The current in a coil changes from 3.50 A to 2.00 A in the same direction in 0.500s. If the average emf induced in the coil is 12.0mV. What is the inductance of the coil?

Options

A. 2.00mH
B. 4.00mH
C. 3.00mH
D. 8.00mH

Answer

To find the inductance of the coil, we can follow these steps:

### Step 1: Identify the given values
- Initial current, \( I_1 = 3.50 \, \text{A} \)
- Final current, \( I_2 = 2.00 \, \text{A} \)
- Time interval, \( t = 0.500 \, \text{s} \)
- Average induced EMF, \( E = 12.0 \, \text{mV} = 12 \times 10^{-3} \, \text{V} \)

### Step 2: Calculate the change in current (\( \Delta I \))
\[ \Delta I = I_2 - I_1 = 2.00 \, \text{A} - 3.50 \, \text{A} = -1.50 \, \text{A} \]

### Step 3: Calculate the rate of change of current (\( \frac{dI}{dt} \))
\[ \frac{dI}{dt} = \frac{\Delta I}{t} = \frac{-1.50 \, \text{A}}{0.500 \, \text{s}} = -3.00 \, \text{A/s} \]

### Step 4: Use the formula for induced EMF
The induced EMF (\( E \)) is related to the inductance (\( L \)) and the rate of change of current by the formula:
\[ E = -L \frac{dI}{dt} \]
Rearranging this gives:
\[ L = -\frac{E}{\frac{dI}{dt}} \]

### Step 5: Substitute the values into the formula
Substituting the values we have:
\[ L = -\frac{12 \times 10^{-3} \, \text{V}}{-3.00 \, \text{A/s}} = \frac{12 \times 10^{-3}}{3.00} \]

### Step 6: Calculate the inductance
\[ L = 4 \times 10^{-3} \, \text{H} = 4 \, \text{mH} \]

### Conclusion
The inductance of the coil is \( 4 \, \text{mH} \).

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