Step 1: Understanding the Concept:
According to Faraday's Law of Electromagnetic Induction, a change in magnetic flux passing through a coil induces an electromotive force (e.m.f.) in it.
The magnitude of the induced e.m.f. is equal to the rate of change of magnetic flux.
Step 2: Key Formula or Approach:
Faraday's law states that the magnitude of average induced e.m.f. is $|e| = \frac{\Delta \Phi}{\Delta t}$.
Here, $\Delta \Phi = |\Phi_{\text{final}} - \Phi_{\text{initial}}|$ is the change in flux, and $\Delta t = t - 0 = t$ is the time interval.
Step 3: Detailed Explanation:
The initial magnetic flux is $\Phi_1 = 4 \times 10^{-4} \text{ Wb}$.
The final flux reduces to $30%$ of the original value:
\[ \Phi_2 = 30% \text{ of } \Phi_1 = 0.30 \times 4 \times 10^{-4} \text{ Wb} \]
\[ \Phi_2 = 1.2 \times 10^{-4} \text{ Wb} \]
Calculate the magnitude of the change in magnetic flux:
\[ \Delta \Phi = \Phi_1 - \Phi_2 = (4 - 1.2) \times 10^{-4} \text{ Wb} \]
\[ \Delta \Phi = 2.8 \times 10^{-4} \text{ Wb} \]
The induced e.m.f. is given as $e = 0.56 \text{ mV}$.
Convert e.m.f. to standard SI units (Volts):
\[ e = 0.56 \times 10^{-3} \text{ V} = 5.6 \times 10^{-4} \text{ V} \]
Apply Faraday's law to find the time $t$:
\[ |e| = \frac{\Delta \Phi}{t} \]
Rearrange to solve for $t$:
\[ t = \frac{\Delta \Phi}{|e|} \]
Substitute the calculated values:
\[ t = \frac{2.8 \times 10^{-4}}{5.6 \times 10^{-4}} \]
The $10^{-4}$ terms cancel out:
\[ t = \frac{2.8}{5.6} = \frac{1}{2} = 0.5 \text{ s} \]
Step 4: Final Answer:
The value of time $t$ is $0.5 \text{ s}$.