, $M$ is the coefficient of mutual induction, $\Delta I$ is the change in primary current, and $\Delta t$ is the time interval over which this change occurs.
Step 3: Detailed Explanation:
Given values:
Mutual inductance, $M = 2\text{ H}$
Induced e.m.f., $|e| = 2\text{ kV} = 2000\text{ V}$
Initial current, $I_1 = 6\text{ A}$
Final current, $I_2 = 3\text{ A}$
Change in current magnitude, $|\Delta I| = |I_2 - I_1| = |3 - 6| = |-3| = 3\text{ A}$
We need to find the time $\Delta t$. Rearrange the formula:
\[ \Delta t = \frac{M \times |\Delta I|}{|e|} \]
Substitute the given values into the equation:
\[ \Delta t = \frac{2 \times 3}{2000} \]
\[ \Delta t = \frac{6}{2000} \]
\[ \Delta t = \frac{3}{1000} \]
Converting to scientific notation:
\[ \Delta t = 3 \times 10^{-3}\text{ s} \]
Step 4: Final Answer:
The time required for the change is $3 \times 10^{-3}\text{ s}$.