Step 1: Understanding the Question:
We need to calculate the self-inductance (\(L\)) of a coil. We are given the induced electromotive force (EMF), the initial and final currents, and the time interval for the change.
Step 2: Key Formula or Approach:
The induced EMF (\(E\)) in a coil due to a change in its own current is related to its self-inductance (\(L\)) by Faraday's law of induction. The formula for the magnitude of the average EMF is:
\[
E = L \left| \frac{\Delta I}{\Delta t} \right|
\]
where \(\Delta I\) is the change in current and \(\Delta t\) is the time interval.
Step 3: Detailed Explanation:
(i) Identify the given quantities:
- Induced EMF, \(E = 30\,\text{V}\).
- Initial current, \(I_{initial} = 5\,\text{A}\).
- Final current, \(I_{final} = 2\,\text{A}\).
- Time interval, \(\Delta t = 0.1\,\text{s}\).
(ii) Calculate the rate of change of current:
The change in current is \(\Delta I = I_{final} - I_{initial} = 2\,\text{A} - 5\,\text{A} = -3\,\text{A}\).
The magnitude of the rate of change of current is:
\[
\left| \frac{\Delta I}{\Delta t} \right| = \left| \frac{-3\,\text{A}}{0.1\,\text{s}} \right| = 30\,\text{A/s}
\]
(iii) Solve for the self-inductance (L):
Rearrange the formula to solve for \(L\):
\[
L = \frac{E}{\left| \frac{\Delta I}{\Delta t} \right|}
\]
Substitute the known values:
\[
L = \frac{30\,\text{V}}{30\,\text{A/s}} = 1\,\text{H}
\]
Step 4: Final Answer:
The self-inductance of the coil is \(1\,\text{H}\).