The average e.m.f. induced in a coil in which the current changes from `2` amperes to `4` amperes in `0.05` seconds is `8` volts. What is the self-inductance of the coil?

Options

A. `0.1 H`
B. `0.2 H`
C. `0.4 H`
D. `0.8 H`

Explanation

To find the self-inductance of the coil, we can use the formula for the average induced electromotive force (e.m.f.) in terms of self-inductance. The formula is given by:

\[ E = L \frac{dI}{dt} \]

Where:
- \(E\) is the average induced e.m.f. (in volts),
- \(L\) is the self-inductance (in henries),
- \(\frac{dI}{dt}\) is the rate of change of current (in amperes per second).

### Step 1: Identify the given values
- The average e.m.f. \(E = 8 \, \text{volts}\)
- Initial current \(I_1 = 2 \, \text{amperes}\)
- Final current \(I_2 = 4 \, \text{amperes}\)
- Time interval \(\Delta t = 0.05 \, \text{seconds}\)

### Step 2: Calculate the change in current (\(dI\))
The change in current is given by:

\[ dI = I_2 - I_1 = 4 \, \text{amperes} - 2 \, \text{amperes} = 2 \, \text{amperes} \]

### Step 3: Calculate the rate of change of current (\(\frac{dI}{dt}\))
The rate of change of current is calculated as:

\[ \frac{dI}{dt} = \frac{dI}{\Delta t} = \frac{2 \, \text{amperes}}{0.05 \, \text{seconds}} = 40 \, \text{amperes/second} \]

### Step 4: Substitute the values into the e.m.f. formula
Now we can substitute the values into the formula:

\[ E = L \frac{dI}{dt} \]

Rearranging for \(L\):

\[ L = \frac{E}{\frac{dI}{dt}} = \frac{8 \, \text{volts}}{40 \, \text{amperes/second}} = 0.2 \, \text{henries} \]

### Step 5: Conclusion
Thus, the self-inductance of the coil is:

\[ L = 0.2 \, \text{henries} \]